https://hilbert.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Valko&feedformat=atomUW-Math Wiki - User contributions [en]2021-06-12T11:38:25ZUser contributionsMediaWiki 1.30.1https://hilbert.math.wisc.edu/wiki/index.php?title=Option_2_packages&diff=21201Option 2 packages2021-05-09T16:43:57Z<p>Valko: </p>
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<div>'''NOTE: in the Fall 2020 semester the Department of Mathematics introduced five new named options (see the named options section in the [https://guide.wisc.edu/undergraduate/letters-science/mathematics/mathematics-bs/#requirementstext Guide page of the major]). The old "Option 2" math major is not available to students anymore. Those who declared the Option 2 math major before Fall 2020 may finish it with the original rules, or they may switch to one of the new named options. <br />
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'''<br />
<br />
<br />
<br />
<br />
The '''Option 2 math major''' requires six math courses and four courses in an area of focus. These four courses are required to have a certain mathematical content. The selection of the four courses, together with the six required math courses must be approved by the student's advisor. This page lists some sample course collections in several popular areas.<br />
<br />
NOTES: <br />
<br />
1) '''These course collections do not include course prerequisites.''' For example, math 310 has stats 302 as a prerequisite. But stat 302 cannot be used as a focus or major course.<br />
<br />
2) '''Courses offered by departments/schools/colleges outside of mathematics may have restricted enrollments.''' For example, an L&S student interested in an option 2 program with finance emphasis may not reliably be able to enroll in fin 300 since it is taught by Business.<br />
<br />
== Economics and Business ==<br />
<br />
=== Actuarial Mathematics ===<br />
Actuaries use techniques in mathematics and statistics to evaluate risk in a variety of areas including insurance, finance, healthcare, and even criminal justice. In recent history the field has been revolutionized by advances in the theory of probability and the ability to access, store, and process very large data sets.<br />
<br />
Professional actuaries are currently in demand, have lucrative pay, and is a growth field [http://www.bls.gov/ooh/math/actuaries.htm]. Similar to some other fields (law, accounting, etc.) there are professional organizations which administer a series of examinations [http://www.beanactuary.org/exams/]. Oftentimes students complete some of these examinations before graduating which allows them to move right into a career (Note: these exams are not required for graduation).<br />
<br />
Students who are interested in actuarial mathematics should consider coursework in probability, statistics, analysis, as well as computational mathematics.<br />
<br />
'''Application Courses'''<br />
* Act Sci 303<br />
<br />
* Act. Sci 650 and 652<br />
<br />
* Act. Sci. 651 or 653<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. <br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
** Has the prerequisite: one of the probability courses mentioned above AND an elementary stats class (Stat 302 is recommended).<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Linear Programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
'''Additional Courses to Consider'''<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
<br />
'''Also:''' Students interested in the areas of mathematics with applications to actuarial science might consider the following as well:<br />
<br />
* Advanced courses offered by the [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Department of Statistics].<br />
<br />
* A [http://bus.wisc.edu/knowledge-expertise/academic-departments/actuarial-science-risk-management-insurance program] offered by the UW-Madison School of Business.<br />
<br />
=== Business ===<br />
Applications of mathematics to business is often referred to as Operations Research or Management Science. Specifically, the goal is to use mathematics to make the best decisions in a variety of areas: searching, routing, scheduling, transport, etc.<br />
<br />
The modern version of the field grew out of the work mathematicians did in order to aid the Allied war effort during world war II.[http://www.history.army.mil/html/books/hist_op_research/CMH_70-102-1.pdf] Since then, the field has grown into a robust and active area of research and scholarship including several journals and professional organizations.[http://www.informs.org/]<br />
<br />
Students interested in applications of mathematics to business can find many employment opportunities in private corporations, government agencies, nonprofit enterprises, and more. Students can also move onto postgraduate programs in mathematics or business.<br />
<br />
'''Application Courses'''<br />
* Linear programming and Optimization: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Note that this course cannot also be used as a core math course.<br />
* Operations Research: OTM 410<br />
* At least two from the following: Gen Bus 306, Gen Bus 307; OTM 451, 411, 633, 654<br />
** Note that OTM 633 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/111 Math/Stat 310]<br />
<br />
* Computational Mathematics: [https://www.math.wisc.edu/514-numerical-analysis Math 514] or [http://www.math.wisc.edu/513-numerical-linear-algebra 513]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastics: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics.<br />
* [http://www.math.wisc.edu/633-queueing-theory-and-stochastic-modeling Math 633].<br />
<br />
'''Also:''' Consider a program in the UW-Madison [http://bus.wisc.edu/bba/academics-and-programs/majors/operations-technology-management School of Business].<br />
<br />
=== Economics ===<br />
Economics is perhaps the most mathematical of the social sciences. Specifically economists wish to model and understand the behavior of individuals (people, countries, animals, etc.). Typically this is done by quantifying some elements of interest to the individuals.<br />
<br />
Due to the quantitative nature of the field, economic theory has begun to move from the classic areas of markets, products, supply, demand, etc. and into many seemingly unrelated areas: law, psychology, political science, biology, and more.[http://en.wikipedia.org/wiki/Economics_imperialism]<br />
<br />
Regardless, the backbone of economics and economic theory is mathematics. The classical area of mathematics most often related with economics is analysis. <br />
<br />
'''Application Courses'''<br />
* Microeconomics: Econ 301 or 311.<br />
* Macroeconomics: Econ 302 or 312.<br />
* Economic Electives: At least two courses from Econ 410, 460, 475, 503, 521, 525, and 666; Math 310 and Math 415.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Linear programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in the [http://www.econ.wisc.edu/undergrad/Reqs%20for%20Major.html Department of Economics].<br />
<br />
=== Finance ===<br />
Financial mathematics is more popular than ever with financial firms hiring "quants" from all areas of mathematics and the natural sciences. Financial markets are of interest to mathematicians due to the difficult nature of modeling the complex systems. The standard tools involved are evolutionary differential equations, measure theory, and stochastic calculus.<br />
<br />
'''Application Courses'''<br />
* Statistics: Econ 410 or Math/Stat 310.<br />
* Finance core: Finance 300, 320, 330.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Mat 619].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522] and [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Linear Programming (optimization): [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in [http://bus.wisc.edu/bba/academics-and-programs/majors/finance Finance] at the the Wisconsin School of Business.<br />
<br />
== Physical Sciences ==<br />
The physical sciences and mathematics have grown hand-in-hand since antiquity.<br />
Students with strong backgrounds in mathematics who are also interested in a branch of the physical sciences can find opportunities in laboratories, engineering firms, education, finance, law, business, and medicine. Those with very strong academic records can find themselves as preferred candidates for graduate study in their choice of field.<br />
<br />
The following sample programs in mathematics have strong relationships with a particular area of interest in the natural sciences.<br />
<br />
=== Atmospheric & Oceanic Sciences ===<br />
Weather and climate is determined by the interaction between two thin layers which cover the planet: The oceans and the atmosphere. Understanding how these two fluids act and interact allow humans to describe historical climate trends, forecast near future weather with incredible accuracy, and hopefully describe long term climate change which will affect the future of human society.<br />
<br />
A student interested in atmospheric and oceanic studies who has a strong mathematics background can find a career working in local, national, and international meteorological laboratories. These include private scientific consulting businesses as well as public enterprises. Students interested in graduate study could find their future studies supported by the National Science Foundation, the Department of Energy, NASA, or others [http://www.nsf.gov/funding/]. There is a large amount of funding available in the area due to the relevance research findings have on energy and economic policy.<br />
<br />
Mathematicians who work in Atmospheric and oceanic studies are drawn to the complexities of the problems and the variety of methods in both pure and applied mathematics which can be brought to bear on them. Students should take coursework in methods of applied mathematics, differential equations, computational mathematics, and differential geometry and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* ATM OCN 310, 311, and 330 [http://www.aos.wisc.edu/education/Syllabus/courses_majors.html]<br />
** 310 and 330 have Physics 208/248 as a prerequisite.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322]<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619]<br />
<br />
'''Also:''' Students who are interested in this area might consider <br />
* A program offered by the [http://www.aos.wisc.edu/education/undergrad_program.htm Department of Atmospheric and Oceanic Sciences].<br />
* The [http://www.math.wisc.edu/amep AMEP] program.<br />
<br />
=== Chemistry ===<br />
The applications of mathematics to chemistry range from the mundane: Ratios for chemical reactants; to the esoteric: Computational methods in quantum chemistry. Research in this latter topic lead to a Nobel Prize in Chemistry to mathematician [http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/index.html John Pople].<br />
<br />
All areas of pure and applied mathematics have applications in modern chemistry. The most accessible track features coursework focusing on applied analysis and computational math. Students with a strong interest in theoretical mathematics should also consider modern algebra (for group theory) and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* Analytical Chemistry: Chem 327 or Chem 329[http://www.chem.wisc.edu/content/courses]<br />
** Prerequisite: Chem 104 or 109 <br />
* Physical Chemistry: Chem 561 and 562<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 320 recommended.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Math 513 or 514 suggested.<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Several higher level courses have connections to theoretical chemistry: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
** Any of these courses are acceptable in lieu of the 500 level courses above.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.chem.wisc.edu/content/undergraduate Department of Chemistry].<br />
<br />
=== Physics ===<br />
Perhaps the subject with the strongest historical ties with mathematics is physics. Certainly some of the great physical theories have been based on novel applications of mathematical theory or the invention of new subjects in the field: Newtonian mechanics and calculus, relativity and Riemannian geometry, quantum theory and functional analysis, etc.<br />
<br />
Nearly all mathematics courses offered here at UW Madison will have some applications to physics. The following is a collection of courses which would support general interest in physics.<br />
<br />
'''Application Courses'''<br />
* Mechanics, Electricity, and Magnetism: [http://www.physics.wisc.edu/academics/undergrads/inter-adv-311 Physics 311] and [http://www.physics.wisc.edu/academics/undergrads/inter-adv-322 Physics 322]<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 421 is suggested to prepare students for math 521.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. See suggested courses below.<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* ODEs: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
* PDEs: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619].<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541].<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
* Differential Geometry [http://www.math.wisc.edu/561-differential-geometry Math 561].<br />
* Complex Analysis: [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/514-numerical-analysis 514].<br />
<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
=== Astronomy ===<br />
The Astronomy package has the same mathematics core, but different suggested application courses:<br />
<br />
'''Application Courses'''<br />
* Astronomy core: Choose two courses from Astron 310, 320, or 335.<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. Suggested courses are: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses above the 500 level.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
== Biological Sciences ==<br />
Applications of mathematics to biology has undergone a recent boom. Historically, the biologist was perhaps most interested in applications of calculus, but now nearly any modern area of mathematical research has an application to some biological field[http://www.ams.org/notices/199509/hoppensteadt.pdf]. The following lists some possibilities.<br />
<br />
=== Bio-Informatics ===<br />
Bioinformatics is the application of computational methods to understand biological information. Of course the most interesting items of biological information is genetic and genomic information. Considering that the human genome has over three billion basepairs [http://www.genome.gov/12011238], it's no wonder that many mathematicians find compelling problems in the area to devote their time.<br />
<br />
Students with strong mathematical backgrounds who are interested in bioinformatics can find careers as a part of research teams in public and private laboratories across the world [http://www.bioinformatics.org/jobs/]. Moreover, many universities have established interdisciplinary graduate programs promoting this intersection of mathematics, biology, and computer science [http://ils.unc.edu/informatics_programs/doc/Bioinformatics_2006.html].<br />
<br />
Students interested in bioinformatics should have a strong background in computational mathematics and probability. Students should also have a strong programming background.<br />
<br />
'''Application Courses'''<br />
* Computer Science: CS 300 and CS 400 (or CS 302 and CS 367).<br />
* Bioinformatics: [http://www.biostat.wisc.edu/content/bmi-576-introduction-bioinformatics BMI/CS 576]<br />
* Genetics: Gen 466<br />
** Note that this class has a prerequisite of a year of chemistry and a year of biology coursework. Please contact the UW-Madison [http://www.genetics.wisc.edu/UndergraduateProgram.htm genetics] program for more information.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: At least three of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
<br />
'''Also''' <br />
* Consider a program in [http://www.cs.wisc.edu/academics/Undergraduate-Programs Computer Science] or [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
* Complete this major with a few additional courses if you are interested in medical school [http://prehealth.wisc.edu/explore/documents/Pre-Med.pdf].<br />
<br />
=== Bio-Statistics ===<br />
Biostatistics is the application of mathematical statistical methods to areas of biology. Historically, one could consider the field to have been founded by Gregor Mendel himself. He used basic principles of statistics and probability to offer a theory for which genetic traits would arise from cross hybridization of plants and animals. His work was forgotten for nearly fifty years before it was rediscovered and become an integral part of modern genetic theory.<br />
<br />
Beyond applications to genetics, applications of biostatistics range from public health policy to evaluating laboratory experimental results to tracking population dynamics in the field. Currently, health organizations consider there to be a shortage of trained biostatisticians[http://www.amstat.org/careers/biostatistics.cfm]. Students interested in this area should find excellent job prospects.<br />
<br />
Students interested in biostatistics should have strong backgrounds in probability, statistics, and computational methods.<br />
<br />
'''Application Courses'''<br />
* Statistics: Any four from Stat 333, 424, 575, 641, and 642 [http://www.stat.wisc.edu/course-listing]<br />
** Stat 333 has as a prerequisite some experience with statistical software. This can be achieved by also registering for Stat 327. Stat 327 is a single credit course which does not count for the mathematics major.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* More courses in computational mathematics listed above.<br />
* [http://www.math.wisc.edu/635-introduction-brownian-motion-and-stochastic-calculus Math 635]<br />
<br />
'''Also'''<br />
* Consider a program with [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Statistics] or in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences].<br />
* Compare this major program to requirements for Medical School.<br />
<br />
=== Ecology, Forestry, Wildlife Ecology ===<br />
Applications of advanced mathematics to ecology has resulted in science's improved ability to track wild animal populations, predict the spread of diseases, model the impact of humans on native wildlife, control invasive species, and more. Modeling in this area is mathematically interesting due to the variety of scales and the inherent difficulty of doing science outside of a laboratory! As such the methods of deterministic and stochastic models are particularly useful.<br />
<br />
'''Application Courses'''<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Computational Methods: [http://www.cs.wisc.edu/courses/412 CS 412].<br />
* Any two courses from [http://zoology.wisc.edu/courses/courselist.htm Zoo 460, 504, and 540]; or [http://forestandwildlifeecology.wisc.edu/undergraduate-study-courses F&W Ecol 300, 410, 460, 531, 652, and 655].<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Stochastic Processes: Either [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
=== Genetics ===<br />
Applications of mathematics in genetics appear on a wide range of scales: chemical processes, cellular processes, organism breeding, and speciation. For applications of mathematics in genetics on the scale of chemical processes you might want to examine the suggested packages for bioinformatics or structural biology. If instead you are interested in the larger scale of organisms you might consider the package in biostatistics or the one below:<br />
<br />
'''Application Courses'''<br />
* Any four courses chosen from: GEN 466, 564, 565, 626, 629, and BMI 563.[http://www.genetics.wisc.edu/UndergraduateProgram.htm]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended for non-honors students.<br />
<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
* Consider a program in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences] such as [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
<br />
=== Structural Biology ===<br />
Structural biologists are primarily interested in the large molecules which are involved in cellular processes: the fundamental chemical building blocks of life. The field lies on the intersection of biology, physics, chemistry, and mathematics and so structural biology is an exciting area of interdisciplinary research.<br />
<br />
In general, the mathematics involved in structural biology is focused on computational methods, probability, and statistics. Note that we offer a specialized course in Mathematics Methods in Structural Biology - Math 606.<br />
<br />
'''Application Courses'''<br />
* Analytical Methods in Chemistry: Chem 327 or 329<br />
* Physical Chemistry: Chem 561 and 562<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/mathematical-methods-structural-biology Math 606]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [https://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
=== Systems Biology ===<br />
Systems biology is the computational and mathematical modeling of biological systems at any scale. The classical example of this may be the [http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation predator-prey] model of differential equations which describe the relative population dynamics of two species. Other systems examples include disease transmission, chemical pathways, cellular processes, and more.<br />
<br />
In general, the mathematics involved in systems biology is focused on computational methods, dynamical systems, differential equations, the mathematics of networks, control theory, and others. Note that we offer a specialized course in Mathematical Methods in Systems Biology - Math 609.<br />
<br />
'''Application Courses'''<br />
* Organic Chemistry: Chem 341 or 343<br />
* Introductory Biochemistry: Biochem 501<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/609-mathematical-methods-systems-biology Math 609]<br />
* One Biochem elective: Any Biochem class numbered above 600. Suggested courses are Biochem 601, 612, 620, 621, 624, and 630.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
<br />
'''Additional Courses to Consider'''<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses in computational mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
== Engineering ==<br />
Engineering is the application of science and mathematics to the invention, improvement, and maintenance of anything and everything. As with many of the sciences, engineers and mathematicians have a symbiotic relationship: Engineers use mathematics to make new things; the new things exhibit novel properties that are mathematically interesting.<br />
<br />
In general all of mathematics can be applied to some field of engineering. However the programs offered below are not substitutes for engineering degrees. That is, student who are interested in an engineering career upon completion of their undergraduate degree should probably enroll in one of the engineering programs offered by the [http://www.engr.wisc.edu/current/undergrad.html College of Engineering]. Similarly, students who are primarily interested in mathematics might instead choose an option I major and concentrate their upper level coursework in applied mathematics. Students who are truly interested in both areas should consider the degree program in [http://www.math.wisc.edu/amep Applied Mathematics, Engineering, and Physics].<br />
<br />
So who do the programs below serve? They serve engineering students who wish to take a second major in mathematics. In general such students are excellent candidates for graduate study in engineering.<br />
<br />
=== Chemical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Chemical Engineering who are interested in pursuing a second major in mathematics.<br />
<br />
'''Application Courses'''<br />
* [http://www.engr.wisc.edu/cmsdocuments/cbe-undergrade-handbook-2009-v7.pdf CBE 320, 326, 426, 470]<br />
** Note: All of these course are required for the undergraduate program in chemical engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), complex analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]), and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
=== Civil Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Civil and Environmental Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Fluid Mechanics and Structural Analysis: [http://courses.engr.wisc.edu/cee/ CIV ENG 310, 311, 340]<br />
** Note: All of these course are required for the undergraduate program in civil engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective Structural Analysis Course: CIV ENG 440, 442, 445, or 447.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]); and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], and [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
=== Electrical and Computer Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Computer and Electrical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core ECE: [http://courses.engr.wisc.edu/ece/ ECE 220, 230, 352]<br />
** Note: All of these course are required for the undergraduate program in electrical and computer engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective: ECE 435, 525, or 533.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
** ECE 435 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* At least two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), linear programming [http://www.math.wisc.edu/525-linear-programming-methods Math 525], modern algebra [http://www.math.wisc.edu/541-modern-algebra Math 541], differential geometry [http://www.math.wisc.edu/561-differential-geometry Math 561], and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
* Error correcting codes: [http://www.math.wisc.edu/641-introduction-error-correcting-codes Math 641]<br />
<br />
===Engineering Mechanics and Astronautics===<br />
The following program details an option 2 package for students in the College of Engineering program in Engineering Mechanics and Astronautics who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Engineering Mechanics: [http://courses.engr.wisc.edu/ema/ EMA 201, 202, 303]<br />
* One elective: EMA 521, 542, 545, or 563<br />
** All of the above courses may be used to satisfy the EMA program requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525]), and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
<br />
=== Industrial Engineering ===<br />
Industrial engineering is the application of engineering principles to create the most effective means of production. In particular, they work to optimize complex systems.<br />
<br />
'''Application Courses[http://www.engr.wisc.edu/isye/isye-curriculum-documents.html]'''<br />
* Core Industrial engineering: I SY E 315, 320, and 323.<br />
* Industrial Engineering Elective: At least one of I SY E 425, 516, 525, 526, 558, 575, 615, 620, 624, 635, or 643.<br />
** Note: ISYE 425 and 525 are both crosslisted with math and cannot be used to complete both the application and core math requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Both 309 and 431 are preferred over stat 311.<br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Numerical Analysis: [http://www.math.wisc.edu/514-numerical-analysis Math 514].<br />
<br />
'''Also:'''<br />
Consider the program in [http://www.engr.wisc.edu/isye/isye-academics-undergraduate-program.html Industrial Engineering] offered by the College of Engineering.<br />
<br />
=== Materials Science ===<br />
The following program details an option 2 package for students in the College of Engineering program in Materials Science and Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Materials Courses: [http://www.engr.wisc.edu/cmsdocuments/Degree_requirements_2014.pdf MSE 330, 331, and 351]<br />
* One Engineering Elective: CBE 255, CS 300, CS 302, CS 310, ECE 230, ECE 376, EMA 303, Phys 321, Stat 424].<br />
** All of the above classes may be used to satisfy the program requirements for MS&E BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Mechanical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Mechanical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Mechanical Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/me-flowchart-spring-2014.pdf ME 340, 361, 363, 364]<br />
** All of the above courses are required by the Mechanical Engineering program.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Nuclear Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Nuclear Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Nuclear Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/NE-UGguide2014.pdf NE 305, 405, and 408]<br />
* One Engineering Elective: Physics 321 or 322, ECE 376, BME 501, or NE 411.<br />
** All of the above classes may be used to satisfy the program requirements for the Nuclear Engineering BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
* Additional courses above the 500 level listed above.<br />
<br />
== Computer Science ==<br />
Computer science as an independent discipline is rather young: The first computer science degree program offered in the United States was formed in 1962 (at Purdue University). Despite its youth, one could argue that no single academic discipline has had more of an effect on human society since the scientific revolution.<br />
<br />
Since computer science is foremost concerned with the theory of computation, its link with mathematics is robust. Historical examples include Alan Turing, A mathematician and WWII cryptoanalyst who's theory of the Universal Turing Machine forms the central framework of modern computation; and John Von Neumann, A mathematician who's name is ascribed to the architecture still used for nearly all computers today.[https://web.archive.org/web/20130314123032/http://qss.stanford.edu/~godfrey/vonNeumann/vnedvac.pdf] There are broad overlaps in reasearch in the two fields. For example, one of the most famous unsolved problems in mathematics, the [http://www.claymath.org/millenium-problems/p-vs-np-problem P vs NP] problem, is also considered an open problem in the theory of computation.<br />
<br />
Since computer science is a full field enveloping philosophy, mathematics, and engineering there are many possible areas of interest which a student of mathematics and computer science might focus on. Below are several examples.<br />
<br />
=== Computational Methods ===<br />
Computational methods are the algorithms a computer follows in order to perform a specific task. Of interest besides the algorithms is methods for evaluating their quality and efficiency. Since computational mathematics is on the interface between pure and applied methods students who concentrate in this area can find many exciting research opportunities available at the undergraduate level. <br />
<br />
The mathematical coursework focuses on combinatorics, analysis, and numerical methods. <br />
<br />
'''Application Courses'''<br />
* Any four courses from: CS 352, 367, 400, 412, 435, 475, 513, 514, 515, 520, 525, 533, 540, 545, 558, 559, and 577.<br />
** Note that 435, 475, 513, 514, 515, and 525 are crosslisted with math. They may not be used as both application courses and core mathematics courses<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Advanced Calculus [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] and/or [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics above.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Analysis II: [http://www.math.wisc.edu/522-advanced-calculus Math 522].<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542].<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567].<br />
* Logic: [http://www.math.wisc.edu/571-mathematical-logic Math 571].<br />
<br />
'''Also:'''<br />
Consider the program in the [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science].<br />
<br />
=== Theoretical Computer Science ===<br />
If you are interested in a more theoretical bend to your studies, follow the program above but with the following changes:<br />
* Include both CS 520 and CS 577 into your core applied courses.<br />
* Replace the two computational methods courses with Math 567 and Math 571.<br />
<br />
=== Cryptography ===<br />
Due to the widespread use of computer storage, platforms, and devices; security is now of singular interest. Students with expertise in the mathematics associated with cryptography can find interesting opportunities after graduation in public and private security sectors.<br />
<br />
The mathematics associated to secure messaging and cryptography is typically centered on combinatorics and number theory.<br />
<br />
'''Application Courses'''<br />
* Programming: CS 300 and CS 400 (or CS 302 and 367).<br />
* One of the following two pairs:<br />
** The CS track: Operating systems (CS 537) and Security (CS 642)<br />
** The ECE track: Digital Systems: (ECE 352) and Error Correcting Codes (ECE 641).<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Cryptography: [http://www.math.wisc.edu/435-introduction-cryptography Math 435]<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567]<br />
<br />
'''Additional Courses to Consider'''<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Also:'''<br />
Consider combining the programs offered by [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science] or [http://www.engr.wisc.edu/ece/ece-academics-undergraduate-program.html Computer Engineering].<br />
<br />
== Secondary Education ==<br />
The so called STEM fields continue to be a major area of interest and investment for education policy makers. In particular secondary education instructors with strong mathematics backgrounds are in demand across the nation in public, private, and charter school environments. <br />
<br />
The following program was designed for a math major who is interested in becoming an educator at the secondary level. Note that successful completion of the coursework outlined below would make a strong candidate for graduate work in mathematics and education at the masters level. Our own School of Education offers a [http://www.uwteach.com/mathematics.html Masters Degree in Secondary Mathematics] which concludes with state certification. <br />
<br />
''Note that a major requires at least two courses at the 500 level. Therefore you should consider the suggestions below carefully.''<br />
<br />
'''Application Courses'''<br />
* History and philosophy of mathematics: [http://www.math.wisc.edu/473-history-mathematics Math 473].<br />
* Math education capstone course: [http://www.math.wisc.edu/371-basic-concepts-mathematics Math 471]<br />
* Two additional courses from Mathematics, Computer Science, Physics, or Economics at the Intermediate or Advanced Level.<br />
** Suggested: CS 300, CS 302, Phys 207, Math 421, Math 475, Math 561, Math 567<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 (or Math 375) suggested.<br />
* College Geometry: [http://www.math.wisc.edu/461-college-geometry-i Math 461]. <br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] <br />
** Math 431 and 309 are equivalent. <br />
** [http://www.math.wisc.edu/531-probability-theory Math 531] can also be considered. This is a proof based introduction to probability and may be taken only after Math 421 or Math 521.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310] (Math 310 has a prerequisite of Math 309 or 431.)<br />
* Modern Algebra: [http://www.math.wisc.edu/441-introduction-modern-algebra Math 441] or [http://www.math.wisc.edu/541-modern-algebra 541].<br />
* Analysis: [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or [http://www.math.wisc.edu/521-advanced-calculus 521].<br />
** Math 521 is strongly suggested for students planning to teach AP Calculus in high school<br />
<br />
'''Additional Courses to Consider'''<br />
* Math 421 can be a useful course to take before the 500 level coursework.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Additional courses at the 500 level in mathematics.<br />
* Courses in computer programming, statistics, physics, economics, and finance can broaden your content areas and qualify you for more subjects.<br />
<br />
== Statistics ==<br />
Statistics is the study of the collection, measuring, and evaluation of data. Recent advances in our ability to collect and parse large amounts of data has made the field more exciting then ever before. Positions in data analysis are becoming common outside of laboratory environments: marketing, education, health, sports, infrastructure, politics, etc.<br />
<br />
Statistics has a strong relationship with mathematics. The areas of mathematics of particular interest are linear algebra, probability, and analysis.<br />
<br />
'''Application Courses'''<br />
* Core Statistics: Stat 333 and Stat 424<br />
* Statistics Electives: At least two from: Stat 349, 351, 411, 421, 456, 471, 609, or 610.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Mathematical Statistics Sequence: [http://www.math.wisc.edu/node/111 Math 309] and [http://www.math.wisc.edu/node/114 Math 310]<br />
** Math 431 may be used for Math 309.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522], [http://www.math.wisc.edu/621-analysis-iii-0 621], or [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Advanced Probability Theory [http://www.math.wisc.edu/531-probability-theory Math 531].<br />
* Algebra: [https://www.math.wisc.edu/541-modern-algebra Math 541].<br />
<br />
'''Also:'''<br />
A student who wishes to complete a major in statistics offered by the [https://www.stat.wisc.edu/undergrad/undergraduate-major-statistics Department of Statistics] should complete the program above and include:<br />
* Stat 302 and 327.<br />
* A course in programming (e.g. CS 300).<br />
* At least one more course from the statistics electives above.</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Option_2_packages&diff=21200Option 2 packages2021-05-09T16:43:22Z<p>Valko: </p>
<hr />
<div>'''NOTE: in the Fall 2020 semester the Department of Mathematics introduced five new named options (see [https://guide.wisc.edu/undergraduate/letters-science/mathematics/mathematics-bs/#requirementstext the Guide page of the major]). The old "Option 2" math major is not available to students anymore. Those who declared the Option 2 math major before Fall 2020 may finish it with the original rules, or they may switch to one of the new named options. <br />
<br />
'''<br />
<br />
<br />
<br />
<br />
The '''Option 2 math major''' requires six math courses and four courses in an area of focus. These four courses are required to have a certain mathematical content. The selection of the four courses, together with the six required math courses must be approved by the student's advisor. This page lists some sample course collections in several popular areas.<br />
<br />
NOTES: <br />
<br />
1) '''These course collections do not include course prerequisites.''' For example, math 310 has stats 302 as a prerequisite. But stat 302 cannot be used as a focus or major course.<br />
<br />
2) '''Courses offered by departments/schools/colleges outside of mathematics may have restricted enrollments.''' For example, an L&S student interested in an option 2 program with finance emphasis may not reliably be able to enroll in fin 300 since it is taught by Business.<br />
<br />
== Economics and Business ==<br />
<br />
=== Actuarial Mathematics ===<br />
Actuaries use techniques in mathematics and statistics to evaluate risk in a variety of areas including insurance, finance, healthcare, and even criminal justice. In recent history the field has been revolutionized by advances in the theory of probability and the ability to access, store, and process very large data sets.<br />
<br />
Professional actuaries are currently in demand, have lucrative pay, and is a growth field [http://www.bls.gov/ooh/math/actuaries.htm]. Similar to some other fields (law, accounting, etc.) there are professional organizations which administer a series of examinations [http://www.beanactuary.org/exams/]. Oftentimes students complete some of these examinations before graduating which allows them to move right into a career (Note: these exams are not required for graduation).<br />
<br />
Students who are interested in actuarial mathematics should consider coursework in probability, statistics, analysis, as well as computational mathematics.<br />
<br />
'''Application Courses'''<br />
* Act Sci 303<br />
<br />
* Act. Sci 650 and 652<br />
<br />
* Act. Sci. 651 or 653<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. <br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
** Has the prerequisite: one of the probability courses mentioned above AND an elementary stats class (Stat 302 is recommended).<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Linear Programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
'''Additional Courses to Consider'''<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
<br />
'''Also:''' Students interested in the areas of mathematics with applications to actuarial science might consider the following as well:<br />
<br />
* Advanced courses offered by the [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Department of Statistics].<br />
<br />
* A [http://bus.wisc.edu/knowledge-expertise/academic-departments/actuarial-science-risk-management-insurance program] offered by the UW-Madison School of Business.<br />
<br />
=== Business ===<br />
Applications of mathematics to business is often referred to as Operations Research or Management Science. Specifically, the goal is to use mathematics to make the best decisions in a variety of areas: searching, routing, scheduling, transport, etc.<br />
<br />
The modern version of the field grew out of the work mathematicians did in order to aid the Allied war effort during world war II.[http://www.history.army.mil/html/books/hist_op_research/CMH_70-102-1.pdf] Since then, the field has grown into a robust and active area of research and scholarship including several journals and professional organizations.[http://www.informs.org/]<br />
<br />
Students interested in applications of mathematics to business can find many employment opportunities in private corporations, government agencies, nonprofit enterprises, and more. Students can also move onto postgraduate programs in mathematics or business.<br />
<br />
'''Application Courses'''<br />
* Linear programming and Optimization: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Note that this course cannot also be used as a core math course.<br />
* Operations Research: OTM 410<br />
* At least two from the following: Gen Bus 306, Gen Bus 307; OTM 451, 411, 633, 654<br />
** Note that OTM 633 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/111 Math/Stat 310]<br />
<br />
* Computational Mathematics: [https://www.math.wisc.edu/514-numerical-analysis Math 514] or [http://www.math.wisc.edu/513-numerical-linear-algebra 513]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastics: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics.<br />
* [http://www.math.wisc.edu/633-queueing-theory-and-stochastic-modeling Math 633].<br />
<br />
'''Also:''' Consider a program in the UW-Madison [http://bus.wisc.edu/bba/academics-and-programs/majors/operations-technology-management School of Business].<br />
<br />
=== Economics ===<br />
Economics is perhaps the most mathematical of the social sciences. Specifically economists wish to model and understand the behavior of individuals (people, countries, animals, etc.). Typically this is done by quantifying some elements of interest to the individuals.<br />
<br />
Due to the quantitative nature of the field, economic theory has begun to move from the classic areas of markets, products, supply, demand, etc. and into many seemingly unrelated areas: law, psychology, political science, biology, and more.[http://en.wikipedia.org/wiki/Economics_imperialism]<br />
<br />
Regardless, the backbone of economics and economic theory is mathematics. The classical area of mathematics most often related with economics is analysis. <br />
<br />
'''Application Courses'''<br />
* Microeconomics: Econ 301 or 311.<br />
* Macroeconomics: Econ 302 or 312.<br />
* Economic Electives: At least two courses from Econ 410, 460, 475, 503, 521, 525, and 666; Math 310 and Math 415.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Linear programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in the [http://www.econ.wisc.edu/undergrad/Reqs%20for%20Major.html Department of Economics].<br />
<br />
=== Finance ===<br />
Financial mathematics is more popular than ever with financial firms hiring "quants" from all areas of mathematics and the natural sciences. Financial markets are of interest to mathematicians due to the difficult nature of modeling the complex systems. The standard tools involved are evolutionary differential equations, measure theory, and stochastic calculus.<br />
<br />
'''Application Courses'''<br />
* Statistics: Econ 410 or Math/Stat 310.<br />
* Finance core: Finance 300, 320, 330.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Mat 619].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522] and [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Linear Programming (optimization): [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in [http://bus.wisc.edu/bba/academics-and-programs/majors/finance Finance] at the the Wisconsin School of Business.<br />
<br />
== Physical Sciences ==<br />
The physical sciences and mathematics have grown hand-in-hand since antiquity.<br />
Students with strong backgrounds in mathematics who are also interested in a branch of the physical sciences can find opportunities in laboratories, engineering firms, education, finance, law, business, and medicine. Those with very strong academic records can find themselves as preferred candidates for graduate study in their choice of field.<br />
<br />
The following sample programs in mathematics have strong relationships with a particular area of interest in the natural sciences.<br />
<br />
=== Atmospheric & Oceanic Sciences ===<br />
Weather and climate is determined by the interaction between two thin layers which cover the planet: The oceans and the atmosphere. Understanding how these two fluids act and interact allow humans to describe historical climate trends, forecast near future weather with incredible accuracy, and hopefully describe long term climate change which will affect the future of human society.<br />
<br />
A student interested in atmospheric and oceanic studies who has a strong mathematics background can find a career working in local, national, and international meteorological laboratories. These include private scientific consulting businesses as well as public enterprises. Students interested in graduate study could find their future studies supported by the National Science Foundation, the Department of Energy, NASA, or others [http://www.nsf.gov/funding/]. There is a large amount of funding available in the area due to the relevance research findings have on energy and economic policy.<br />
<br />
Mathematicians who work in Atmospheric and oceanic studies are drawn to the complexities of the problems and the variety of methods in both pure and applied mathematics which can be brought to bear on them. Students should take coursework in methods of applied mathematics, differential equations, computational mathematics, and differential geometry and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* ATM OCN 310, 311, and 330 [http://www.aos.wisc.edu/education/Syllabus/courses_majors.html]<br />
** 310 and 330 have Physics 208/248 as a prerequisite.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322]<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619]<br />
<br />
'''Also:''' Students who are interested in this area might consider <br />
* A program offered by the [http://www.aos.wisc.edu/education/undergrad_program.htm Department of Atmospheric and Oceanic Sciences].<br />
* The [http://www.math.wisc.edu/amep AMEP] program.<br />
<br />
=== Chemistry ===<br />
The applications of mathematics to chemistry range from the mundane: Ratios for chemical reactants; to the esoteric: Computational methods in quantum chemistry. Research in this latter topic lead to a Nobel Prize in Chemistry to mathematician [http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/index.html John Pople].<br />
<br />
All areas of pure and applied mathematics have applications in modern chemistry. The most accessible track features coursework focusing on applied analysis and computational math. Students with a strong interest in theoretical mathematics should also consider modern algebra (for group theory) and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* Analytical Chemistry: Chem 327 or Chem 329[http://www.chem.wisc.edu/content/courses]<br />
** Prerequisite: Chem 104 or 109 <br />
* Physical Chemistry: Chem 561 and 562<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 320 recommended.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Math 513 or 514 suggested.<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Several higher level courses have connections to theoretical chemistry: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
** Any of these courses are acceptable in lieu of the 500 level courses above.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.chem.wisc.edu/content/undergraduate Department of Chemistry].<br />
<br />
=== Physics ===<br />
Perhaps the subject with the strongest historical ties with mathematics is physics. Certainly some of the great physical theories have been based on novel applications of mathematical theory or the invention of new subjects in the field: Newtonian mechanics and calculus, relativity and Riemannian geometry, quantum theory and functional analysis, etc.<br />
<br />
Nearly all mathematics courses offered here at UW Madison will have some applications to physics. The following is a collection of courses which would support general interest in physics.<br />
<br />
'''Application Courses'''<br />
* Mechanics, Electricity, and Magnetism: [http://www.physics.wisc.edu/academics/undergrads/inter-adv-311 Physics 311] and [http://www.physics.wisc.edu/academics/undergrads/inter-adv-322 Physics 322]<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 421 is suggested to prepare students for math 521.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. See suggested courses below.<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* ODEs: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
* PDEs: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619].<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541].<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
* Differential Geometry [http://www.math.wisc.edu/561-differential-geometry Math 561].<br />
* Complex Analysis: [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/514-numerical-analysis 514].<br />
<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
=== Astronomy ===<br />
The Astronomy package has the same mathematics core, but different suggested application courses:<br />
<br />
'''Application Courses'''<br />
* Astronomy core: Choose two courses from Astron 310, 320, or 335.<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. Suggested courses are: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses above the 500 level.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
== Biological Sciences ==<br />
Applications of mathematics to biology has undergone a recent boom. Historically, the biologist was perhaps most interested in applications of calculus, but now nearly any modern area of mathematical research has an application to some biological field[http://www.ams.org/notices/199509/hoppensteadt.pdf]. The following lists some possibilities.<br />
<br />
=== Bio-Informatics ===<br />
Bioinformatics is the application of computational methods to understand biological information. Of course the most interesting items of biological information is genetic and genomic information. Considering that the human genome has over three billion basepairs [http://www.genome.gov/12011238], it's no wonder that many mathematicians find compelling problems in the area to devote their time.<br />
<br />
Students with strong mathematical backgrounds who are interested in bioinformatics can find careers as a part of research teams in public and private laboratories across the world [http://www.bioinformatics.org/jobs/]. Moreover, many universities have established interdisciplinary graduate programs promoting this intersection of mathematics, biology, and computer science [http://ils.unc.edu/informatics_programs/doc/Bioinformatics_2006.html].<br />
<br />
Students interested in bioinformatics should have a strong background in computational mathematics and probability. Students should also have a strong programming background.<br />
<br />
'''Application Courses'''<br />
* Computer Science: CS 300 and CS 400 (or CS 302 and CS 367).<br />
* Bioinformatics: [http://www.biostat.wisc.edu/content/bmi-576-introduction-bioinformatics BMI/CS 576]<br />
* Genetics: Gen 466<br />
** Note that this class has a prerequisite of a year of chemistry and a year of biology coursework. Please contact the UW-Madison [http://www.genetics.wisc.edu/UndergraduateProgram.htm genetics] program for more information.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: At least three of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
<br />
'''Also''' <br />
* Consider a program in [http://www.cs.wisc.edu/academics/Undergraduate-Programs Computer Science] or [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
* Complete this major with a few additional courses if you are interested in medical school [http://prehealth.wisc.edu/explore/documents/Pre-Med.pdf].<br />
<br />
=== Bio-Statistics ===<br />
Biostatistics is the application of mathematical statistical methods to areas of biology. Historically, one could consider the field to have been founded by Gregor Mendel himself. He used basic principles of statistics and probability to offer a theory for which genetic traits would arise from cross hybridization of plants and animals. His work was forgotten for nearly fifty years before it was rediscovered and become an integral part of modern genetic theory.<br />
<br />
Beyond applications to genetics, applications of biostatistics range from public health policy to evaluating laboratory experimental results to tracking population dynamics in the field. Currently, health organizations consider there to be a shortage of trained biostatisticians[http://www.amstat.org/careers/biostatistics.cfm]. Students interested in this area should find excellent job prospects.<br />
<br />
Students interested in biostatistics should have strong backgrounds in probability, statistics, and computational methods.<br />
<br />
'''Application Courses'''<br />
* Statistics: Any four from Stat 333, 424, 575, 641, and 642 [http://www.stat.wisc.edu/course-listing]<br />
** Stat 333 has as a prerequisite some experience with statistical software. This can be achieved by also registering for Stat 327. Stat 327 is a single credit course which does not count for the mathematics major.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* More courses in computational mathematics listed above.<br />
* [http://www.math.wisc.edu/635-introduction-brownian-motion-and-stochastic-calculus Math 635]<br />
<br />
'''Also'''<br />
* Consider a program with [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Statistics] or in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences].<br />
* Compare this major program to requirements for Medical School.<br />
<br />
=== Ecology, Forestry, Wildlife Ecology ===<br />
Applications of advanced mathematics to ecology has resulted in science's improved ability to track wild animal populations, predict the spread of diseases, model the impact of humans on native wildlife, control invasive species, and more. Modeling in this area is mathematically interesting due to the variety of scales and the inherent difficulty of doing science outside of a laboratory! As such the methods of deterministic and stochastic models are particularly useful.<br />
<br />
'''Application Courses'''<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Computational Methods: [http://www.cs.wisc.edu/courses/412 CS 412].<br />
* Any two courses from [http://zoology.wisc.edu/courses/courselist.htm Zoo 460, 504, and 540]; or [http://forestandwildlifeecology.wisc.edu/undergraduate-study-courses F&W Ecol 300, 410, 460, 531, 652, and 655].<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Stochastic Processes: Either [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
=== Genetics ===<br />
Applications of mathematics in genetics appear on a wide range of scales: chemical processes, cellular processes, organism breeding, and speciation. For applications of mathematics in genetics on the scale of chemical processes you might want to examine the suggested packages for bioinformatics or structural biology. If instead you are interested in the larger scale of organisms you might consider the package in biostatistics or the one below:<br />
<br />
'''Application Courses'''<br />
* Any four courses chosen from: GEN 466, 564, 565, 626, 629, and BMI 563.[http://www.genetics.wisc.edu/UndergraduateProgram.htm]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended for non-honors students.<br />
<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
* Consider a program in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences] such as [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
<br />
=== Structural Biology ===<br />
Structural biologists are primarily interested in the large molecules which are involved in cellular processes: the fundamental chemical building blocks of life. The field lies on the intersection of biology, physics, chemistry, and mathematics and so structural biology is an exciting area of interdisciplinary research.<br />
<br />
In general, the mathematics involved in structural biology is focused on computational methods, probability, and statistics. Note that we offer a specialized course in Mathematics Methods in Structural Biology - Math 606.<br />
<br />
'''Application Courses'''<br />
* Analytical Methods in Chemistry: Chem 327 or 329<br />
* Physical Chemistry: Chem 561 and 562<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/mathematical-methods-structural-biology Math 606]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [https://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
=== Systems Biology ===<br />
Systems biology is the computational and mathematical modeling of biological systems at any scale. The classical example of this may be the [http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation predator-prey] model of differential equations which describe the relative population dynamics of two species. Other systems examples include disease transmission, chemical pathways, cellular processes, and more.<br />
<br />
In general, the mathematics involved in systems biology is focused on computational methods, dynamical systems, differential equations, the mathematics of networks, control theory, and others. Note that we offer a specialized course in Mathematical Methods in Systems Biology - Math 609.<br />
<br />
'''Application Courses'''<br />
* Organic Chemistry: Chem 341 or 343<br />
* Introductory Biochemistry: Biochem 501<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/609-mathematical-methods-systems-biology Math 609]<br />
* One Biochem elective: Any Biochem class numbered above 600. Suggested courses are Biochem 601, 612, 620, 621, 624, and 630.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
<br />
'''Additional Courses to Consider'''<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses in computational mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
== Engineering ==<br />
Engineering is the application of science and mathematics to the invention, improvement, and maintenance of anything and everything. As with many of the sciences, engineers and mathematicians have a symbiotic relationship: Engineers use mathematics to make new things; the new things exhibit novel properties that are mathematically interesting.<br />
<br />
In general all of mathematics can be applied to some field of engineering. However the programs offered below are not substitutes for engineering degrees. That is, student who are interested in an engineering career upon completion of their undergraduate degree should probably enroll in one of the engineering programs offered by the [http://www.engr.wisc.edu/current/undergrad.html College of Engineering]. Similarly, students who are primarily interested in mathematics might instead choose an option I major and concentrate their upper level coursework in applied mathematics. Students who are truly interested in both areas should consider the degree program in [http://www.math.wisc.edu/amep Applied Mathematics, Engineering, and Physics].<br />
<br />
So who do the programs below serve? They serve engineering students who wish to take a second major in mathematics. In general such students are excellent candidates for graduate study in engineering.<br />
<br />
=== Chemical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Chemical Engineering who are interested in pursuing a second major in mathematics.<br />
<br />
'''Application Courses'''<br />
* [http://www.engr.wisc.edu/cmsdocuments/cbe-undergrade-handbook-2009-v7.pdf CBE 320, 326, 426, 470]<br />
** Note: All of these course are required for the undergraduate program in chemical engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), complex analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]), and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
=== Civil Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Civil and Environmental Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Fluid Mechanics and Structural Analysis: [http://courses.engr.wisc.edu/cee/ CIV ENG 310, 311, 340]<br />
** Note: All of these course are required for the undergraduate program in civil engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective Structural Analysis Course: CIV ENG 440, 442, 445, or 447.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]); and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], and [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
=== Electrical and Computer Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Computer and Electrical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core ECE: [http://courses.engr.wisc.edu/ece/ ECE 220, 230, 352]<br />
** Note: All of these course are required for the undergraduate program in electrical and computer engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective: ECE 435, 525, or 533.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
** ECE 435 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* At least two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), linear programming [http://www.math.wisc.edu/525-linear-programming-methods Math 525], modern algebra [http://www.math.wisc.edu/541-modern-algebra Math 541], differential geometry [http://www.math.wisc.edu/561-differential-geometry Math 561], and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
* Error correcting codes: [http://www.math.wisc.edu/641-introduction-error-correcting-codes Math 641]<br />
<br />
===Engineering Mechanics and Astronautics===<br />
The following program details an option 2 package for students in the College of Engineering program in Engineering Mechanics and Astronautics who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Engineering Mechanics: [http://courses.engr.wisc.edu/ema/ EMA 201, 202, 303]<br />
* One elective: EMA 521, 542, 545, or 563<br />
** All of the above courses may be used to satisfy the EMA program requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525]), and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
<br />
=== Industrial Engineering ===<br />
Industrial engineering is the application of engineering principles to create the most effective means of production. In particular, they work to optimize complex systems.<br />
<br />
'''Application Courses[http://www.engr.wisc.edu/isye/isye-curriculum-documents.html]'''<br />
* Core Industrial engineering: I SY E 315, 320, and 323.<br />
* Industrial Engineering Elective: At least one of I SY E 425, 516, 525, 526, 558, 575, 615, 620, 624, 635, or 643.<br />
** Note: ISYE 425 and 525 are both crosslisted with math and cannot be used to complete both the application and core math requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Both 309 and 431 are preferred over stat 311.<br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Numerical Analysis: [http://www.math.wisc.edu/514-numerical-analysis Math 514].<br />
<br />
'''Also:'''<br />
Consider the program in [http://www.engr.wisc.edu/isye/isye-academics-undergraduate-program.html Industrial Engineering] offered by the College of Engineering.<br />
<br />
=== Materials Science ===<br />
The following program details an option 2 package for students in the College of Engineering program in Materials Science and Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Materials Courses: [http://www.engr.wisc.edu/cmsdocuments/Degree_requirements_2014.pdf MSE 330, 331, and 351]<br />
* One Engineering Elective: CBE 255, CS 300, CS 302, CS 310, ECE 230, ECE 376, EMA 303, Phys 321, Stat 424].<br />
** All of the above classes may be used to satisfy the program requirements for MS&E BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Mechanical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Mechanical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Mechanical Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/me-flowchart-spring-2014.pdf ME 340, 361, 363, 364]<br />
** All of the above courses are required by the Mechanical Engineering program.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Nuclear Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Nuclear Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Nuclear Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/NE-UGguide2014.pdf NE 305, 405, and 408]<br />
* One Engineering Elective: Physics 321 or 322, ECE 376, BME 501, or NE 411.<br />
** All of the above classes may be used to satisfy the program requirements for the Nuclear Engineering BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
* Additional courses above the 500 level listed above.<br />
<br />
== Computer Science ==<br />
Computer science as an independent discipline is rather young: The first computer science degree program offered in the United States was formed in 1962 (at Purdue University). Despite its youth, one could argue that no single academic discipline has had more of an effect on human society since the scientific revolution.<br />
<br />
Since computer science is foremost concerned with the theory of computation, its link with mathematics is robust. Historical examples include Alan Turing, A mathematician and WWII cryptoanalyst who's theory of the Universal Turing Machine forms the central framework of modern computation; and John Von Neumann, A mathematician who's name is ascribed to the architecture still used for nearly all computers today.[https://web.archive.org/web/20130314123032/http://qss.stanford.edu/~godfrey/vonNeumann/vnedvac.pdf] There are broad overlaps in reasearch in the two fields. For example, one of the most famous unsolved problems in mathematics, the [http://www.claymath.org/millenium-problems/p-vs-np-problem P vs NP] problem, is also considered an open problem in the theory of computation.<br />
<br />
Since computer science is a full field enveloping philosophy, mathematics, and engineering there are many possible areas of interest which a student of mathematics and computer science might focus on. Below are several examples.<br />
<br />
=== Computational Methods ===<br />
Computational methods are the algorithms a computer follows in order to perform a specific task. Of interest besides the algorithms is methods for evaluating their quality and efficiency. Since computational mathematics is on the interface between pure and applied methods students who concentrate in this area can find many exciting research opportunities available at the undergraduate level. <br />
<br />
The mathematical coursework focuses on combinatorics, analysis, and numerical methods. <br />
<br />
'''Application Courses'''<br />
* Any four courses from: CS 352, 367, 400, 412, 435, 475, 513, 514, 515, 520, 525, 533, 540, 545, 558, 559, and 577.<br />
** Note that 435, 475, 513, 514, 515, and 525 are crosslisted with math. They may not be used as both application courses and core mathematics courses<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Advanced Calculus [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] and/or [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics above.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Analysis II: [http://www.math.wisc.edu/522-advanced-calculus Math 522].<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542].<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567].<br />
* Logic: [http://www.math.wisc.edu/571-mathematical-logic Math 571].<br />
<br />
'''Also:'''<br />
Consider the program in the [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science].<br />
<br />
=== Theoretical Computer Science ===<br />
If you are interested in a more theoretical bend to your studies, follow the program above but with the following changes:<br />
* Include both CS 520 and CS 577 into your core applied courses.<br />
* Replace the two computational methods courses with Math 567 and Math 571.<br />
<br />
=== Cryptography ===<br />
Due to the widespread use of computer storage, platforms, and devices; security is now of singular interest. Students with expertise in the mathematics associated with cryptography can find interesting opportunities after graduation in public and private security sectors.<br />
<br />
The mathematics associated to secure messaging and cryptography is typically centered on combinatorics and number theory.<br />
<br />
'''Application Courses'''<br />
* Programming: CS 300 and CS 400 (or CS 302 and 367).<br />
* One of the following two pairs:<br />
** The CS track: Operating systems (CS 537) and Security (CS 642)<br />
** The ECE track: Digital Systems: (ECE 352) and Error Correcting Codes (ECE 641).<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Cryptography: [http://www.math.wisc.edu/435-introduction-cryptography Math 435]<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567]<br />
<br />
'''Additional Courses to Consider'''<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Also:'''<br />
Consider combining the programs offered by [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science] or [http://www.engr.wisc.edu/ece/ece-academics-undergraduate-program.html Computer Engineering].<br />
<br />
== Secondary Education ==<br />
The so called STEM fields continue to be a major area of interest and investment for education policy makers. In particular secondary education instructors with strong mathematics backgrounds are in demand across the nation in public, private, and charter school environments. <br />
<br />
The following program was designed for a math major who is interested in becoming an educator at the secondary level. Note that successful completion of the coursework outlined below would make a strong candidate for graduate work in mathematics and education at the masters level. Our own School of Education offers a [http://www.uwteach.com/mathematics.html Masters Degree in Secondary Mathematics] which concludes with state certification. <br />
<br />
''Note that a major requires at least two courses at the 500 level. Therefore you should consider the suggestions below carefully.''<br />
<br />
'''Application Courses'''<br />
* History and philosophy of mathematics: [http://www.math.wisc.edu/473-history-mathematics Math 473].<br />
* Math education capstone course: [http://www.math.wisc.edu/371-basic-concepts-mathematics Math 471]<br />
* Two additional courses from Mathematics, Computer Science, Physics, or Economics at the Intermediate or Advanced Level.<br />
** Suggested: CS 300, CS 302, Phys 207, Math 421, Math 475, Math 561, Math 567<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 (or Math 375) suggested.<br />
* College Geometry: [http://www.math.wisc.edu/461-college-geometry-i Math 461]. <br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] <br />
** Math 431 and 309 are equivalent. <br />
** [http://www.math.wisc.edu/531-probability-theory Math 531] can also be considered. This is a proof based introduction to probability and may be taken only after Math 421 or Math 521.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310] (Math 310 has a prerequisite of Math 309 or 431.)<br />
* Modern Algebra: [http://www.math.wisc.edu/441-introduction-modern-algebra Math 441] or [http://www.math.wisc.edu/541-modern-algebra 541].<br />
* Analysis: [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or [http://www.math.wisc.edu/521-advanced-calculus 521].<br />
** Math 521 is strongly suggested for students planning to teach AP Calculus in high school<br />
<br />
'''Additional Courses to Consider'''<br />
* Math 421 can be a useful course to take before the 500 level coursework.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Additional courses at the 500 level in mathematics.<br />
* Courses in computer programming, statistics, physics, economics, and finance can broaden your content areas and qualify you for more subjects.<br />
<br />
== Statistics ==<br />
Statistics is the study of the collection, measuring, and evaluation of data. Recent advances in our ability to collect and parse large amounts of data has made the field more exciting then ever before. Positions in data analysis are becoming common outside of laboratory environments: marketing, education, health, sports, infrastructure, politics, etc.<br />
<br />
Statistics has a strong relationship with mathematics. The areas of mathematics of particular interest are linear algebra, probability, and analysis.<br />
<br />
'''Application Courses'''<br />
* Core Statistics: Stat 333 and Stat 424<br />
* Statistics Electives: At least two from: Stat 349, 351, 411, 421, 456, 471, 609, or 610.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Mathematical Statistics Sequence: [http://www.math.wisc.edu/node/111 Math 309] and [http://www.math.wisc.edu/node/114 Math 310]<br />
** Math 431 may be used for Math 309.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522], [http://www.math.wisc.edu/621-analysis-iii-0 621], or [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Advanced Probability Theory [http://www.math.wisc.edu/531-probability-theory Math 531].<br />
* Algebra: [https://www.math.wisc.edu/541-modern-algebra Math 541].<br />
<br />
'''Also:'''<br />
A student who wishes to complete a major in statistics offered by the [https://www.stat.wisc.edu/undergrad/undergraduate-major-statistics Department of Statistics] should complete the program above and include:<br />
* Stat 302 and 327.<br />
* A course in programming (e.g. CS 300).<br />
* At least one more course from the statistics electives above.</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Option_2_packages&diff=21199Option 2 packages2021-05-09T16:43:10Z<p>Valko: </p>
<hr />
<div>'''NOTE: in the Fall 2020 semester the Department of Mathematics introduced five new named options (see [https://guide.wisc.edu/undergraduate/letters-science/mathematics/mathematics-bs/#requirementstext the Guide page of the major]. The old "Option 2" math major is not available to students anymore. Those who declared the Option 2 math major before Fall 2020 may finish it with the original rules, or they may switch to one of the new named options. <br />
<br />
'''<br />
<br />
<br />
<br />
<br />
The '''Option 2 math major''' requires six math courses and four courses in an area of focus. These four courses are required to have a certain mathematical content. The selection of the four courses, together with the six required math courses must be approved by the student's advisor. This page lists some sample course collections in several popular areas.<br />
<br />
NOTES: <br />
<br />
1) '''These course collections do not include course prerequisites.''' For example, math 310 has stats 302 as a prerequisite. But stat 302 cannot be used as a focus or major course.<br />
<br />
2) '''Courses offered by departments/schools/colleges outside of mathematics may have restricted enrollments.''' For example, an L&S student interested in an option 2 program with finance emphasis may not reliably be able to enroll in fin 300 since it is taught by Business.<br />
<br />
== Economics and Business ==<br />
<br />
=== Actuarial Mathematics ===<br />
Actuaries use techniques in mathematics and statistics to evaluate risk in a variety of areas including insurance, finance, healthcare, and even criminal justice. In recent history the field has been revolutionized by advances in the theory of probability and the ability to access, store, and process very large data sets.<br />
<br />
Professional actuaries are currently in demand, have lucrative pay, and is a growth field [http://www.bls.gov/ooh/math/actuaries.htm]. Similar to some other fields (law, accounting, etc.) there are professional organizations which administer a series of examinations [http://www.beanactuary.org/exams/]. Oftentimes students complete some of these examinations before graduating which allows them to move right into a career (Note: these exams are not required for graduation).<br />
<br />
Students who are interested in actuarial mathematics should consider coursework in probability, statistics, analysis, as well as computational mathematics.<br />
<br />
'''Application Courses'''<br />
* Act Sci 303<br />
<br />
* Act. Sci 650 and 652<br />
<br />
* Act. Sci. 651 or 653<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. <br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
** Has the prerequisite: one of the probability courses mentioned above AND an elementary stats class (Stat 302 is recommended).<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Linear Programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
'''Additional Courses to Consider'''<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
<br />
'''Also:''' Students interested in the areas of mathematics with applications to actuarial science might consider the following as well:<br />
<br />
* Advanced courses offered by the [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Department of Statistics].<br />
<br />
* A [http://bus.wisc.edu/knowledge-expertise/academic-departments/actuarial-science-risk-management-insurance program] offered by the UW-Madison School of Business.<br />
<br />
=== Business ===<br />
Applications of mathematics to business is often referred to as Operations Research or Management Science. Specifically, the goal is to use mathematics to make the best decisions in a variety of areas: searching, routing, scheduling, transport, etc.<br />
<br />
The modern version of the field grew out of the work mathematicians did in order to aid the Allied war effort during world war II.[http://www.history.army.mil/html/books/hist_op_research/CMH_70-102-1.pdf] Since then, the field has grown into a robust and active area of research and scholarship including several journals and professional organizations.[http://www.informs.org/]<br />
<br />
Students interested in applications of mathematics to business can find many employment opportunities in private corporations, government agencies, nonprofit enterprises, and more. Students can also move onto postgraduate programs in mathematics or business.<br />
<br />
'''Application Courses'''<br />
* Linear programming and Optimization: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Note that this course cannot also be used as a core math course.<br />
* Operations Research: OTM 410<br />
* At least two from the following: Gen Bus 306, Gen Bus 307; OTM 451, 411, 633, 654<br />
** Note that OTM 633 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/111 Math/Stat 310]<br />
<br />
* Computational Mathematics: [https://www.math.wisc.edu/514-numerical-analysis Math 514] or [http://www.math.wisc.edu/513-numerical-linear-algebra 513]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastics: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics.<br />
* [http://www.math.wisc.edu/633-queueing-theory-and-stochastic-modeling Math 633].<br />
<br />
'''Also:''' Consider a program in the UW-Madison [http://bus.wisc.edu/bba/academics-and-programs/majors/operations-technology-management School of Business].<br />
<br />
=== Economics ===<br />
Economics is perhaps the most mathematical of the social sciences. Specifically economists wish to model and understand the behavior of individuals (people, countries, animals, etc.). Typically this is done by quantifying some elements of interest to the individuals.<br />
<br />
Due to the quantitative nature of the field, economic theory has begun to move from the classic areas of markets, products, supply, demand, etc. and into many seemingly unrelated areas: law, psychology, political science, biology, and more.[http://en.wikipedia.org/wiki/Economics_imperialism]<br />
<br />
Regardless, the backbone of economics and economic theory is mathematics. The classical area of mathematics most often related with economics is analysis. <br />
<br />
'''Application Courses'''<br />
* Microeconomics: Econ 301 or 311.<br />
* Macroeconomics: Econ 302 or 312.<br />
* Economic Electives: At least two courses from Econ 410, 460, 475, 503, 521, 525, and 666; Math 310 and Math 415.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Linear programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in the [http://www.econ.wisc.edu/undergrad/Reqs%20for%20Major.html Department of Economics].<br />
<br />
=== Finance ===<br />
Financial mathematics is more popular than ever with financial firms hiring "quants" from all areas of mathematics and the natural sciences. Financial markets are of interest to mathematicians due to the difficult nature of modeling the complex systems. The standard tools involved are evolutionary differential equations, measure theory, and stochastic calculus.<br />
<br />
'''Application Courses'''<br />
* Statistics: Econ 410 or Math/Stat 310.<br />
* Finance core: Finance 300, 320, 330.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Mat 619].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522] and [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Linear Programming (optimization): [http://www.math.wisc.edu/525-linear-programming-methods Math 525].<br />
<br />
'''Also:'''<br />
Consider a program in [http://bus.wisc.edu/bba/academics-and-programs/majors/finance Finance] at the the Wisconsin School of Business.<br />
<br />
== Physical Sciences ==<br />
The physical sciences and mathematics have grown hand-in-hand since antiquity.<br />
Students with strong backgrounds in mathematics who are also interested in a branch of the physical sciences can find opportunities in laboratories, engineering firms, education, finance, law, business, and medicine. Those with very strong academic records can find themselves as preferred candidates for graduate study in their choice of field.<br />
<br />
The following sample programs in mathematics have strong relationships with a particular area of interest in the natural sciences.<br />
<br />
=== Atmospheric & Oceanic Sciences ===<br />
Weather and climate is determined by the interaction between two thin layers which cover the planet: The oceans and the atmosphere. Understanding how these two fluids act and interact allow humans to describe historical climate trends, forecast near future weather with incredible accuracy, and hopefully describe long term climate change which will affect the future of human society.<br />
<br />
A student interested in atmospheric and oceanic studies who has a strong mathematics background can find a career working in local, national, and international meteorological laboratories. These include private scientific consulting businesses as well as public enterprises. Students interested in graduate study could find their future studies supported by the National Science Foundation, the Department of Energy, NASA, or others [http://www.nsf.gov/funding/]. There is a large amount of funding available in the area due to the relevance research findings have on energy and economic policy.<br />
<br />
Mathematicians who work in Atmospheric and oceanic studies are drawn to the complexities of the problems and the variety of methods in both pure and applied mathematics which can be brought to bear on them. Students should take coursework in methods of applied mathematics, differential equations, computational mathematics, and differential geometry and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* ATM OCN 310, 311, and 330 [http://www.aos.wisc.edu/education/Syllabus/courses_majors.html]<br />
** 310 and 330 have Physics 208/248 as a prerequisite.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322]<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619]<br />
<br />
'''Also:''' Students who are interested in this area might consider <br />
* A program offered by the [http://www.aos.wisc.edu/education/undergrad_program.htm Department of Atmospheric and Oceanic Sciences].<br />
* The [http://www.math.wisc.edu/amep AMEP] program.<br />
<br />
=== Chemistry ===<br />
The applications of mathematics to chemistry range from the mundane: Ratios for chemical reactants; to the esoteric: Computational methods in quantum chemistry. Research in this latter topic lead to a Nobel Prize in Chemistry to mathematician [http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/index.html John Pople].<br />
<br />
All areas of pure and applied mathematics have applications in modern chemistry. The most accessible track features coursework focusing on applied analysis and computational math. Students with a strong interest in theoretical mathematics should also consider modern algebra (for group theory) and topology.<br />
<br />
'''Application Courses'''<br />
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]<br />
** Both of these classes have prerequisites (Physics 207/247).<br />
* Analytical Chemistry: Chem 327 or Chem 329[http://www.chem.wisc.edu/content/courses]<br />
** Prerequisite: Chem 104 or 109 <br />
* Physical Chemistry: Chem 561 and 562<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 320 recommended.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
** Math 513 or 514 suggested.<br />
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Several higher level courses have connections to theoretical chemistry: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
** Any of these courses are acceptable in lieu of the 500 level courses above.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.chem.wisc.edu/content/undergraduate Department of Chemistry].<br />
<br />
=== Physics ===<br />
Perhaps the subject with the strongest historical ties with mathematics is physics. Certainly some of the great physical theories have been based on novel applications of mathematical theory or the invention of new subjects in the field: Newtonian mechanics and calculus, relativity and Riemannian geometry, quantum theory and functional analysis, etc.<br />
<br />
Nearly all mathematics courses offered here at UW Madison will have some applications to physics. The following is a collection of courses which would support general interest in physics.<br />
<br />
'''Application Courses'''<br />
* Mechanics, Electricity, and Magnetism: [http://www.physics.wisc.edu/academics/undergrads/inter-adv-311 Physics 311] and [http://www.physics.wisc.edu/academics/undergrads/inter-adv-322 Physics 322]<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 421 is suggested to prepare students for math 521.<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. See suggested courses below.<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* ODEs: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
* PDEs: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619].<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541].<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
* Differential Geometry [http://www.math.wisc.edu/561-differential-geometry Math 561].<br />
* Complex Analysis: [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/514-numerical-analysis 514].<br />
<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
=== Astronomy ===<br />
The Astronomy package has the same mathematics core, but different suggested application courses:<br />
<br />
'''Application Courses'''<br />
* Astronomy core: Choose two courses from Astron 310, 320, or 335.<br />
* Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* At least one more 500 level course. Suggested courses are: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses above the 500 level.<br />
<br />
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].<br />
<br />
== Biological Sciences ==<br />
Applications of mathematics to biology has undergone a recent boom. Historically, the biologist was perhaps most interested in applications of calculus, but now nearly any modern area of mathematical research has an application to some biological field[http://www.ams.org/notices/199509/hoppensteadt.pdf]. The following lists some possibilities.<br />
<br />
=== Bio-Informatics ===<br />
Bioinformatics is the application of computational methods to understand biological information. Of course the most interesting items of biological information is genetic and genomic information. Considering that the human genome has over three billion basepairs [http://www.genome.gov/12011238], it's no wonder that many mathematicians find compelling problems in the area to devote their time.<br />
<br />
Students with strong mathematical backgrounds who are interested in bioinformatics can find careers as a part of research teams in public and private laboratories across the world [http://www.bioinformatics.org/jobs/]. Moreover, many universities have established interdisciplinary graduate programs promoting this intersection of mathematics, biology, and computer science [http://ils.unc.edu/informatics_programs/doc/Bioinformatics_2006.html].<br />
<br />
Students interested in bioinformatics should have a strong background in computational mathematics and probability. Students should also have a strong programming background.<br />
<br />
'''Application Courses'''<br />
* Computer Science: CS 300 and CS 400 (or CS 302 and CS 367).<br />
* Bioinformatics: [http://www.biostat.wisc.edu/content/bmi-576-introduction-bioinformatics BMI/CS 576]<br />
* Genetics: Gen 466<br />
** Note that this class has a prerequisite of a year of chemistry and a year of biology coursework. Please contact the UW-Madison [http://www.genetics.wisc.edu/UndergraduateProgram.htm genetics] program for more information.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Computational Mathematics: At least three of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
<br />
'''Also''' <br />
* Consider a program in [http://www.cs.wisc.edu/academics/Undergraduate-Programs Computer Science] or [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
* Complete this major with a few additional courses if you are interested in medical school [http://prehealth.wisc.edu/explore/documents/Pre-Med.pdf].<br />
<br />
=== Bio-Statistics ===<br />
Biostatistics is the application of mathematical statistical methods to areas of biology. Historically, one could consider the field to have been founded by Gregor Mendel himself. He used basic principles of statistics and probability to offer a theory for which genetic traits would arise from cross hybridization of plants and animals. His work was forgotten for nearly fifty years before it was rediscovered and become an integral part of modern genetic theory.<br />
<br />
Beyond applications to genetics, applications of biostatistics range from public health policy to evaluating laboratory experimental results to tracking population dynamics in the field. Currently, health organizations consider there to be a shortage of trained biostatisticians[http://www.amstat.org/careers/biostatistics.cfm]. Students interested in this area should find excellent job prospects.<br />
<br />
Students interested in biostatistics should have strong backgrounds in probability, statistics, and computational methods.<br />
<br />
'''Application Courses'''<br />
* Statistics: Any four from Stat 333, 424, 575, 641, and 642 [http://www.stat.wisc.edu/course-listing]<br />
** Stat 333 has as a prerequisite some experience with statistical software. This can be achieved by also registering for Stat 327. Stat 327 is a single credit course which does not count for the mathematics major.<br />
<br />
'''Core Mathematics Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* More courses in computational mathematics listed above.<br />
* [http://www.math.wisc.edu/635-introduction-brownian-motion-and-stochastic-calculus Math 635]<br />
<br />
'''Also'''<br />
* Consider a program with [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Statistics] or in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences].<br />
* Compare this major program to requirements for Medical School.<br />
<br />
=== Ecology, Forestry, Wildlife Ecology ===<br />
Applications of advanced mathematics to ecology has resulted in science's improved ability to track wild animal populations, predict the spread of diseases, model the impact of humans on native wildlife, control invasive species, and more. Modeling in this area is mathematically interesting due to the variety of scales and the inherent difficulty of doing science outside of a laboratory! As such the methods of deterministic and stochastic models are particularly useful.<br />
<br />
'''Application Courses'''<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Computational Methods: [http://www.cs.wisc.edu/courses/412 CS 412].<br />
* Any two courses from [http://zoology.wisc.edu/courses/courselist.htm Zoo 460, 504, and 540]; or [http://forestandwildlifeecology.wisc.edu/undergraduate-study-courses F&W Ecol 300, 410, 460, 531, 652, and 655].<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Stochastic Processes: Either [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
=== Genetics ===<br />
Applications of mathematics in genetics appear on a wide range of scales: chemical processes, cellular processes, organism breeding, and speciation. For applications of mathematics in genetics on the scale of chemical processes you might want to examine the suggested packages for bioinformatics or structural biology. If instead you are interested in the larger scale of organisms you might consider the package in biostatistics or the one below:<br />
<br />
'''Application Courses'''<br />
* Any four courses chosen from: GEN 466, 564, 565, 626, 629, and BMI 563.[http://www.genetics.wisc.edu/UndergraduateProgram.htm]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended for non-honors students.<br />
<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
* Consider a program in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences] such as [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].<br />
<br />
=== Structural Biology ===<br />
Structural biologists are primarily interested in the large molecules which are involved in cellular processes: the fundamental chemical building blocks of life. The field lies on the intersection of biology, physics, chemistry, and mathematics and so structural biology is an exciting area of interdisciplinary research.<br />
<br />
In general, the mathematics involved in structural biology is focused on computational methods, probability, and statistics. Note that we offer a specialized course in Mathematics Methods in Structural Biology - Math 606.<br />
<br />
'''Application Courses'''<br />
* Analytical Methods in Chemistry: Chem 327 or 329<br />
* Physical Chemistry: Chem 561 and 562<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/mathematical-methods-structural-biology Math 606]<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [https://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]<br />
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
=== Systems Biology ===<br />
Systems biology is the computational and mathematical modeling of biological systems at any scale. The classical example of this may be the [http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation predator-prey] model of differential equations which describe the relative population dynamics of two species. Other systems examples include disease transmission, chemical pathways, cellular processes, and more.<br />
<br />
In general, the mathematics involved in systems biology is focused on computational methods, dynamical systems, differential equations, the mathematics of networks, control theory, and others. Note that we offer a specialized course in Mathematical Methods in Systems Biology - Math 609.<br />
<br />
'''Application Courses'''<br />
* Organic Chemistry: Chem 341 or 343<br />
* Introductory Biochemistry: Biochem 501<br />
* Mathematical Methods in Structural Biology: [http://www.math.wisc.edu/609-mathematical-methods-systems-biology Math 609]<br />
* One Biochem elective: Any Biochem class numbered above 600. Suggested courses are Biochem 601, 612, 620, 621, 624, and 630.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].<br />
<br />
'''Additional Courses to Consider'''<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Additional courses in computational mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Also:'''<br />
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].<br />
<br />
== Engineering ==<br />
Engineering is the application of science and mathematics to the invention, improvement, and maintenance of anything and everything. As with many of the sciences, engineers and mathematicians have a symbiotic relationship: Engineers use mathematics to make new things; the new things exhibit novel properties that are mathematically interesting.<br />
<br />
In general all of mathematics can be applied to some field of engineering. However the programs offered below are not substitutes for engineering degrees. That is, student who are interested in an engineering career upon completion of their undergraduate degree should probably enroll in one of the engineering programs offered by the [http://www.engr.wisc.edu/current/undergrad.html College of Engineering]. Similarly, students who are primarily interested in mathematics might instead choose an option I major and concentrate their upper level coursework in applied mathematics. Students who are truly interested in both areas should consider the degree program in [http://www.math.wisc.edu/amep Applied Mathematics, Engineering, and Physics].<br />
<br />
So who do the programs below serve? They serve engineering students who wish to take a second major in mathematics. In general such students are excellent candidates for graduate study in engineering.<br />
<br />
=== Chemical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Chemical Engineering who are interested in pursuing a second major in mathematics.<br />
<br />
'''Application Courses'''<br />
* [http://www.engr.wisc.edu/cmsdocuments/cbe-undergrade-handbook-2009-v7.pdf CBE 320, 326, 426, 470]<br />
** Note: All of these course are required for the undergraduate program in chemical engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), complex analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]), and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
<br />
=== Civil Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Civil and Environmental Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Fluid Mechanics and Structural Analysis: [http://courses.engr.wisc.edu/cee/ CIV ENG 310, 311, 340]<br />
** Note: All of these course are required for the undergraduate program in civil engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective Structural Analysis Course: CIV ENG 440, 442, 445, or 447.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]); and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], and [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
<br />
=== Electrical and Computer Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Computer and Electrical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core ECE: [http://courses.engr.wisc.edu/ece/ ECE 220, 230, 352]<br />
** Note: All of these course are required for the undergraduate program in electrical and computer engineering.<br />
** Several of these courses have as a prerequisite other engineering and science courses.<br />
* One elective: ECE 435, 525, or 533.<br />
** Each of these courses may be used as an elective in the undergraduate program in civil engineering.<br />
** ECE 435 is crosslisted with math. It cannot be used as both an application course AND a core math course.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* At least two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), linear programming [http://www.math.wisc.edu/525-linear-programming-methods Math 525], modern algebra [http://www.math.wisc.edu/541-modern-algebra Math 541], differential geometry [http://www.math.wisc.edu/561-differential-geometry Math 561], and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses above the 500 level listed above.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]<br />
* Error correcting codes: [http://www.math.wisc.edu/641-introduction-error-correcting-codes Math 641]<br />
<br />
===Engineering Mechanics and Astronautics===<br />
The following program details an option 2 package for students in the College of Engineering program in Engineering Mechanics and Astronautics who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Engineering Mechanics: [http://courses.engr.wisc.edu/ema/ EMA 201, 202, 303]<br />
* One elective: EMA 521, 542, 545, or 563<br />
** All of the above courses may be used to satisfy the EMA program requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525]), and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
<br />
=== Industrial Engineering ===<br />
Industrial engineering is the application of engineering principles to create the most effective means of production. In particular, they work to optimize complex systems.<br />
<br />
'''Application Courses[http://www.engr.wisc.edu/isye/isye-curriculum-documents.html]'''<br />
* Core Industrial engineering: I SY E 315, 320, and 323.<br />
* Industrial Engineering Elective: At least one of I SY E 425, 516, 525, 526, 558, 575, 615, 620, 624, 635, or 643.<br />
** Note: ISYE 425 and 525 are both crosslisted with math and cannot be used to complete both the application and core math requirements.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 recommended.<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Math 376 is an honors course.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Both 309 and 431 are preferred over stat 311.<br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Numerical Analysis: [http://www.math.wisc.edu/514-numerical-analysis Math 514].<br />
<br />
'''Also:'''<br />
Consider the program in [http://www.engr.wisc.edu/isye/isye-academics-undergraduate-program.html Industrial Engineering] offered by the College of Engineering.<br />
<br />
=== Materials Science ===<br />
The following program details an option 2 package for students in the College of Engineering program in Materials Science and Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Materials Courses: [http://www.engr.wisc.edu/cmsdocuments/Degree_requirements_2014.pdf MSE 330, 331, and 351]<br />
* One Engineering Elective: CBE 255, CS 300, CS 302, CS 310, ECE 230, ECE 376, EMA 303, Phys 321, Stat 424].<br />
** All of the above classes may be used to satisfy the program requirements for MS&E BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Mechanical Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Mechanical Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Mechanical Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/me-flowchart-spring-2014.pdf ME 340, 361, 363, 364]<br />
** All of the above courses are required by the Mechanical Engineering program.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525])<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]<br />
* Additional courses above the 500 level listed above.<br />
<br />
=== Nuclear Engineering ===<br />
The following program details an option 2 package for students in the College of Engineering program in Nuclear Engineering who are interested in pursuing a second major in mathematics. <br />
<br />
'''Application Courses'''<br />
* Core Nuclear Engineering Courses: [http://www.engr.wisc.edu/cmsdocuments/NE-UGguide2014.pdf NE 305, 405, and 408]<br />
* One Engineering Elective: Physics 321 or 322, ECE 376, BME 501, or NE 411.<br />
** All of the above classes may be used to satisfy the program requirements for the Nuclear Engineering BS degree.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]<br />
** Students who take math 320 should instead consider an additional course below.<br />
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322]<br />
* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]).<br />
<br />
'''Additional Courses to Consider'''<br />
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
* Additional courses above the 500 level listed above.<br />
<br />
== Computer Science ==<br />
Computer science as an independent discipline is rather young: The first computer science degree program offered in the United States was formed in 1962 (at Purdue University). Despite its youth, one could argue that no single academic discipline has had more of an effect on human society since the scientific revolution.<br />
<br />
Since computer science is foremost concerned with the theory of computation, its link with mathematics is robust. Historical examples include Alan Turing, A mathematician and WWII cryptoanalyst who's theory of the Universal Turing Machine forms the central framework of modern computation; and John Von Neumann, A mathematician who's name is ascribed to the architecture still used for nearly all computers today.[https://web.archive.org/web/20130314123032/http://qss.stanford.edu/~godfrey/vonNeumann/vnedvac.pdf] There are broad overlaps in reasearch in the two fields. For example, one of the most famous unsolved problems in mathematics, the [http://www.claymath.org/millenium-problems/p-vs-np-problem P vs NP] problem, is also considered an open problem in the theory of computation.<br />
<br />
Since computer science is a full field enveloping philosophy, mathematics, and engineering there are many possible areas of interest which a student of mathematics and computer science might focus on. Below are several examples.<br />
<br />
=== Computational Methods ===<br />
Computational methods are the algorithms a computer follows in order to perform a specific task. Of interest besides the algorithms is methods for evaluating their quality and efficiency. Since computational mathematics is on the interface between pure and applied methods students who concentrate in this area can find many exciting research opportunities available at the undergraduate level. <br />
<br />
The mathematical coursework focuses on combinatorics, analysis, and numerical methods. <br />
<br />
'''Application Courses'''<br />
* Any four courses from: CS 352, 367, 400, 412, 435, 475, 513, 514, 515, 520, 525, 533, 540, 545, 558, 559, and 577.<br />
** Note that 435, 475, 513, 514, 515, and 525 are crosslisted with math. They may not be used as both application courses and core mathematics courses<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Advanced Calculus [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] and/or [http://www.math.wisc.edu/521-advanced-calculus Math 521]<br />
* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
<br />
'''Additional Courses to Consider'''<br />
* Additional courses in computational mathematics above.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].<br />
* Analysis II: [http://www.math.wisc.edu/522-advanced-calculus Math 522].<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542].<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567].<br />
* Logic: [http://www.math.wisc.edu/571-mathematical-logic Math 571].<br />
<br />
'''Also:'''<br />
Consider the program in the [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science].<br />
<br />
=== Theoretical Computer Science ===<br />
If you are interested in a more theoretical bend to your studies, follow the program above but with the following changes:<br />
* Include both CS 520 and CS 577 into your core applied courses.<br />
* Replace the two computational methods courses with Math 567 and Math 571.<br />
<br />
=== Cryptography ===<br />
Due to the widespread use of computer storage, platforms, and devices; security is now of singular interest. Students with expertise in the mathematics associated with cryptography can find interesting opportunities after graduation in public and private security sectors.<br />
<br />
The mathematics associated to secure messaging and cryptography is typically centered on combinatorics and number theory.<br />
<br />
'''Application Courses'''<br />
* Programming: CS 300 and CS 400 (or CS 302 and 367).<br />
* One of the following two pairs:<br />
** The CS track: Operating systems (CS 537) and Security (CS 642)<br />
** The ECE track: Digital Systems: (ECE 352) and Error Correcting Codes (ECE 641).<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531]<br />
** Math 431 and 309 are equivalent. <br />
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.<br />
* Cryptography: [http://www.math.wisc.edu/435-introduction-cryptography Math 435]<br />
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]<br />
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567]<br />
<br />
'''Additional Courses to Consider'''<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Also:'''<br />
Consider combining the programs offered by [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science] or [http://www.engr.wisc.edu/ece/ece-academics-undergraduate-program.html Computer Engineering].<br />
<br />
== Secondary Education ==<br />
The so called STEM fields continue to be a major area of interest and investment for education policy makers. In particular secondary education instructors with strong mathematics backgrounds are in demand across the nation in public, private, and charter school environments. <br />
<br />
The following program was designed for a math major who is interested in becoming an educator at the secondary level. Note that successful completion of the coursework outlined below would make a strong candidate for graduate work in mathematics and education at the masters level. Our own School of Education offers a [http://www.uwteach.com/mathematics.html Masters Degree in Secondary Mathematics] which concludes with state certification. <br />
<br />
''Note that a major requires at least two courses at the 500 level. Therefore you should consider the suggestions below carefully.''<br />
<br />
'''Application Courses'''<br />
* History and philosophy of mathematics: [http://www.math.wisc.edu/473-history-mathematics Math 473].<br />
* Math education capstone course: [http://www.math.wisc.edu/371-basic-concepts-mathematics Math 471]<br />
* Two additional courses from Mathematics, Computer Science, Physics, or Economics at the Intermediate or Advanced Level.<br />
** Suggested: CS 300, CS 302, Phys 207, Math 421, Math 475, Math 561, Math 567<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 (or Math 375) suggested.<br />
* College Geometry: [http://www.math.wisc.edu/461-college-geometry-i Math 461]. <br />
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] <br />
** Math 431 and 309 are equivalent. <br />
** [http://www.math.wisc.edu/531-probability-theory Math 531] can also be considered. This is a proof based introduction to probability and may be taken only after Math 421 or Math 521.<br />
* Statistics: [http://www.math.wisc.edu/node/114 Math 310] (Math 310 has a prerequisite of Math 309 or 431.)<br />
* Modern Algebra: [http://www.math.wisc.edu/441-introduction-modern-algebra Math 441] or [http://www.math.wisc.edu/541-modern-algebra 541].<br />
* Analysis: [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or [http://www.math.wisc.edu/521-advanced-calculus 521].<br />
** Math 521 is strongly suggested for students planning to teach AP Calculus in high school<br />
<br />
'''Additional Courses to Consider'''<br />
* Math 421 can be a useful course to take before the 500 level coursework.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]<br />
* Additional courses at the 500 level in mathematics.<br />
* Courses in computer programming, statistics, physics, economics, and finance can broaden your content areas and qualify you for more subjects.<br />
<br />
== Statistics ==<br />
Statistics is the study of the collection, measuring, and evaluation of data. Recent advances in our ability to collect and parse large amounts of data has made the field more exciting then ever before. Positions in data analysis are becoming common outside of laboratory environments: marketing, education, health, sports, infrastructure, politics, etc.<br />
<br />
Statistics has a strong relationship with mathematics. The areas of mathematics of particular interest are linear algebra, probability, and analysis.<br />
<br />
'''Application Courses'''<br />
* Core Statistics: Stat 333 and Stat 424<br />
* Statistics Electives: At least two from: Stat 349, 351, 411, 421, 456, 471, 609, or 610.<br />
<br />
'''Core Math Courses'''<br />
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course<br />
** Math 341 suggested.<br />
* Mathematical Statistics Sequence: [http://www.math.wisc.edu/node/111 Math 309] and [http://www.math.wisc.edu/node/114 Math 310]<br />
** Math 431 may be used for Math 309.<br />
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].<br />
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].<br />
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].<br />
<br />
'''Additional Courses to Consider'''<br />
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]<br />
* Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522], [http://www.math.wisc.edu/621-analysis-iii-0 621], or [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].<br />
* Advanced Probability Theory [http://www.math.wisc.edu/531-probability-theory Math 531].<br />
* Algebra: [https://www.math.wisc.edu/541-modern-algebra Math 541].<br />
<br />
'''Also:'''<br />
A student who wishes to complete a major in statistics offered by the [https://www.stat.wisc.edu/undergrad/undergraduate-major-statistics Department of Statistics] should complete the program above and include:<br />
* Stat 302 and 327.<br />
* A course in programming (e.g. CS 300).<br />
* At least one more course from the statistics electives above.</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=20963Probability Seminar2021-03-09T13:26:45Z<p>Valko: /* March 18, 2021, Theo Assiotis (Edinburgh) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2021 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
<b> IMPORTANT: </b> In Spring 2021 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
== January 28, 2021, no seminar ==<br />
<br />
== February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==<br />
<br />
'''Dynamic polymers: invariant measures and ordering by noise'''<br />
<br />
We develop a dynamical approach to infinite volume polymer measures (IVPM) in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow, and establish ordering by noise. We prove that, for a fixed asymptotic slope, the polymer dynamics has a unique invariant distribution given by a unique IVPM. Moreover, One Force-One Solution principle holds.<br />
<br />
== February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford) ==<br />
<br />
'''Non-stationary fluctuations for some non-integrable models'''<br />
<br />
We will discuss recent progress on weak KPZ universality and non-integrable particle systems, including long-range models and slow bond models. The approach is based on a preliminary step in a non-stationary (first-order) Boltzmann-Gibbs principle. We will also discuss the full non-stationary Boltzmann-Gibbs principle itself and pieces of its proof.<br />
<br />
== February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) ==<br />
<br />
'''Signature moments to characterize laws of stochastic processes'''<br />
<br />
The normalized sequence of moments characterizes the law of any finite-dimensional random variable. In this talk, I will describe an extension of this result to path-valued random variables, i.e. stochastic processes, by using the normalized sequence of signature moments. I will show how these moments define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, I will describe a non-parametric two-sample hypothesis test for laws of stochastic processes.<br />
<br />
== February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT) ==<br />
<br />
'''Random matrices, random groups, singular values, and symmetric functions'''<br />
<br />
Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.<br />
<br />
== March 4, 2021, [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==<br />
<br />
'''The Coleman correspondence at the free fermion point'''<br />
<br />
Two-dimensional statistical and quantum field theories are special in many ways. One striking instance of this is the equivalence of certain bosonic and fermionic fields, known as bosonization. <br />
I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions. <br />
I will then present a result that extends this correspondence to the non-Gaussian `massless' sine-Gordon field on $R^2$ at $\beta=4\pi$ and massive Dirac fermions.<br />
This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent. <br />
We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$. <br />
This is joint work with C. Webb (arXiv:2010.07096).<br />
<br />
== March 11, 2021, [https://people.math.rochester.edu/faculty/smkrtchy/ Sevak Mkrtchyan] (Rochester) ==<br />
<br />
'''The limit shape of the Leaky Abelian Sandpile Model'''<br />
<br />
The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion 1-1/d of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case d=1.<br />
<br />
We will discuss how, by connecting the model to a certain killed random walk on the square lattice, for any fixed d>1, an explicit limit shape can be computed for the region visited by the sandpile when it stabilizes.<br />
<br />
We will also discuss the limit shape in the regime when the dissipation parameter d converges to 1 as n grows, as this is related to the ordinary ASM with a modified initial configuration.<br />
<br />
== March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) ==<br />
<br />
'''On the joint moments of characteristic polynomials of random unitary matrices'''<br />
<br />
I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.<br />
<br />
== March 25, 2021, [https://homepages.uc.edu/~brycwz/ Wlodzimierz Bryc] (Cincinnati) ==<br />
'''Fluctuations of particle density for open ASEP'''<br />
<br />
I will review results on fluctuations of particle density for the open Asymmetric Simple Exclusion Process. I will explain the statements and the Laplace transform duality arguments that appear in the proofs.<br />
<br />
The talk is based on past and ongoing projects with Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.<br />
<br />
== April 1, 2021, [https://sites.google.com/view/xiangying-huangs-home-page/home Zoe Huang] (Duke University) ==<br />
<br />
<br />
== April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==<br />
<br />
== April 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics) ==<br />
<br />
== April 16, 2021, [http://www.mathjunge.com/ Matthew Junge] (CUNY) <span style="color:red">FRIDAY at 2:25pm, joint with</span> [https://www.math.wisc.edu/wiki/index.php/Applied/ACMS ACMS] ==<br />
<br />
== April 22, 2021, TBA ==<br />
<br />
== April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford) ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=20962Probability Seminar2021-03-09T13:26:24Z<p>Valko: /* March 18, 2021, Theo Assiotis (Edinburgh) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2021 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
<b> IMPORTANT: </b> In Spring 2021 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
== January 28, 2021, no seminar ==<br />
<br />
== February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==<br />
<br />
'''Dynamic polymers: invariant measures and ordering by noise'''<br />
<br />
We develop a dynamical approach to infinite volume polymer measures (IVPM) in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow, and establish ordering by noise. We prove that, for a fixed asymptotic slope, the polymer dynamics has a unique invariant distribution given by a unique IVPM. Moreover, One Force-One Solution principle holds.<br />
<br />
== February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford) ==<br />
<br />
'''Non-stationary fluctuations for some non-integrable models'''<br />
<br />
We will discuss recent progress on weak KPZ universality and non-integrable particle systems, including long-range models and slow bond models. The approach is based on a preliminary step in a non-stationary (first-order) Boltzmann-Gibbs principle. We will also discuss the full non-stationary Boltzmann-Gibbs principle itself and pieces of its proof.<br />
<br />
== February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) ==<br />
<br />
'''Signature moments to characterize laws of stochastic processes'''<br />
<br />
The normalized sequence of moments characterizes the law of any finite-dimensional random variable. In this talk, I will describe an extension of this result to path-valued random variables, i.e. stochastic processes, by using the normalized sequence of signature moments. I will show how these moments define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, I will describe a non-parametric two-sample hypothesis test for laws of stochastic processes.<br />
<br />
== February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT) ==<br />
<br />
'''Random matrices, random groups, singular values, and symmetric functions'''<br />
<br />
Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.<br />
<br />
== March 4, 2021, [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==<br />
<br />
'''The Coleman correspondence at the free fermion point'''<br />
<br />
Two-dimensional statistical and quantum field theories are special in many ways. One striking instance of this is the equivalence of certain bosonic and fermionic fields, known as bosonization. <br />
I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions. <br />
I will then present a result that extends this correspondence to the non-Gaussian `massless' sine-Gordon field on $R^2$ at $\beta=4\pi$ and massive Dirac fermions.<br />
This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent. <br />
We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$. <br />
This is joint work with C. Webb (arXiv:2010.07096).<br />
<br />
== March 11, 2021, [https://people.math.rochester.edu/faculty/smkrtchy/ Sevak Mkrtchyan] (Rochester) ==<br />
<br />
'''The limit shape of the Leaky Abelian Sandpile Model'''<br />
<br />
The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion 1-1/d of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case d=1.<br />
<br />
We will discuss how, by connecting the model to a certain killed random walk on the square lattice, for any fixed d>1, an explicit limit shape can be computed for the region visited by the sandpile when it stabilizes.<br />
<br />
We will also discuss the limit shape in the regime when the dissipation parameter d converges to 1 as n grows, as this is related to the ordinary ASM with a modified initial configuration.<br />
<br />
== March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) ==<br />
<br />
Title: On the joint moments of characteristic polynomials of random unitary matrices.<br />
<br />
Abstract: I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.<br />
<br />
== March 25, 2021, [https://homepages.uc.edu/~brycwz/ Wlodzimierz Bryc] (Cincinnati) ==<br />
'''Fluctuations of particle density for open ASEP'''<br />
<br />
I will review results on fluctuations of particle density for the open Asymmetric Simple Exclusion Process. I will explain the statements and the Laplace transform duality arguments that appear in the proofs.<br />
<br />
The talk is based on past and ongoing projects with Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.<br />
<br />
== April 1, 2021, [https://sites.google.com/view/xiangying-huangs-home-page/home Zoe Huang] (Duke University) ==<br />
<br />
<br />
== April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==<br />
<br />
== April 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics) ==<br />
<br />
== April 16, 2021, [http://www.mathjunge.com/ Matthew Junge] (CUNY) <span style="color:red">FRIDAY at 2:25pm, joint with</span> [https://www.math.wisc.edu/wiki/index.php/Applied/ACMS ACMS] ==<br />
<br />
== April 22, 2021, TBA ==<br />
<br />
== April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford) ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=20161Graduate student reading seminar2020-10-18T15:41:42Z<p>Valko: /* 2020 Fall */</p>
<hr />
<div>(... in probability)<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list] <br />
<br />
==2020 Fall==<br />
<br />
The graduate probability seminar will be on Zoom this semester. Please sign up for the email list if you would like to receive notifications about the talks.<br />
<br />
==2020 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/4, 2/11: Edwin<br />
<br />
2/18, 2/25: Chaojie<br />
<br />
3/3. 3/10: Yu Sun<br />
<br />
3/24, 3/31: Tony<br />
<br />
4/7, 4/14: Tung<br />
<br />
4/21, 4/28: Tung<br />
<br />
==2019 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Jane Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=20113Probability Seminar2020-10-12T18:43:04Z<p>Valko: /* October 22, 2020, Balint Virag (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2020 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
<b> IMPORTANT: </b> In Fall 2020 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==<br />
<br />
'''Pre-Talk: (1:00pm)'''<br />
<br />
'''Neural Networks for Probabilists''' <br />
<br />
Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.<br />
<br />
'''Talk: (2:30pm)'''<br />
<br />
'''Effective Theory of Deep Neural Networks''' <br />
<br />
Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.<br />
<br />
== September 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin) ==<br />
<br />
'''Some new perspectives on moments of random matrices'''<br />
<br />
The study of `moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.<br />
<br />
== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen] (UIC) ==<br />
<br />
'''Roots of random polynomials near the unit circle'''<br />
<br />
It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle. Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei. We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle. Based on joint work with Julian Sahasrabudhe.<br />
<br />
== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen] (Harvard) ==<br />
<br />
'''Large deviations for dense random graphs: beyond mean-field'''<br />
<br />
In a seminal paper, Chatterjee and Varadhan derived an Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.<br />
<br />
In this talk, we will explore large deviations for dense random graphs, beyond the “mean-field” setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model<br />
random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.<br />
<br />
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.<br />
<br />
== October 15, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==<br />
<br />
Title: '''Concentration in integrable polymer models'''<br />
<br />
I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the<br />
central moments of the partition function and the occupation lengths for each level of the polymer system. The method was developed<br />
for the O'Connell-Yor polymer, but was subsequently extended to discrete integrable polymers. As an application, we obtain<br />
localization of the OY polymer paths along a straight line on the scale O(n^{2/3+o(1)}). <br />
<br />
Joint work with Christian Noack.<br />
<br />
==October 22, 2020, [http://www.math.toronto.edu/balint/ Balint Virag] (Toronto) ==<br />
<br />
Title: '''The heat and the landscape'''<br />
<br />
Abstract: The directed landscape is the conjectured universal scaling limit of the<br />
most common random planar metrics. Examples are planar first passage<br />
percolation, directed last passage percolation, distances in percolation<br />
clusters, random polymer models, and exclusion processes. The limit laws of distances of objects are given by the KPZ fixed point.<br />
<br />
We show that the KPZ fixed point is characterized by the Baik Ben-Arous<br />
Peche statistics well-known from random matrix theory.<br />
<br />
This provides a general and elementary method for showing convergence to<br />
the KPZ fixed point. We apply this method to two models related to<br />
random heat flow: the O'Connell-Yor polymer and the KPZ equation.<br />
<br />
Note: there will be a follow-up talk with details about the proofs at 11am, Friday, October 23.<br />
<br />
==October 29, 2020, [https://www.math.wisc.edu/node/80 Yun Li] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 5, 2020, [http://sayan.web.unc.edu/ Sayan Banerjee] (UNC at Chapel Hill) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 12, 2020, [https://cims.nyu.edu/~ajd594/ Alexander Dunlap] (NYU Courant Institute) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 19, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== December 3, 2020, [https://www.math.wisc.edu/people/faculty-directory Tatyana Shcherbina] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== December 10, 2020, [https://www.ewbates.com/ Erik Bates] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=20112Probability Seminar2020-10-12T18:41:34Z<p>Valko: /* October 22, 2020, Balint Virag (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2020 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
<b> IMPORTANT: </b> In Fall 2020 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]<br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].<br />
<br />
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==<br />
<br />
'''Pre-Talk: (1:00pm)'''<br />
<br />
'''Neural Networks for Probabilists''' <br />
<br />
Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.<br />
<br />
'''Talk: (2:30pm)'''<br />
<br />
'''Effective Theory of Deep Neural Networks''' <br />
<br />
Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.<br />
<br />
== September 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin) ==<br />
<br />
'''Some new perspectives on moments of random matrices'''<br />
<br />
The study of `moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.<br />
<br />
== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen] (UIC) ==<br />
<br />
'''Roots of random polynomials near the unit circle'''<br />
<br />
It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle. Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei. We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle. Based on joint work with Julian Sahasrabudhe.<br />
<br />
== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen] (Harvard) ==<br />
<br />
'''Large deviations for dense random graphs: beyond mean-field'''<br />
<br />
In a seminal paper, Chatterjee and Varadhan derived an Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.<br />
<br />
In this talk, we will explore large deviations for dense random graphs, beyond the “mean-field” setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model<br />
random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.<br />
<br />
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.<br />
<br />
== October 15, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==<br />
<br />
Title: '''Concentration in integrable polymer models'''<br />
<br />
I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the<br />
central moments of the partition function and the occupation lengths for each level of the polymer system. The method was developed<br />
for the O'Connell-Yor polymer, but was subsequently extended to discrete integrable polymers. As an application, we obtain<br />
localization of the OY polymer paths along a straight line on the scale O(n^{2/3+o(1)}). <br />
<br />
Joint work with Christian Noack.<br />
<br />
==October 22, 2020, [http://www.math.toronto.edu/balint/ Balint Virag] (Toronto) ==<br />
<br />
Title: '''The heat and the landscape'''<br />
<br />
Abstract: The directed landscape is the conjectured universal scaling limit of the<br />
most common random planar metrics. Examples are planar first passage<br />
percolation, directed last passage percolation, distances in percolation<br />
clusters, random polymer models, and exclusion processes. The limit laws of distances of objects are given by the KPZ fixed point.<br />
<br />
We show that the KPZ fixed point is characterized by the Baik Ben-Arous<br />
Peche statistics well-known from random matrix theory.<br />
<br />
This provides a general and elementary method for showing convergence to<br />
the KPZ fixed point. We apply this method to two models related to<br />
random heat flow: the O'Connell-Yor polymer and the KPZ equation.<br />
<br />
==October 29, 2020, [https://www.math.wisc.edu/node/80 Yun Li] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 5, 2020, [http://sayan.web.unc.edu/ Sayan Banerjee] (UNC at Chapel Hill) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 12, 2020, [https://cims.nyu.edu/~ajd594/ Alexander Dunlap] (NYU Courant Institute) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== November 19, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== December 3, 2020, [https://www.math.wisc.edu/people/faculty-directory Tatyana Shcherbina] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
== December 10, 2020, [https://www.ewbates.com/ Erik Bates] (UW-Madison) ==<br />
<br />
Title: '''TBA'''<br />
<br />
Abstract: TBA<br />
<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=19627Probability group timetable2020-09-01T20:33:59Z<p>Valko: </p>
<hr />
<div>2020 Fall<br />
<br />
<br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| || || || || <br />
|- <br />
| 10-11|| || || || || <br />
|-<br />
| 11-12|| || || || ||<br />
|-<br />
| 12-1|| || || || || <br />
|-<br />
| 1-2|| || || || ||<br />
|-<br />
| 2-3|| || graduate probability seminar (2:25) || || probability seminar (2:25) || <br />
|-<br />
| 3-4|| || || || || <br />
|-<br />
| 4-5|| || || || || colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
<br />
<br />
<!-- <br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Timo 431, Kurt 222|| Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431, Kurt 222 || Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431<br />
|-<br />
| 10-11|| Kurt 222, Hans 234 || Phil out all day, Kurt 735 || Kurt 222, Hans 234 || Kurt 735 || Phil out all day, Hans 234 <br />
|-<br />
| 11-12|| Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Hans 846, Christian 846<br />
|-<br />
| 12-1|| Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431 <br />
|-<br />
| 1-2|| || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 || || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 ||<br />
|-<br />
| 2-3|| Daniele 431 (2:25) || graduate probability seminar (2:25) || Daniele 431 (2:25) || probability seminar (2:25) || Daniele 431 (2:25)<br />
|-<br />
| 3-4|| || Kurt 222, Hans 234 || || Kurt 222, Hans 234 || <br />
|-<br />
| 4-5|| || || || || colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
--><br />
<br />
<!--<br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Phil out all day || Benedek 531 (9:30)|| || Benedek 531 (9:30) || Phil out all day<br />
|-<br />
| 10-11||Jinsu 722, Louis 431 || || Jinsu 722, Louis 431|| ||Jinsu 722, Louis 431<br />
|-<br />
| 11-12|| || Hans 820 || || Hans 820 ||<br />
|-<br />
| 12-1|| Jinsu 222, Louis 632 || ||Jinsu 222, Louis 632 || || Jinsu 222, Louis 632<br />
|-<br />
| 1-2|| Jinsu 222, Hans 851 || Benedek OH, Hans 843 || Jinsu 222, Hans 851|| Hans 843 ||Jinsu 222, Hans 851<br />
|-<br />
| 2-3|| || graduate probability seminar (2:25) || Louis (Seb) || probability seminar (2:25) ||<br />
|-<br />
| 3-4|| ||Benedek (OH (3:30) || Benedek OH || || <br />
|-<br />
| 4-5|| || || Louis (OH 4:30)|| Louis (OH 4:30)|| colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
--></div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19626Probability2020-09-01T20:33:44Z<p>Valko: /* Graduate Courses in Probability */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem General email list]<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/lunchwithprobsemspeaker Email list for lunch/dinner with a speaker]<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list] <br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2020 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
Math/Stat 735 Stochastic Analysis<br />
<br />
Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
<br />
<br />
'''2021 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability: Integrable probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19625Probability2020-09-01T20:33:26Z<p>Valko: /* Graduate students */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem General email list]<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/lunchwithprobsemspeaker Email list for lunch/dinner with a speaker]<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
[https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/grad_prob_seminar Email list] <br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2020 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
Math/Stat 735 Stochastic Analysis<br />
<br />
Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
<br />
<br />
'''2021 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19520Probability2020-08-04T21:49:25Z<p>Valko: /* Graduate Courses in Probability */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2020 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
Math/Stat 735 Stochastic Analysis<br />
<br />
Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
<br />
<br />
'''2021 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19519Probability2020-08-04T21:49:16Z<p>Valko: /* Graduate Courses in Probability */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2020 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
Math/Stat 735 Stochastic Analysis<br />
Math 833 Topics in Probability: Modern Discrete Probability<br />
<br />
<br />
<br />
'''2021 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19518Probability2020-08-04T21:47:32Z<p>Valko: /* Postdocs */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19517Probability2020-08-04T21:47:22Z<p>Valko: /* Postdocs */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Erik Bates (Stanford, 2019)<br />
Scott Smith (Maryland, 2016)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19516Probability2020-08-04T21:46:43Z<p>Valko: /* Tenured and tenure-track faculty */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19515Probability2020-08-04T21:46:26Z<p>Valko: /* Tenured and tenure-track faculty */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/??? Tatyana Shcherbyna] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19514Probability2020-08-04T21:46:14Z<p>Valko: /* Tenured and tenure-track faculty */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[[http://www.math.wisc.edu/??? Tatyana Shcherbyna]] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=19513Probability2020-08-04T21:45:56Z<p>Valko: /* Tenured and tenure-track faculty */</p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
[http://www.math.wisc.edu/~vadicgor/ Vadim Gorin] (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[[???? Tatyana Shcherbyna]] (Kharkiv, 2012) mathematical physics, random matrices<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:30pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Transition_to_proof_courses&diff=18950Transition to proof courses2020-02-07T15:03:48Z<p>Valko: Created page with "Under construction Math 341 Math 375 Math 421 Math 467"</p>
<hr />
<div>Under construction<br />
<br />
<br />
Math 341<br />
Math 375<br />
Math 421<br />
Math 467</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=18747Graduate student reading seminar2020-01-22T22:57:27Z<p>Valko: /* 2020 Spring */</p>
<hr />
<div>(... in probability)<br />
<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
==2020 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/4, 2/11: Edwin<br />
<br />
2/18, 2/25: Chaojie<br />
<br />
3/3. 3/10: Yu Sun<br />
<br />
3/24, 3/31: Tony<br />
<br />
4/7, 4/14: Tung<br />
<br />
4/21, 4/28: Tung<br />
<br />
==2019 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Jane Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=18621Graduate student reading seminar2020-01-13T23:16:14Z<p>Valko: </p>
<hr />
<div>(... in probability)<br />
<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
==2020 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
==2019 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Jane Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=18620Graduate student reading seminar2020-01-13T23:13:55Z<p>Valko: /* 2018 Fall */</p>
<hr />
<div>(... in probability)<br />
<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
==2019 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Jane Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=18607Past Probability Seminars Spring 20202020-01-09T15:29:42Z<p>Valko: Replaced content with "__NOTOC__ = Spring 2020 = <b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <b>We usually end for questions at 3:20 PM.</b> If you would like..."</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2020 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== January 31, 2020, TBA ==<br />
'''TBA'''<br />
<br />
<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Fall_2019&diff=18606Past Probability Seminars Fall 20192020-01-09T15:26:36Z<p>Valko: Created page with " Back to Current Probability Seminar Schedule Back to Past Seminars = Fall 2019 = <b>Thursdays in 901 Van Vleck Hall at 2:30..."</p>
<hr />
<div>[[Probability Seminar | Back to Current Probability Seminar Schedule ]]<br />
<br />
<br />
[[Past Seminars | Back to Past Seminars]]<br />
<br />
<br />
= Fall 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==<br />
'''Furstenberg theorem: now with a parameter!'''<br />
<br />
The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. <br />
It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes.<br />
Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.<br />
<br />
== September 19, 2019, [http://math.columbia.edu/~xuanw Xuan Wu], Columbia University==<br />
<br />
'''A Gibbs resampling method for discrete log-gamma line ensemble.'''<br />
<br />
In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.<br />
<br />
== October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==<br />
<br />
== October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==<br />
<br />
''' Simplified dynamics for noisy systems with delays.'''<br />
<br />
Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.<br />
<br />
== October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==<br />
<br />
'''A general beta crossover ensemble'''<br />
<br />
I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known "soft" and "hard" edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).<br />
<br />
== October 31, 2019, Vadim Gorin, UW Madison==<br />
<br />
'''Shift invariance for the six-vertex model and directed polymers.'''<br />
<br />
I will explain a recently discovered mysterious property in a variety of stochastic systems ranging from the six-vertex model and to the directed polymers, last passage percolation, Kardar-Parisi-Zhang equation, and Airy sheet. Vaguely speaking, the property says that the multi-point joint distributions are unchanged when some (but not necessarily all!) points of observations are shifted. The property leads to explicit computations for the previously inaccessible joint distributions in all these settings.<br />
<br />
== November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==<br />
'''Domino tilings of the Aztec diamond with doubly periodic weightings'''<br />
<br />
This talk will be centered around domino tilings of the Aztec diamond with doubly periodic weightings. In particular asymptotic results of the $ 2 \times k $-periodic Aztec diamond will be discussed, both in the macroscopic and microscopic scale. The macroscopic picture is described using a close connection to a Riemann surface. For instance, the number of smooth regions (also called gas regions) is the same as the genus of the mentioned Riemann surface. <br />
<br />
The starting point of the asymptotic analysis is a non-intersecting path formulation and a double integral formula for the correlation kernel. The proof of this double integral formula is based on joint work with M. Duits, which will be discuss briefly if time permits.<br />
<br />
== November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==<br />
'''Universality of extremal eigenvalue statistics of random matrices'''<br />
<br />
The past decade has seen significant progress on the understanding of universality of various eigenvalue statistics of random matrix theory. However, the behavior of certain ``extremal'' or ``critical'' observables is not fully understood. Towards the former, we discuss progress on the universality of the largest gap between consecutive eigenvalues. With regards to the latter, we discuss the central limit theorem for the eigenvalue counting function, which can be viewed as a linear spectral statistic with critical regularity and has logarithmically growing variance.<br />
<br />
== November 21, 2019, Tung Nguyen, UW Madison ==<br />
<br />
'''Prevalence of deficiency zero reaction networks under an Erdos-Renyi framework<br />
'''<br />
<br />
Reaction network models, which are used to model many types of systems in biology, have grown dramatically in popularity over the past decade. This popularity has translated into a number of mathematical results that relate the topological features of the network to different qualitative behaviors of the associated dynamical system. One of the main topological features studied in the field is ''deficiency'' of a network. A reaction network which has strong connectivity in its connected components and a deficiency of zero is stable in both the deterministic and stochastic dynamical models.<br />
<br />
This leads to the question: how prevalent are deficiency zero models among all such network models. In this talk, I will quantify the prevalence of deficiency zero networks among random reaction networks generated under an Erdos-Renyi framework. Specifically, with n being the number of species, I will uncover a threshold function r(n) such that the probability of the random network being deficiency zero converges to 1 if the edge probability p_n << r(n) and converges to 0 if p_n >> r(n).</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Seminars&diff=18605Past Seminars2020-01-09T15:25:25Z<p>Valko: </p>
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<!-- [http://www.math.wisc.edu/~probsem/list-old-sem.html Webpage for older past probability seminars] --></div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Problem_Solver%27s_Toolbox&diff=18232Problem Solver's Toolbox2019-10-22T18:05:09Z<p>Valko: /* General ideas */</p>
<hr />
<div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br />
This page and the discussed topics can be used as a starting point for future exploration.<br />
<br />
<br />
== General ideas ==<br />
<br />
<br />
There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br />
* Make sure that you understand the problem. <br />
* If possible, draw a figure. <br />
* Can you connect the problem to a problem you have solved before? <br />
* If you have to show something for all numbers (or up to a large number) then try to check the statement for small values first.<br />
* Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br />
* Is there some symmetry in the problem that you can exploit? <br />
* Is it possible to work backward? <br />
* Does it help to consider an extreme case of the problem?<br />
* Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br />
<br />
== Modular arithmetic ==<br />
<br />
<br />
When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br />
It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same "modulo <math>m</math>" if they have the same remainder when divided by <math>m</math>. If <math>a</math> and <math>x</math> are the same modulo <math>m</math>, and <math>b</math> and <math>y</math> are the same modulo <math>m</math>, then <math>a+b</math> and <math>x+y</math> are the same modulo <math>m</math>, and similarly for subtraction and multiplication. <br />
<br />
For example, 5 is the same as 1 modulo 4, and hence <math>5\cdot 5 \cdot 5 \cdot 5=5^4</math> is the same as <math>1\cdot 1\cdot 1\cdot 1=1</math> modulo <math>4</math>. Same way you can show that <math>5^{1000}</math> has a remainder of 1 when we divide it by 4.<br />
<br />
Modular arithmetic often makes calculation much simpler. For example, see [https://www.math.wisc.edu/talent/sites/default/files/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br />
<br />
See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br />
<br />
== Mathematical induction ==<br />
<br />
Suppose that you want to prove a statement for all positive integers, for example that for each positive integer <math>n</math> the following is true: <math display="block">1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n+1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) </math><br />
<br />
Mathematical induction provides a tool for doing this. You need to show the following two things:<br />
# (Base case) The statement is true for <math>n=1</math>. <br />
# (Induction step) If the statement is true for <math>n</math> then it must be true for <math>n+1</math> as well.<br />
<br />
If we can show both of these parts, then it follows that the statement is true for all positive integer <math>n</math>. Why? The first part (the base case) shows that the statement is true for <math>n=1</math>. But then by the second part (the induction step) the statement must be true for <math>n=2</math> as well. Using the second part again and again we see that the statement is true for <math>n=3, 4, 5, \cdots</math> and repeating this sufficiently times we can prove that the statement is true for any fixed value of <math>n</math>. <br />
<br />
Often the idea of induction is demonstrated as a version of `Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if <math>n</math>th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br />
<br />
* Try to use induction to show the identity <math>(*)</math> above for all positive integer <math>n</math>.<br />
* You can also use induction to show a statement for all integers <math>n\ge 5</math>. Then for your base case you have to show that the statement is true for <math>n=5</math>. (The induction step is the same.)<br />
<br />
See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br />
<br />
== Proof by contradiction ==<br />
<br />
This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br />
<br />
Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer <math>n</math> so that <math>n(n+1)(n+2)</math> is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence <math>n(n+1)(n+2)</math> cannot be a prime. This contradicts our assumption that <math>n(n+1)(n+2)</math> is a prime, which shows that our assumption had to be incorrect. <br />
<br />
Proof by contradiction can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br />
<br />
== Pigeonhole Principle ==<br />
<br />
The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that <math>n<m</math> are positive integers, and we have <math>m</math> objects and <math>n</math> boxes. The Pigeonhole Principle states that If we place each of the <math>m</math> objects into one of the <math>n</math> boxes then there must be at least one box with at least two objects in it. <br />
The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most <math>n</math> objects all together. This is a contradiction, which means that the original statement must be correct. <br />
<br />
The Pigeonhole Principle is often used in the following, more general form. Suppose that <math>n, m, k</math> are positive integers with <math>n k< m </math>. If we place each of <math>m</math> objects into one of <math>n</math> boxes then there must be at least one box with at least <math>k+1</math> objects in it. Try to prove this version by contradiction.<br />
<br />
Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since <math>2\cdot 6<13</math>, we must have a box with at least <math>2+1=3</math> objects. In other words: there will be number that appears at least three times. <br />
<br />
Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br />
<br />
== Angles in the circle ==<br />
<br />
The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br />
<br />
'''Thales' Theorem''' <br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle, and <math>AB</math> is a diameter of of the circle. Then the angle <math>ACB</math> is <math>90^{\text{o}}</math>. In other words: the triangle <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math>. <br />
<br />
The theorem can be proved with a little bit of `angle-chasing'. Denote the center of the circle by <math>O</math>. Then <math>AO, BO, CO</math> are all radii of the circle, so they have the same length. Thus <math>\triangle AOC</math> and <math>\triangle BOC</math> are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br />
<br />
The converse of Thales's theorem states that if <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math> then we can draw a circle with a center that is the midpoint of <math>AB</math> that passes through <math>A, B, C</math>.<br />
<br />
<br />
The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br />
<br />
<br />
'''The Inscribed Angle Theorem'''<br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle and let <math>O</math> be the center of the circle. Then depending on the position of these points we have the following statements:<br />
<br />
* If <math>O</math> is on the line <math>AB</math> then <math>\angle ACB=90^{\text{o}}</math>. (This is just Thales' theorem again.)<br />
* If <math>O</math> and <math>C</math> are both on the same side of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of <math>360^{\text{o}}</math> minus the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= \angle AOB.</math><br />
* If <math>O</math> and <math>C</math> are on the opposite sides of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= 360^{\text{o}}-\angle AOB.</math><br />
<br />
If we measure the central angle <math>\angle AOB</math> the `right way' then we don't need to separate the three cases. In the first case the central angle is just <math>180^{\text{o}}</math>, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be <math>360^{\text{o}}-\angle AOB</math> then again we get that the inscribed angle is half of the central angle. <br />
<br />
<br />
The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br />
<br />
<br />
'''Applications to cyclic quadrilaterals'''<br />
<br />
The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br />
<br />
<br />
1. Suppose that the points <math>A, B, C, D</math> form a cyclic quadrilateral, this means that we can draw a circle going through the four points. <math>AB</math> divides the circle into two arcs. If the points <math>C</math> and <math>D</math> are in the same arc (meaning that they are on the same side of <math>AB</math>) then <br />
<math display="block"> \angle ACB= \angle ADB.</math><br />
The converse of this statement is also true: if <math>A, B, C, D</math> are distinct points, the points <math>C, D</math> are on the same side of the line <math>AB</math> and <math>\angle ACB= \angle ADB<br />
</math> then we can draw a circle around <math>A, B, C, D</math>, in other words <math>ABCD</math> is a cyclic quadrilateral.<br />
<br />
2. Suppose that <math>ABCD</math> is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to <math>180^{\text{o}}</math>. This means that <br />
<math display="block"> \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)</math><br />
<br />
The converse of the previous statement is also true: suppose that <math>ABCD</math> is a quadrilateral with angles satisfying the equations <math>(**)</math>. Then <math>ABCD</math> is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br />
<br />
The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=18226Past Probability Seminars Spring 20202019-10-21T21:56:54Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==<br />
'''Furstenberg theorem: now with a parameter!'''<br />
<br />
The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. <br />
It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes.<br />
Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.<br />
<br />
== September 19, 2019, [http://math.columbia.edu/~xuanw Xuan Wu], Columbia University==<br />
<br />
'''A Gibbs resampling method for discrete log-gamma line ensemble.'''<br />
<br />
In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.<br />
<br />
== October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==<br />
<br />
== October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==<br />
<br />
''' Simplified dynamics for noisy systems with delays.'''<br />
<br />
Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.<br />
<br />
== October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==<br />
<br />
'''A general beta crossover ensemble'''<br />
<br />
I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known "soft" and "hard" edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).<br />
<br />
== October 31, 2019, Vadim Gorin, UW Madison==<br />
<br />
'''Shift invariance for the six-vertex model and directed polymers.'''<br />
<br />
I will explain a recently discovered mysterious property in a variety of stochastic systems ranging from the six-vertex model and to the directed polymers, last passage percolation, Kardar-Parisi-Zhang equation, and Airy sheet. Vaguely speaking, the property says that the multi-point joint distributions are unchanged when some (but not necessarily all!) points of observations are shifted. The property leads to explicit computations for the previously inaccessible joint distributions in all these settings.<br />
<br />
== November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==<br />
<br />
== November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==<br />
<br />
== November 21, 2019, Tung Nguyen, UW Madison ==<br />
<br />
== November 28, 2019, Thanksgiving (no seminar) ==<br />
<br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Problem_Solver%27s_Toolbox&diff=18214Problem Solver's Toolbox2019-10-19T15:13:41Z<p>Valko: /* Mathematical induction */</p>
<hr />
<div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br />
This page and the discussed topics can be used as a starting point for future exploration.<br />
<br />
<br />
== General ideas ==<br />
<br />
<br />
There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br />
* Make sure that you understand the problem. <br />
* If possible, draw a figure. <br />
* Can you connect the problem to a problem you have solved before? <br />
* If you have to show something for all numbers (or a large number) then try to check the statement for small values first.<br />
* Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br />
* Is there some symmetry in the problem that you can exploit? <br />
* Is it possible to work backward? <br />
* Does it help to consider an extreme case of the problem?<br />
* Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br />
<br />
== Modular arithmetic ==<br />
<br />
<br />
When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br />
It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same "modulo <math>m</math>" if they have the same remainder when divided by <math>m</math>. If <math>a</math> and <math>x</math> are the same modulo <math>m</math>, and <math>b</math> and <math>y</math> are the same modulo <math>m</math>, then <math>a+b</math> and <math>x+y</math> are the same modulo <math>m</math>, and similarly for subtraction and multiplication. <br />
<br />
For example, 5 is the same as 1 modulo 4, and hence <math>5\cdot 5 \cdot 5 \cdot 5=5^4</math> is the same as <math>1\cdot 1\cdot 1\cdot 1=1</math> modulo <math>4</math>. Same way you can show that <math>5^{1000}</math> has a remainder of 1 when we divide it by 4.<br />
<br />
Modular arithmetic often makes calculation much simpler. For example, see [https://www.math.wisc.edu/talent/sites/default/files/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br />
<br />
See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br />
<br />
== Mathematical induction ==<br />
<br />
Suppose that you want to prove a statement for all positive integers, for example that for each positive integer <math>n</math> the following is true: <math display="block">1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n+1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) </math><br />
<br />
Mathematical induction provides a tool for doing this. You need to show the following two things:<br />
# (Base case) The statement is true for <math>n=1</math>. <br />
# (Induction step) If the statement is true for <math>n</math> then it must be true for <math>n+1</math> as well.<br />
<br />
If we can show both of these parts, then it follows that the statement is true for all positive integer <math>n</math>. Why? The first part (the base case) shows that the statement is true for <math>n=1</math>. But then by the second part (the induction step) the statement must be true for <math>n=2</math> as well. Using the second part again and again we see that the statement is true for <math>n=3, 4, 5, \cdots</math> and repeating this sufficiently times we can prove that the statement is true for any fixed value of <math>n</math>. <br />
<br />
Often the idea of induction is demonstrated as a version of `Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if <math>n</math>th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br />
<br />
* Try to use induction to show the identity <math>(*)</math> above for all positive integer <math>n</math>.<br />
* You can also use induction to show a statement for all integers <math>n\ge 5</math>. Then for your base case you have to show that the statement is true for <math>n=5</math>. (The induction step is the same.)<br />
<br />
See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br />
<br />
== Proof by contradiction ==<br />
<br />
This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br />
<br />
Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer <math>n</math> so that <math>n(n+1)(n+2)</math> is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence <math>n(n+1)(n+2)</math> cannot be a prime. This contradicts our assumption that <math>n(n+1)(n+2)</math> is a prime, which shows that our assumption had to be incorrect. <br />
<br />
Proof by contradiction can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br />
<br />
== Pigeonhole Principle ==<br />
<br />
The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that <math>n<m</math> are positive integers, and we have <math>m</math> objects and <math>n</math> boxes. The Pigeonhole Principle states that If we place each of the <math>m</math> objects into one of the <math>n</math> boxes then there must be at least one box with at least two objects in it. <br />
The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most <math>n</math> objects all together. This is a contradiction, which means that the original statement must be correct. <br />
<br />
The Pigeonhole Principle is often used in the following, more general form. Suppose that <math>n, m, k</math> are positive integers with <math>n k< m </math>. If we place each of <math>m</math> objects into one of <math>n</math> boxes then there must be at least one box with at least <math>k+1</math> objects in it. Try to prove this version by contradiction.<br />
<br />
Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since <math>2\cdot 6<13</math>, we must have a box with at least <math>2+1=3</math> objects. In other words: there will be number that appears at least three times. <br />
<br />
Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br />
<br />
== Angles in the circle ==<br />
<br />
The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br />
<br />
'''Thales' Theorem''' <br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle, and <math>AB</math> is a diameter of of the circle. Then the angle <math>ACB</math> is <math>90^{\text{o}}</math>. In other words: the triangle <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math>. <br />
<br />
The theorem can be proved with a little bit of `angle-chasing'. Denote the center of the circle by <math>O</math>. Then <math>AO, BO, CO</math> are all radii of the circle, so they have the same length. Thus <math>\triangle AOC</math> and <math>\triangle BOC</math> are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br />
<br />
The converse of Thales's theorem states that if <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math> then we can draw a circle with a center that is the midpoint of <math>AB</math> that passes through <math>A, B, C</math>.<br />
<br />
<br />
The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br />
<br />
<br />
'''The Inscribed Angle Theorem'''<br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle and let <math>O</math> be the center of the circle. Then depending on the position of these points we have the following statements:<br />
<br />
* If <math>O</math> is on the line <math>AB</math> then <math>\angle ACB=90^{\text{o}}</math>. (This is just Thales' theorem again.)<br />
* If <math>O</math> and <math>C</math> are both on the same side of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of <math>360^{\text{o}}</math> minus the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= \angle AOB.</math><br />
* If <math>O</math> and <math>C</math> are on the opposite sides of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= 360^{\text{o}}-\angle AOB.</math><br />
<br />
If we measure the central angle <math>\angle AOB</math> the `right way' then we don't need to separate the three cases. In the first case the central angle is just <math>180^{\text{o}}</math>, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be <math>360^{\text{o}}-\angle AOB</math> then again we get that the inscribed angle is half of the central angle. <br />
<br />
<br />
The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br />
<br />
<br />
'''Applications to cyclic quadrilaterals'''<br />
<br />
The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br />
<br />
<br />
1. Suppose that the points <math>A, B, C, D</math> form a cyclic quadrilateral, this means that we can draw a circle going through the four points. <math>AB</math> divides the circle into two arcs. If the points <math>C</math> and <math>D</math> are in the same arc (meaning that they are on the same side of <math>AB</math>) then <br />
<math display="block"> \angle ACB= \angle ADB.</math><br />
The converse of this statement is also true: if <math>A, B, C, D</math> are distinct points, the points <math>C, D</math> are on the same side of the line <math>AB</math> and <math>\angle ACB= \angle ADB<br />
</math> then we can draw a circle around <math>A, B, C, D</math>, in other words <math>ABCD</math> is a cyclic quadrilateral.<br />
<br />
2. Suppose that <math>ABCD</math> is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to <math>180^{\text{o}}</math>. This means that <br />
<math display="block"> \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)</math><br />
<br />
The converse of the previous statement is also true: suppose that <math>ABCD</math> is a quadrilateral with angles satisfying the equations <math>(**)</math>. Then <math>ABCD</math> is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br />
<br />
The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=18142Past Probability Seminars Spring 20202019-10-11T00:04:17Z<p>Valko: /* October 24, 2019, Brian Rider, Temple University */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==<br />
'''Furstenberg theorem: now with a parameter!'''<br />
<br />
The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. <br />
It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes.<br />
Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.<br />
<br />
== September 19, 2019, [http://math.columbia.edu/~xuanw Xuan Wu], Columbia University==<br />
<br />
'''A Gibbs resampling method for discrete log-gamma line ensemble.'''<br />
<br />
In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.<br />
<br />
== October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==<br />
<br />
== October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==<br />
<br />
''' Simplified dynamics for noisy systems with delays.'''<br />
<br />
Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.<br />
<br />
== October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==<br />
<br />
'''A general beta crossover ensemble'''<br />
<br />
I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known "soft" and "hard" edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).<br />
<br />
== October 31, 2019, [http://math.mit.edu/~elmos/ Elchanan Mossel], MIT ==<br />
<br />
== November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==<br />
<br />
== November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==<br />
<br />
== November 21, 2019, Tung Nguyen, UW Madison ==<br />
<br />
== November 28, 2019, Thanksgiving (no seminar) ==<br />
<br />
== December 5, 2019, Vadim Gorin, UW Madison ==<br />
<br />
<br />
<br />
<!--<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=18141Past Probability Seminars Spring 20202019-10-11T00:04:01Z<p>Valko: /* October 24, 2019, Brian Rider, Temple University */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:20 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==<br />
'''Furstenberg theorem: now with a parameter!'''<br />
<br />
The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. <br />
It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes.<br />
Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.<br />
<br />
== September 19, 2019, [http://math.columbia.edu/~xuanw Xuan Wu], Columbia University==<br />
<br />
'''A Gibbs resampling method for discrete log-gamma line ensemble.'''<br />
<br />
In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.<br />
<br />
== October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==<br />
<br />
== October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==<br />
<br />
''' Simplified dynamics for noisy systems with delays.'''<br />
<br />
Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.<br />
<br />
== October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==<br />
<br />
Title: A general beta crossover ensemble<br />
<br />
Abstract: I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known "soft" and "hard" edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).<br />
<br />
== October 31, 2019, [http://math.mit.edu/~elmos/ Elchanan Mossel], MIT ==<br />
<br />
== November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==<br />
<br />
== November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==<br />
<br />
== November 21, 2019, Tung Nguyen, UW Madison ==<br />
<br />
== November 28, 2019, Thanksgiving (no seminar) ==<br />
<br />
== December 5, 2019, Vadim Gorin, UW Madison ==<br />
<br />
<br />
<br />
<!--<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=17903Probability2019-09-16T19:56:41Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
Vadim Gorin (Moscow, 2011) integrable probability, random matrices, asymptotic representation theory<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
<br />
== Postdocs ==<br />
<br />
Scott Smith (Maryland, 2016)<br />
<br />
<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:25pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=17813Graduate student reading seminar2019-09-11T18:11:13Z<p>Valko: </p>
<hr />
<div>(... in probability)<br />
<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
9/24, 10/1: Xiao<br />
<br />
10/8, 10/15: Jakwang<br />
<br />
10/22, 10/29: Evan<br />
<br />
11/5, 11/12: Chaojie<br />
<br />
12/3, 12/10: Tung<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Stephen Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Talk:Past_Probability_Seminars_Spring_2020&diff=17779Talk:Past Probability Seminars Spring 20202019-09-06T19:19:27Z<p>Valko: Blanked the page</p>
<hr />
<div></div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Problem_Solver%27s_Toolbox&diff=17683Problem Solver's Toolbox2019-08-27T19:04:18Z<p>Valko: </p>
<hr />
<div>The goal of this page is to collect simple problem solving strategies and tools. We hope that students interested in the Wisconsin Math Talent Search would find the described ideas useful. <br />
This page and the discussed topics can be used as a starting point for future exploration.<br />
<br />
<br />
== General ideas ==<br />
<br />
<br />
There is no universal recipe for math problems that would work every time, that's what makes math fun! There are however a number of general strategies that could be useful in most cases, here is a short list of them. (Many of these ideas were popularized by the Hungarian born mathematician George Pólya in his book [https://en.wikipedia.org/wiki/How_to_Solve_It How to Solve It].)<br />
* Make sure that you understand the problem. <br />
* If possible, draw a figure. <br />
* Can you connect the problem to a problem you have solved before? <br />
* If you have to show something for all numbers (or a large number) then try to check the statement for small values first.<br />
* Can you solve the problem in a special case first? Can you solve a modified version of the problem first? <br />
* Is there some symmetry in the problem that you can exploit? <br />
* Is it possible to work backward? <br />
* Does it help to consider an extreme case of the problem?<br />
* Is it possible to generalize the problem? (Sometimes the generalized is easier to solve.)<br />
<br />
== Modular arithmetic ==<br />
<br />
<br />
When we have to divide two integers, they don't always divide evenly, and there is a quotient and a remainder. For example when we divide 10 by 3 we get a remainder of 1.<br />
It turns out that these remainders behave very well under addition, subtraction, and multiplication. We say two numbers are the same "modulo <math>m</math>" if they have the same remainder when divided by <math>m</math>. If <math>a</math> and <math>x</math> are the same modulo <math>m</math>, and <math>b</math> and <math>y</math> are the same modulo <math>m</math>, then <math>a+b</math> and <math>x+y</math> are the same modulo <math>m</math>, and similarly for subtraction and multiplication. <br />
<br />
For example, 5 is the same as 1 modulo 4, and hence <math>5\cdot 5 \cdot 5 \cdot 5=5^4</math> is the same as <math>1\cdot 1\cdot 1\cdot 1=1</math> modulo <math>4</math>. Same way you can show that <math>5^{1000}</math> has a remainder of 1 when we divide it by 4.<br />
<br />
Modular arithmetic often makes calculation much simpler. For example, see [https://www.math.wisc.edu/talent/sites/default/files/Talent16-2q.pdf 2016-17 Set #2 Problem 3].<br />
<br />
See [http://artofproblemsolving.com/wiki/index.php?title=Modular_arithmetic/Introduction Art of Problem Solving's introduction to modular arithmetic] for more information.<br />
<br />
== Mathematical induction ==<br />
<br />
Suppose that you want to prove a statement for all positive integers, for example that for each positive integer <math>n</math> the following is true: <math display="block">1\cdot 2+2\cdot 3+3\cdot 4+\cdots+n\cdot (n-1)=\frac{n(n+1)(n+2)}{3}.\qquad\qquad(*) </math><br />
Mathematical induction provides a tool for doing this. You need to show the following two things:<br />
# (Base case) The statement is true for <math>n=1</math>. <br />
# (Induction step) If the statement is true for <math>n</math> then it must be true for <math>n+1</math> as well.<br />
<br />
If we can show both of these parts, then it follows that the statement is true for all positive integer <math>n</math>. Why? The first part (the base case) shows that the statement is true for <math>n=1</math>. But then by the second part (the induction step) the statement must be true for <math>n=2</math> as well. Using the second part again and again we see that the statement is true for <math>n=3, 4, 5, \cdots</math> and repeating this sufficiently times we can prove that the statement is true for any fixed value of <math>n</math>. <br />
<br />
Often the idea of induction is demonstrated as a version of `Domino effect'. Imagine that you have an infinite row of dominos numbered with the positive integers, where if <math>n</math>th domino falls then the next one will fall as well (this is the induction step). If we make the first domino fall (this is the base case) then eventually all other dominos will fall as well. <br />
<br />
* Try to use induction to show the identity <math>(*)</math> above for all positive integer <math>n</math>.<br />
* You can also use induction to show a statement for all integers <math>n\ge 5</math>. Then for your base case you have to show that the statement is true for <math>n=5</math>. (The induction step is the same.)<br />
<br />
See this page from [https://www.mathsisfun.com/algebra/mathematical-induction.html Math Is Fun] for some simple applications of induction.<br />
<br />
== Proof by contradiction ==<br />
<br />
This is a commonly used problem solving method. Suppose that you have to prove a certain statement. Now pretend that the statement is not true and try to derive (as a consequence) a false statement. The found false statement shows that your assumption about the original statement was incorrect: thus the original statement must be true. <br />
<br />
Here is a simple example: we will prove that the product of three consecutive positive integers cannot be a prime number. Assume the opposite: that means that there is a positive integer <math>n</math> so that <math>n(n+1)(n+2)</math> is a prime. But among three consecutive integers we will always have a multiple of 2, and also a multiple of 3. Thus the product of the three numbers must be divisible by both 2 and 3, and hence <math>n(n+1)(n+2)</math> cannot be a prime. This contradicts our assumption that <math>n(n+1)(n+2)</math> is a prime, which shows that our assumption had to be incorrect. <br />
<br />
Proof by contradiction can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent16-1q.pdf 2016-17 Set #1 Problem 4].<br />
<br />
== Pigeonhole Principle ==<br />
<br />
The Pigeonhole Principle is one of the simplest tools in mathematics, but it can be very powerful. Suppose that <math>n<m</math> are positive integers, and we have <math>m</math> objects and <math>n</math> boxes. The Pigeonhole Principle states that If we place each of the <math>m</math> objects into one of the <math>n</math> boxes then there must be at least one box with at least two objects in it. <br />
The statement can be proved by contradiction: if we can find an arrangement of objects so that each box has less than two objects in it, then each box would contain at most one object, and hence we had at most <math>n</math> objects all together. This is a contradiction, which means that the original statement must be correct. <br />
<br />
The Pigeonhole Principle is often used in the following, more general form. Suppose that <math>n, m, k</math> are positive integers with <math>n k< m </math>. If we place each of <math>m</math> objects into one of <math>n</math> boxes then there must be at least one box with at least <math>k+1</math> objects in it. Try to prove this version by contradiction.<br />
<br />
Here is a simple application: if we roll a die 13 times then there must be a number that appears at least three times. Here each die roll correspond to an object, each of the 6 possible outcomes correspond to a possible box. Since <math>2\cdot 6<13</math>, we must have a box with at least <math>2+1=3</math> objects. In other words: there will be number that appears at least three times. <br />
<br />
Pigeonhole Principle can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/T14-1q_0_0.pdf 2014-15 Set #1 Problem 4].<br />
<br />
== Angles in the circle ==<br />
<br />
The following theorems are often useful when working with geometry problems. [[File:Thales_thm.jpg|250px|thumb|right|An illustration of Thales' Theorem. O is the center of the circle.]] <br />
<br />
'''Thales' Theorem''' <br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle, and <math>AB</math> is a diameter of of the circle. Then the angle <math>ACB</math> is <math>90^{\text{o}}</math>. In other words: the triangle <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math>. <br />
<br />
The theorem can be proved with a little bit of `angle-chasing'. Denote the center of the circle by <math>O</math>. Then <math>AO, BO, CO</math> are all radii of the circle, so they have the same length. Thus <math>\triangle AOC</math> and <math>\triangle BOC</math> are both isosceles triangles. Now try labeling the various angles in the picture and you should quickly arrive to a proof. (You can find the worked out proof at the [https://en.wikipedia.org/wiki/Thales%27_theorem wiki page] of the theorem, but it is more fun if you figure it out on your own!)<br />
<br />
The converse of Thales's theorem states that if <math>\triangle ABC</math> is a right triangle with hypotenuse <math>AB</math> then we can draw a circle with a center that is the midpoint of <math>AB</math> that passes through <math>A, B, C</math>.<br />
<br />
<br />
The Inscribed Angle Theorem below is a generalization of Thales' Theorem. <br />
<br />
<br />
'''The Inscribed Angle Theorem'''<br />
<br />
Suppose that the distinct points <math>A, B, C</math> are all on a circle and let <math>O</math> be the center of the circle. Then depending on the position of these points we have the following statements:<br />
<br />
* If <math>O</math> is on the line <math>AB</math> then <math>\angle ACB=90^{\text{o}}</math>. (This is just Thales' theorem again.)<br />
* If <math>O</math> and <math>C</math> are both on the same side of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of <math>360^{\text{o}}</math> minus the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= \angle AOB.</math><br />
* If <math>O</math> and <math>C</math> are on the opposite sides of the line <math>AB</math> then the inscribed angle <math>\angle ACB</math> is half of the central angle <math>\angle AOB</math>: <br />
<math display="block"> 2 \angle ACB= 360^{\text{o}}-\angle AOB.</math><br />
<br />
If we measure the central angle <math>\angle AOB</math> the `right way' then we don't need to separate the three cases. In the first case the central angle is just <math>180^{\text{o}}</math>, and the inscribed angle is exactly the half of that. In the third case if we define the central angle to be <math>360^{\text{o}}-\angle AOB</math> then again we get that the inscribed angle is half of the central angle. <br />
<br />
<br />
The theorem can be proved with angle-chasing, using the same idea that was described for Thales' theorem. See the [https://en.wikipedia.org/wiki/Inscribed_angle wiki page] for the proof (but first try to do it on your own!).<br />
<br />
<br />
'''Applications to cyclic quadrilaterals'''<br />
<br />
The following statements (and their converses) are useful applications of the Inscribed Angle theorem.<br />
<br />
<br />
1. Suppose that the points <math>A, B, C, D</math> form a cyclic quadrilateral, this means that we can draw a circle going through the four points. <math>AB</math> divides the circle into two arcs. If the points <math>C</math> and <math>D</math> are in the same arc (meaning that they are on the same side of <math>AB</math>) then <br />
<math display="block"> \angle ACB= \angle ADB.</math><br />
The converse of this statement is also true: if <math>A, B, C, D</math> are distinct points, the points <math>C, D</math> are on the same side of the line <math>AB</math> and <math>\angle ACB= \angle ADB<br />
</math> then we can draw a circle around <math>A, B, C, D</math>, in other words <math>ABCD</math> is a cyclic quadrilateral.<br />
<br />
2. Suppose that <math>ABCD</math> is a cyclic quadrilateral. Then the sum of any two opposite angles is equal to <math>180^{\text{o}}</math>. This means that <br />
<math display="block"> \angle ABC+\angle CDA= 180^{\text{o}}, \quad \text{and}\quad \angle BCD+\angle DAB= 180^{\text{o}}. \qquad\qquad (**)</math><br />
<br />
The converse of the previous statement is also true: suppose that <math>ABCD</math> is a quadrilateral with angles satisfying the equations <math>(**)</math>. Then <math>ABCD</math> is a cyclic quadrilateral: we can draw a circle that passes through the four points.<br />
<br />
The Inscribed Angle Theorem and the statements about cyclic quadrilaterals can be used for example in [https://www.math.wisc.edu/talent/sites/default/files/Talent15-4q.pdf 2015-16 Set #4 Problem 5].</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Undergraduate_courses_in_probability&diff=17642Undergraduate courses in probability2019-08-15T20:20:06Z<p>Valko: </p>
<hr />
<div>'''431 - Introduction to the theory of probability'''<br />
<br />
Math 431 is an introduction to probability theory, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include methods of counting (combinatorics), axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.<br />
<br />
Probability theory is ubiquitous in natural science, social science and engineering, so this course can be valuable in conjunction with many different majors. 431 is not a course in statistics. Statistics is a discipline mainly concerned with analyzing and representing data. Probability theory forms the mathematical foundation of statistics, but the two disciplines are separate.<br />
<br />
The course is offered every semester, including the summer. <br />
<br />
''Prerequisite'': Math 234. <br />
<br />
<span style="color:#0000FF"> '''Who should take this class?'''</span> A well rounded undergraduate experience in math should include some probability theory. Math 431 is our introductory probability class with no high level prerequisites. <br />
<br />
<br />
<br />
'''531 - Probability theory'''<br />
<br />
The course is a rigorous introduction to probability theory on an advanced undergraduate level. Only a minimal amount of measure theory is used (in particular, Lebesgue integrals will not be needed). The course gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and discusses some of the classical results of probability theory with proofs (DeMoivre-Laplace limit theorems, the study of simple random walk on Z, applications of generating functions).<br />
<br />
The course is offered every spring.<br />
<br />
''Prerequisite'': a proof based analysis course (Math 376, Math 421 or Math 521). <br />
<br />
<span style="color:#0000FF"> '''Who should take this class?'''</span> Students who would like to get a rigorous introduction to probability. It could also provide a stepping stone for our 600 level stochastic processes courses. (The course can be taken even after taking Math 431.)<br />
<br />
<br />
<br />
'''605 - Stochastic methods in biology'''<br />
<br />
Math 605 provides an introduction to stochastic processes. It introduces both discrete and continuous time Markov chains, and some aspects of renewal theory. The course focuses on biological applications of these mathematical models including: the Wright-Fischer model, birth and death processes, branching processes, and many models from intracellular biochemistry. This course is similar to Math 632 in content. However, unlike in Math 632, simulation plays a vital role in the study of the requisite processes in Math 605, with Matlab the software package of choice. <br />
<br />
The course is offered every two years in the fall semester. <br />
<br />
''Prerequisite'': Math 431, a basic knowledge of linear algebra and linear differential equations (e.g. Math 319, Math 340, Math 341)<br />
<br />
<span style="color:#0000FF"> '''Who should take this class?'''</span> Anybody who is interested in stochastic processes and would like to learn more about applications in the biosciences, and especially intracellular biochemical processes.<br />
<br />
<br />
'''632 - Introduction to stochastic processes'''<br />
<br />
Math 632 gives an introduction to Markov chains and Markov processes with discrete state spaces and their applications. Particular models studied include birth-death chains, queuing models, random walks and branching processes. Selected topics from renewal theory, martingales, and Brownian motion are also included, but vary from semester to semester to meet the needs of different audiences. <br />
<br />
''Prerequisite'': Intro to probability (Math 309, 431 or 531)+ a linear algebra or an intro to proofs class (320, 340, 341, 375, 421)<br />
<br />
<span style="color:#0000FF"> '''Who should take this class?'''</span> Math 632 is the natural next step after an introductory probability course. It could be useful for an Option 1 math major interested in higher level probability and it is also a great fit for many of our [https://www.math.wisc.edu/undergraduate/option-2-sample-packages Option 2 packages]. <br />
<br />
<br />
<br />
'''635 - Introduction to Brownian motion and stochastic calculus'''<br />
<br />
Math 635 is an introduction to Brownian motion and stochastic calculus without a measure theory prerequisite. Topics touched upon include sample path properties of Brownian motion, Itô stochastic integrals, Itô's formula, stochastic differential equations and their solutions. As an application we will discuss the Black-Scholes formula of mathematical finance.<br />
<br />
The course is offered every two years in the spring semester. <br />
<br />
''Prerequisite'': Math 521 and Math 632<br />
<br />
<span style="color:#0000FF"> '''Who should take this class?'''</span> Anybody with an interest in higher level probability. It is especially useful for those who are planning to study financial math on a graduate level. <br />
<br />
<br />
<!--[[File:Probability_courses_1.jpg|600px]]--></div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability&diff=17641Probability2019-08-15T20:17:07Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
<br />
= '''Probability at UW-Madison''' =<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson] (Duke, 2005) applied probability, numerical methods, mathematical biology.<br />
<br />
Vadim Gorin<br />
<br />
[http://www.math.wisc.edu/~roch/ Sebastien Roch] (UC Berkeley, 2007) applied probability, mathematical biology, theoretical computer science.<br />
<br />
[http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen] (Minnesota, 1991) motion in a random medium, random growth models, interacting particle systems, large deviation theory.<br />
<br />
[http://www.math.wisc.edu/~hshen3/ Hao Shen] (Princeton, 2013) stochastic partial differential equations, mathematical physics, integrable probability<br />
<br />
[http://www.math.wisc.edu/~valko/ Benedek Valko] (Budapest, 2004) interacting particle systems, random matrices.<br />
<br />
<br />
<br />
<br />
== Emeriti ==<br />
<br />
[http://psoup.math.wisc.edu/kitchen.html David Griffeath] (Cornell, 1976)<br />
<br />
[http://www.math.wisc.edu/~kuelbs Jim Kuelbs] (Minnesota, 1965)<br />
<br />
[http://www.math.wisc.edu/~kurtz Tom Kurtz] (Stanford, 1967)<br />
<br />
Peter Ney (Columbia, 1961)<br />
<br />
Josh Chover (Michigan, 1952)<br />
<br />
== Graduate students ==<br />
<br />
<br />
[http://www.math.wisc.edu/~kehlert/ Kurt Ehlert] <br />
<br />
[http://www.math.wisc.edu/~kang Dae Han Kang]<br />
<br />
[https://sites.google.com/a/wisc.edu/brandon-legried/ Brandon Legried]<br />
<br />
Yun Li<br />
<br />
[http://sites.google.com/a/wisc.edu/tung-nguyen/ Tung Nguyen]<br />
<br />
[http://www.math.wisc.edu/~cyuan25/ Chaojie Yuan]<br />
<br />
<br />
<br />
== [[Probability Seminar]] ==<br />
<br />
Thursdays at 2:25pm, VV901<br />
<br />
==[[Graduate student reading seminar]]==<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
Tuesdays, 2:30pm, 901 Van Vleck<br />
<br />
== [[Probability group timetable]]==<br />
<br />
== [[Undergraduate courses in probability]]==<br />
<br />
== Graduate Courses in Probability ==<br />
<br />
<br />
<br />
'''2019 Fall'''<br />
<br />
Math/Stat 733 Theory of Probability I<br />
<br />
<br />
<br />
<br />
'''2020 Spring'''<br />
<br />
Math/Stat 734 Theory of Probability II <br />
<br />
Math 833 Topics in Probability</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=17585Probability group timetable2019-07-25T18:00:15Z<p>Valko: </p>
<hr />
<div>2019 Fall<br />
<br />
<br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| || || || || <br />
|- <br />
| 10-11|| || || || || <br />
|-<br />
| 11-12|| || || || ||<br />
|-<br />
| 12-1|| || || || || <br />
|-<br />
| 1-2|| || || || ||<br />
|-<br />
| 2-3|| || graduate probability seminar (2:25) || || probability seminar (2:25) || <br />
|-<br />
| 3-4|| || || || || <br />
|-<br />
| 4-5|| || || || || colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
<br />
<br />
<!-- <br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Timo 431, Kurt 222|| Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431, Kurt 222 || Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431<br />
|-<br />
| 10-11|| Kurt 222, Hans 234 || Phil out all day, Kurt 735 || Kurt 222, Hans 234 || Kurt 735 || Phil out all day, Hans 234 <br />
|-<br />
| 11-12|| Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Hans 846, Christian 846<br />
|-<br />
| 12-1|| Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431 <br />
|-<br />
| 1-2|| || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 || || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 ||<br />
|-<br />
| 2-3|| Daniele 431 (2:25) || graduate probability seminar (2:25) || Daniele 431 (2:25) || probability seminar (2:25) || Daniele 431 (2:25)<br />
|-<br />
| 3-4|| || Kurt 222, Hans 234 || || Kurt 222, Hans 234 || <br />
|-<br />
| 4-5|| || || || || colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
--><br />
<br />
<!--<br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Phil out all day || Benedek 531 (9:30)|| || Benedek 531 (9:30) || Phil out all day<br />
|-<br />
| 10-11||Jinsu 722, Louis 431 || || Jinsu 722, Louis 431|| ||Jinsu 722, Louis 431<br />
|-<br />
| 11-12|| || Hans 820 || || Hans 820 ||<br />
|-<br />
| 12-1|| Jinsu 222, Louis 632 || ||Jinsu 222, Louis 632 || || Jinsu 222, Louis 632<br />
|-<br />
| 1-2|| Jinsu 222, Hans 851 || Benedek OH, Hans 843 || Jinsu 222, Hans 851|| Hans 843 ||Jinsu 222, Hans 851<br />
|-<br />
| 2-3|| || graduate probability seminar (2:25) || Louis (Seb) || probability seminar (2:25) ||<br />
|-<br />
| 3-4|| ||Benedek (OH (3:30) || Benedek OH || || <br />
|-<br />
| 4-5|| || || Louis (OH 4:30)|| Louis (OH 4:30)|| colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
--></div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=17413Past Probability Seminars Spring 20202019-05-01T14:46:06Z<p>Valko: /* Tuesday , May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:220px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=17412Past Probability Seminars Spring 20202019-05-01T14:45:55Z<p>Valko: /* Tuesday , May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:220px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: The directed landscape<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=17344Past Probability Seminars Spring 20202019-04-18T18:07:12Z<p>Valko: /* May 2, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== May 7, '''Tuesday''', Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=17343Past Probability Seminars Spring 20202019-04-18T18:06:12Z<p>Valko: /* April 26, Colloquium, Kavita Ramanan, Brown */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=17342Past Probability Seminars Spring 20202019-04-18T18:05:59Z<p>Valko: /* April 26, Colloquium, Kavita Ramanan, Brown */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: Tales of Random Projections<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=16989Graduate student reading seminar2019-02-18T23:57:30Z<p>Valko: /* 2019 Spring */</p>
<hr />
<div>(... in probability)<br />
<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
2/5: Timo<br />
<br />
2/12, 2/19: Evan<br />
<br />
2/26, 3/5: Chaojie<br />
<br />
3/12, 3/26: Kurt<br />
<br />
4/2, 4/9: Yu<br />
<br />
4/16, 4/23: Max<br />
<br />
4/30, 5/7: Xiao<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Stephen Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=16899Past Probability Seminars Spring 20202019-02-11T00:03:12Z<p>Valko: /* March 28, Shamgar Gurevitch UW-Madison */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=16898Past Probability Seminars Spring 20202019-02-11T00:02:31Z<p>Valko: /* March 28, Shamgar Gurevitch UW-Madison */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=16897Past Probability Seminars Spring 20202019-02-11T00:01:40Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=16894Past Probability Seminars Spring 20202019-02-10T23:31:49Z<p>Valko: /* February 14, Timo Seppäläinen */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2020&diff=16893Past Probability Seminars Spring 20202019-02-10T23:31:31Z<p>Valko: </p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen]==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah). <br />
<br />
<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Valkohttps://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_student_reading_seminar&diff=16689Graduate student reading seminar2019-01-23T17:06:13Z<p>Valko: </p>
<hr />
<div>(... in probability)<br />
<br />
<br />
Email list: join-grad_prob_seminar@lists.wisc.edu<br />
<br />
==2019 Spring==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
==2018 Fall==<br />
<br />
Tuesday 2:30pm, 901 Van Vleck<br />
<br />
<br />
The topic this semester is large deviation theory. Send me (BV) an email, if you want access to the shared Box folder with some reading material. <br />
<br />
<br />
9/25, 10/2: Dae Han<br />
<br />
10/9, 10/16: Kurt<br />
<br />
10/23, 10/30: Stephen Davis<br />
<br />
11/6, 11/13: Brandon Legried <br />
<br />
11/20, 11/27: Shuqi Yu<br />
<br />
12/4, 12/11: Yun Li<br />
<br />
==2018 Spring==<br />
<br />
Tuesday 2:30pm, B135 Van Vleck<br />
<br />
<br />
Preliminary schedule:<br />
<br />
2/20, 2/27: Yun<br />
<br />
3/6, 3/13: Greg<br />
<br />
3/20, 4/3: Yu<br />
<br />
4/10, 4/17: Shuqi<br />
<br />
4/24, 5/1: Tony<br />
<br />
==2017 Fall==<br />
<br />
Tuesday 2:30pm, 214 Ingraham Hall<br />
<br />
<br />
Preliminary schedule: <br />
<br />
9/26, 10/3: Hans<br />
<br />
10/10, 10/17: Guo<br />
<br />
10/24, 10/31: Chaoji<br />
<br />
11/7, 11/14: Yun <br />
<br />
11/21, 11/28: Kurt<br />
<br />
12/5, 12/12: Christian<br />
<br />
<br />
<br />
<br />
==2017 Spring==<br />
<br />
Tuesday 2:25pm, B211<br />
<br />
1/31, 2/7: Fan<br />
<br />
I will talk about the Hanson-Wright inequality, which is a large deviation estimate for random variable of the form X^* A X, where X is a random vector with independent subgaussian entries and A is an arbitrary deterministic matrix. In the first talk, I will present a beautiful proof given by Mark Rudelson and Roman Vershynin. In the second talk, I will talk about some applications of this inequality.<br />
<br />
Reference: M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-gaussian concentration, Electron. Commun. Probab. Volume 18 (2013).<br />
<br />
3/7, 3/14 : Jinsu<br />
<br />
Title : Donsker's Theorem and its application.<br />
Donsker's Theorem roughly says normalized random walk with linear interpolation on time interval [0,1] weakly converges to the Brownian motion B[0,1] in C([0,1]). It is sometimes called Donsker's invariance principle or the functional central limit theorem. I will show main ideas for the proof of this theorem tomorrow and show a couple of applications in my 2nd talk.<br />
<br />
Reference : https://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf<br />
<br />
==2016 Fall==<br />
<br />
9/27 Daniele<br />
<br />
Stochastic reaction networks.<br />
<br />
Stochastic reaction networks are continuous time Markov chain models used primarily in biochemistry. I will define them, prove some results that connect them to related deterministic models and introduce some open questions. <br />
<br />
10/4 Jessica<br />
<br />
10/11, 10/18: Dae Han<br />
<br />
10/25, 11/1: Jinsu<br />
<br />
Coupling of Markov processes.<br />
<br />
When we have two distributions on same probability space, we can think of a pair whose marginal probability is each of two distributions.<br />
This pairing can be used to estimate the total variation distance between two distributions. This idea is called coupling method.<br />
I am going to introduce basic concepts,ideas and applications of coupling for Markov processes.<br />
<br />
Links of References<br />
<br />
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf<br />
<br />
http://websites.math.leidenuniv.nl/probability/lecturenotes/CouplingLectures.pdf<br />
<br />
11/8, 11/15: Hans<br />
<br />
11/22, 11/29: Keith<br />
<br />
Surprisingly Determinental: DPPs and some asymptotics of ASEP <br />
<br />
I'll be reading and presenting some recent papers of Alexei Borodin and a few collaborators which have uncovered certain equivalences between determinental point processes and non-determinental processes.<br />
<br />
<br />
==2016 Spring==<br />
<br />
Tuesday, 2:25pm, B321 Van Vleck<br />
<br />
<br />
3/29, 4/5: Fan Yang<br />
<br />
I will talk about the ergodic decomposition theorem (EDT). More specifically, given a compact metric space X and a continuous transformation T on it, the theorem shows that any T-invariant measure on X can be decomposed into a convex combination of ergodic measures. In the first talk I introduced the EDT and some related facts. In the second talk, I will talk about the conditional measures, and prove that the ergodic measures in EDT are indeed the conditional measures.<br />
<br />
<br />
2/16 : Jinsu<br />
<br />
Lyapunov function for Markov Processes.<br />
<br />
For ODE, we can show stability of the trajectory using Lyapunov functions.<br />
<br />
There is an analogy for Markov Processes. I'd like to talk about the existence of stationary distribution with Lyapunov function.<br />
<br />
In some cases, it is also possible to show the rate of convergence to the stationary distribution.<br />
<br />
==2015 Fall==<br />
<br />
This semester we will focus on tools and methods.<br />
<br />
[https://www.math.wisc.edu/wiki/images/a/ac/Reading_seminar_2015.pdf Seminar notes] ([https://www.dropbox.com/s/f4km7pevwfb1vbm/Reading%20seminar%202015.tex?dl=1 tex file], [https://www.dropbox.com/s/lg7kcgyf3nsukbx/Reading_seminar_2015.bib?dl=1 bib file])<br />
<br />
9/15, 9/22: Elnur<br />
<br />
I will talk about large deviation theory and its applications. For the first talk, my plan is to introduce Gartner-Ellis theorem and show a few applications of it to finite state discrete time Markov chains.<br />
<br />
9/29, 10/6, 10/13 :Dae Han<br />
<br />
10/20, 10/27, 11/3: Jessica<br />
<br />
I will first present an overview of concentration of measure and concentration inequalities with a focus on the connection with related topics in analysis and geometry. Then, I will present Log-Sobolev inequalities and their connection to concentration of measure. <br />
<br />
11/10, 11/17: Hao Kai<br />
<br />
11/24, 12/1, 12/8, 12/15: Chris<br />
<br />
: <br />
<br />
<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
2/2, 2/9: Louis<br />
<br />
<br />
2/16, 2/23: Jinsu<br />
<br />
3/1, 3/8: Hans<br />
<br />
==2015 Spring==<br />
<br />
<br />
2/3, 2/10: Scott<br />
<br />
An Introduction to Entropy for Random Variables<br />
<br />
In these lectures I will introduce entropy for random variables and present some simple, finite state-space, examples to gain some intuition. We will prove the <br />
MacMillan Theorem using entropy and the law of large numbers. Then I will introduce relative entropy and prove the Markov Chain Convergence Theorem. Finally I will <br />
define entropy for a discrete time process. The lecture notes can be found at http://www.math.wisc.edu/~shottovy/EntropyLecture.pdf.<br />
<br />
2/17, 2/24: Dae Han<br />
<br />
3/3, 3/10: Hans<br />
<br />
3/17, 3/24: In Gun<br />
<br />
4/7, 4/14: Jinsu<br />
<br />
4/21, 4/28: Chris N.<br />
<br />
<br />
<br />
<br />
<br />
<br />
==2014 Fall==<br />
<br />
9/23: Dave<br />
<br />
I will go over Mike Giles’ 2008 paper “Multi-level Monte Carlo path simulation.” This paper introduced a new Monte Carlo method to approximate expectations of SDEs (driven by Brownian motions) that is significantly more efficient than what was the state of the art. This work opened up a whole new field in the numerical analysis of stochastic processes as the basic idea is quite flexible and has found a variety of applications including SDEs driven by Brownian motions, Levy-driven SDEs, SPDEs, and models from biology<br />
<br />
9/30: Benedek<br />
<br />
A very quick introduction to Stein's method. <br />
<br />
I will give a brief introduction to Stein's method, mostly based on the the first couple of sections of the following survey article:<br />
<br />
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210-293. <br />
<br />
The following webpage has a huge collection of resources if you want to go deeper: https://sites.google.com/site/yvikswan/about-stein-s-method<br />
<br />
<br />
Note that the Midwest Probability Colloquium (http://www.math.northwestern.edu/mwp/) will have a tutorial program on Stein's method this year. <br />
<br />
10/7, 10/14: Chris J.<br />
[http://www.math.wisc.edu/~janjigia/research/MartingaleProblemNotes.pdf An introduction to the (local) martingale problem.]<br />
<br />
<br />
10/21, 10/28: Dae Han<br />
<br />
11/4, 11/11: Elnur<br />
<br />
11/18, 11/25: Chris N. Free Probability with an emphasis on C* and Von Neumann Algebras<br />
<br />
12/2, 12/9: Yun Zhai<br />
<br />
==2014 Spring==<br />
<br />
<br />
1/28: Greg<br />
<br />
2/04, 2/11: Scott <br />
<br />
[http://www.math.wisc.edu/~shottovy/BLT.pdf Reflected Brownian motion, Occupation time, and applications.] <br />
<br />
2/18: Phil-- Examples of structure results in probability theory.<br />
<br />
2/25, 3/4: Beth-- Derivative estimation for discrete time Markov chains<br />
<br />
3/11, 3/25: Chris J [http://www.math.wisc.edu/~janjigia/research/stationarytalk.pdf Some classical results on stationary distributions of Markov processes]<br />
<br />
4/1, 4/8: Chris N <br />
<br />
4/15, 4/22: Yu Sun<br />
<br />
4/29. 5/6: Diane<br />
<br />
==2013 Fall==<br />
<br />
9/24, 10/1: Chris<br />
[http://www.math.wisc.edu/~janjigia/research/metastabilitytalk.pdf A light introduction to metastability]<br />
<br />
10/8, Dae Han<br />
Majoring multiplicative cascades for directed polymers in random media<br />
<br />
10/15, 10/22: no reading seminar<br />
<br />
10/29, 11/5: Elnur<br />
Limit fluctuations of last passage times <br />
<br />
11/12: Yun<br />
Helffer-Sjostrand representation and Brascamp-Lieb inequality for stochastic interface models<br />
<br />
11/19, 11/26: Yu Sun<br />
<br />
12/3, 12/10: Jason<br />
<br />
==2013 Spring==<br />
<br />
2/13: Elnur <br />
<br />
Young diagrams, RSK correspondence, corner growth models, distribution of last passage times. <br />
<br />
2/20: Elnur<br />
<br />
2/27: Chris<br />
<br />
A brief introduction to enlargement of filtration and the Dufresne identity<br />
[http://www.math.wisc.edu/~janjigia/research/Presentation%20Notes.pdf Notes]<br />
<br />
3/6: Chris<br />
<br />
3/13: Dae Han<br />
<br />
An introduction to random polymers<br />
<br />
3/20: Dae Han<br />
<br />
Directed polymers in a random environment: path localization and strong disorder<br />
<br />
4/3: Diane<br />
<br />
Scale and Speed for honest 1 dimensional diffusions<br />
<br />
References: <br><br />
Rogers & Williams - Diffusions, Markov Processes and Martingales <br><br />
Ito & McKean - Diffusion Processes and their Sample Paths <br><br />
Breiman - Probability <br><br />
http://www.statslab.cam.ac.uk/~beresty/Articles/diffusions.pdf<br />
<br />
4/10: Diane<br />
<br />
4/17: Yun<br />
<br />
Introduction to stochastic interface models<br />
<br />
4/24: Yun<br />
<br />
Dynamics and Gaussian equilibrium sytems<br />
<br />
5/1: This reading seminar will be shifted because of a probability seminar.<br />
<br />
<br />
5/8: Greg, Maso<br />
<br />
The Bethe ansatz vs. The Replica Trick. This lecture is an overview of the two <br />
approaches. See [http://arxiv.org/abs/1212.2267] for a nice overview.<br />
<br />
5/15: Greg, Maso<br />
<br />
Rigorous use of the replica trick.</div>Valko