https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Jessica&feedformat=atomUW-Math Wiki - User contributions [en]2024-03-29T01:19:34ZUser contributionsMediaWiki 1.39.5https://wiki.math.wisc.edu/index.php?title=Fall_2021_and_Spring_2022_Analysis_Seminars&diff=13666Fall 2021 and Spring 2022 Analysis Seminars2017-04-14T18:10:57Z<p>Jessica: /* Analysis Seminar Schedule Spring 2017 */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
[http://www.math.wisc.edu/~seeger/curr.html Current Semester]<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Andreas at seeger(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule Spring 2017 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 17, Math Department Colloquium<br />
| Fabio Pusateri (Princeton) <br />
|[[#Fabio Pusateri | The Water Waves Problem ]]<br />
| Sigurd Angenent<br />
|<br />
|-<br />
|January 24, Joint Analysis/Geometry Seminar<br />
| Tamás Darvas (Maryland) <br />
|[[#Tamás Darvas | Existence of constant scalar curvature Kähler metrics and properness of the K-energy ]]<br />
| Jeff Viaclovsky<br />
|<br />
|-<br />
|Monday, January 30, 3:30, VV901 (PDE Seminar)<br />
| Serguei Denissov (UW Madison)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid ]]<br />
| <br />
|-<br />
|February 7<br />
| Andreas Seeger (UW Madison)<br />
|[[#Andreas Seeger| The Haar system in Sobolev spaces]]<br />
|<br />
|-<br />
|February 21<br />
| Jongchon Kim (UW Madison)<br />
|[[#Jongchon Kim | Some remarks on Fourier restriction estimates ]]<br />
| Andreas Seeger<br />
|-<br />
|March 7, Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana) <br />
|[[#Roger Temam (Colloquium) | On the mathematical modeling of the humid atmosphere ]]<br />
| Leslie Smith<br />
|-<br />
|Wednesday, March 8, Joint Applied Math/PDE/Analysis Seminar<br />
| Roger Temam (Indiana) <br />
|[[#Roger Temam (Seminar) | Weak solutions of the Shigesada-Kawasaki-Teramoto system]]<br />
| Leslie Smith<br />
|-<br />
|March 14<br />
| Xianghong Chen (UW Milwaukee)<br />
|[[#Xianghong Chen | Restricting the Fourier transform to some oscillating curves ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|March 21<br />
| SPRING BREAK<br />
|[[#linktoabstract | ]]<br />
<br />
<br />
|-<br />
|Monday, March 27 (joint PDE/Analysis Seminar), 3:30, VV901<br />
| Sylvia Serfaty (NYU)<br />
|[[#Sylvia Serfaty |Mean Field Limits for Ginzburg Landau Vortices ]]<br />
| Hung Tran<br />
|<br />
|-<br />
|March 28<br />
| Brian Cook (Fields Institute) <br />
|[[#Brian Cook |Twists on the twisted ergodic theorems ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|Friday, March 31, 4:00 p.m., B139<br />
| Laura Cladek (UBC) <br />
|[[#Laura Cladek | Endpoint bounds for the lacunary spherical maximal operator ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|April 4<br />
| Francesco Di Plinio (Virginia)<br />
|[[#Francesco di Plinio | Sparse domination of singular integral operators ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|April 11<br />
| Xianghong Gong (UW Madison)<br />
|[[#lXianghong Gong | Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary ]]<br />
| <br />
|<br />
|-<br />
|April 25 (joint PDE/Analysis Seminar)<br />
| Chris Henderson (University of Chicago)<br />
|[[#lChris Henderson | A local-in-time Harnack inequality and applications to reaction-diffusion equations ]]<br />
| Jessica Lin<br />
|<br />
|-<br />
<br />
=Abstracts=<br />
<br />
=== Fabio Pusateri ===<br />
''The Water Waves problem''<br />
<br />
We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas.<br />
<br />
=== Tamás Darvas ===<br />
''Existence of constant scalar curvature Kähler metrics and properness of the K-energy''<br />
<br />
Given a compact Kähler manifold $(X,\omega)$, we show that if there exists a constant<br />
scalar curvature Kähler metric cohomologous to $\omega$ then Mabuchi's K-energy is J-proper in an<br />
appropriate sense, confirming a conjecture of Tian from the nineties. The proof involves a careful<br />
study of weak minimizers of the K-energy, and involves a surprising amount of analysis. This is<br />
joint work with Robert Berman and Chinh H. Lu.<br />
<br />
=== Serguei Denissov ===<br />
''Instability in 2D Euler equation of incompressible inviscid fluid''<br />
<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
=== Andreas Seeger ===<br />
''The Haar system in Sobolev spaces''<br />
<br />
We consider the Haar system on Sobolev spaces and ask:<br />
When is it a Schauder basis?<br />
When is it an unconditional basis?<br />
Some answers are given in recent joint work Tino Ullrich and Gustavo Garrigós.<br />
<br />
=== Jongchon Kim ===<br />
''Some remarks on Fourier restriction estimates''<br />
<br />
The Fourier restriction problem, raised by Stein in the 1960’s, is a hard open problem in harmonic analysis. Recently, Guth made some impressive progress on this problem using polynomial partitioning, a divide and conquer technique developed by Guth and Katz for some problems in incidence geometry.<br />
In this talk, I will introduce the restriction problem and the polynomial partitioning method. In addition, I will present some sharp L^p to L^q estimates for the Fourier extension operator that use an estimate of Guth as a black box.<br />
<br />
=== Roger Temam (Colloquium) ===<br />
''On the mathematical modeling of the humid atmosphere''<br />
<br />
The humid atmosphere is a multi-phase system, made of air, water vapor, cloud-condensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, non-continuous (and non-monotone) in the framework of nonlinear partial differential equations.<br />
We will discuss some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasi-variational inequalities.<br />
<br />
=== Roger Temam (Seminar) ===<br />
''Weak solutions of the Shigesada-Kawasaki-Teramoto system''<br />
<br />
We will present a result of existence of weak solutions to the Shigesada-Kawasaki-Teramoto system, in all dimensions. The method is based on new a priori estimates, the construction of approximate solutions and passage to the limit. The proof of existence is completely self-contained and does not rely on any earlier result.<br />
Based on an article with Du Pham, to appear in Nonlinear Analysis.<br />
<br />
=== Xianghong Chen ===<br />
''Restricting the Fourier transform to some oscillating curves''<br />
<br />
I will talk about Fourier restriction to some compact smooth curves. The problem is relatively well understood for curves with nonvanishing torsion due to work of Drury from the 80's, but is less so for curves that contain 'flat' points (i.e. vanishing torsion). Sharp results are known for some monomial-like or finite type curves by work of Bak-Oberlin-Seeger, Dendrinos-Mueller, and Stovall, where a geometric inequality (among others) plays an important role. Such an inequality fails to hold if the torsion demonstrates strong sign-changing behavior, in which case endpoint restriction bounds may fail. In this talk I will present how one could obtain sharp non-endpoint results for certain space curves of this kind. Our approach uses a covering lemma for smooth functions that strengthens a variation bound of Sjolin, who used it to obtain a similar result for plane curves. This is joint work with Dashan Fan and Lifeng Wang.<br />
<br />
===Sylvia Serfaty ===<br />
<br />
''Mean Field Limits for Ginzburg Landau Vortices''<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
<br />
=== Brian Cook ===<br />
''Twists on the twisted ergodic theorems''<br />
<br />
The classical pointwise ergodic theorem has been adapted to include averages twisted by a phase polynomial, primary examples being the ergodic theorems of Wiener-Wintner and Lesigne. Certain uniform versions of these results are also known. Here uniformity refers to the collection of polynomials of degree less than some prescribed number. In this talk we wish to consider weakening the hypothesis in these latter results by considering uniformity over a smaller class of polynomials, which is naturally motivated when considering certain applications related to the circle method.<br />
<br />
<br />
=== Laura Cladek ===<br />
''Endpoint bounds for the lacunary spherical maximal operator''<br />
<br />
Define the lacunary spherical maximal operator as the maximal operator corresponding to averages over spheres of radius 2^k for k an integer. This operator may be viewed as a model case for studying more general classes of singular maximal operators and Radon transforms. It is a classical result in harmonic analysis that this operator is bounded on L^p for p>1, but the question of weak-type (1, 1) boundedness (which would correspond to pointwise convergence of lacunary spherical averages for functions in L^1 has remained open. Although this question still remains open, we discuss some new endpoint bounds for the operator near L^1 that allows us to conclude almost everywhere pointwise convergence of lacunary spherical means for functions in a slightly smaller space than L\log\log\log L. This is based on joint work with Ben Krause.<br />
<br />
<br />
=== Francesco di Plinio ===<br />
''Sparse domination of singular integral operators''<br />
<br />
Singular integral operators, which are a priori signed and non-local, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means, and singular integrals along submanifolds with curvature. Collaborators: Amalia Culiuc, Laura Cladek, Jose Manuel Conde-Alonso, Yen Do, Yumeng Ou and Gennady Uraltsev.<br />
<br />
<br />
===Xianghong Gong===<br />
''Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary''<br />
<br />
Abstract: We derive a new homotopy formula for a bounded strictly pseudoconvex domain of C^2 boundary by using a method of Lieb and Range, and we obtain estimates for its homotopy operator. We show that the d-bar equation on the domain admits a solution gaining half-derivative in the Hoelder-Zygmund spaces. The estimates are also applied to obtain a boundary regularity for D-solutions on a suitable product domain in the Levi-flat Euclidean spaces.<br />
<br />
===Chris Henderson===<br />
''A local-in-time Harnack inequality and applications to reaction-diffusion equations''<br />
<br />
Abstract: The classical Harnack inequality requires one to look back in time to obtain a uniform lower bound on the solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Fall_2021_and_Spring_2022_Analysis_Seminars&diff=13665Fall 2021 and Spring 2022 Analysis Seminars2017-04-14T18:10:06Z<p>Jessica: /* Analysis Seminar Schedule Spring 2017 */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
[http://www.math.wisc.edu/~seeger/curr.html Current Semester]<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Andreas at seeger(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule Spring 2017 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 17, Math Department Colloquium<br />
| Fabio Pusateri (Princeton) <br />
|[[#Fabio Pusateri | The Water Waves Problem ]]<br />
| Sigurd Angenent<br />
|<br />
|-<br />
|January 24, Joint Analysis/Geometry Seminar<br />
| Tamás Darvas (Maryland) <br />
|[[#Tamás Darvas | Existence of constant scalar curvature Kähler metrics and properness of the K-energy ]]<br />
| Jeff Viaclovsky<br />
|<br />
|-<br />
|Monday, January 30, 3:30, VV901 (PDE Seminar)<br />
| Serguei Denissov (UW Madison)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid ]]<br />
| <br />
|-<br />
|February 7<br />
| Andreas Seeger (UW Madison)<br />
|[[#Andreas Seeger| The Haar system in Sobolev spaces]]<br />
|<br />
|-<br />
|February 21<br />
| Jongchon Kim (UW Madison)<br />
|[[#Jongchon Kim | Some remarks on Fourier restriction estimates ]]<br />
| Andreas Seeger<br />
|-<br />
|March 7, Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana) <br />
|[[#Roger Temam (Colloquium) | On the mathematical modeling of the humid atmosphere ]]<br />
| Leslie Smith<br />
|-<br />
|Wednesday, March 8, Joint Applied Math/PDE/Analysis Seminar<br />
| Roger Temam (Indiana) <br />
|[[#Roger Temam (Seminar) | Weak solutions of the Shigesada-Kawasaki-Teramoto system]]<br />
| Leslie Smith<br />
|-<br />
|March 14<br />
| Xianghong Chen (UW Milwaukee)<br />
|[[#Xianghong Chen | Restricting the Fourier transform to some oscillating curves ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|March 21<br />
| SPRING BREAK<br />
|[[#linktoabstract | ]]<br />
<br />
<br />
|-<br />
|Monday, March 27 (joint PDE/Analysis Seminar), 3:30, VV901<br />
| Sylvia Serfaty (NYU)<br />
|[[#Sylvia Serfaty |Mean Field Limits for Ginzburg Landau Vortices ]]<br />
| Hung Tran<br />
|<br />
|-<br />
|March 28<br />
| Brian Cook (Fields Institute) <br />
|[[#Brian Cook |Twists on the twisted ergodic theorems ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|Friday, March 31, 4:00 p.m., B139<br />
| Laura Cladek (UBC) <br />
|[[#Laura Cladek | Endpoint bounds for the lacunary spherical maximal operator ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|April 4<br />
| Francesco Di Plinio (Virginia)<br />
|[[#Francesco di Plinio | Sparse domination of singular integral operators ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|April 11<br />
| Xianghong Gong (UW Madison)<br />
|[[#lXianghong Gong | Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary ]]<br />
| <br />
|<br />
|-<br />
|April 25 (joint PDE/Analysis Seminar)<br />
| Chris Henderson (University of Chicago)<br />
|[[#|Chris Henderson | A local-in-time Harnack inequality and applications to reaction-diffusion equations]<br />
| Jessica Lin<br />
|}<br />
<br />
=Abstracts=<br />
<br />
=== Fabio Pusateri ===<br />
''The Water Waves problem''<br />
<br />
We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas.<br />
<br />
=== Tamás Darvas ===<br />
''Existence of constant scalar curvature Kähler metrics and properness of the K-energy''<br />
<br />
Given a compact Kähler manifold $(X,\omega)$, we show that if there exists a constant<br />
scalar curvature Kähler metric cohomologous to $\omega$ then Mabuchi's K-energy is J-proper in an<br />
appropriate sense, confirming a conjecture of Tian from the nineties. The proof involves a careful<br />
study of weak minimizers of the K-energy, and involves a surprising amount of analysis. This is<br />
joint work with Robert Berman and Chinh H. Lu.<br />
<br />
=== Serguei Denissov ===<br />
''Instability in 2D Euler equation of incompressible inviscid fluid''<br />
<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
=== Andreas Seeger ===<br />
''The Haar system in Sobolev spaces''<br />
<br />
We consider the Haar system on Sobolev spaces and ask:<br />
When is it a Schauder basis?<br />
When is it an unconditional basis?<br />
Some answers are given in recent joint work Tino Ullrich and Gustavo Garrigós.<br />
<br />
=== Jongchon Kim ===<br />
''Some remarks on Fourier restriction estimates''<br />
<br />
The Fourier restriction problem, raised by Stein in the 1960’s, is a hard open problem in harmonic analysis. Recently, Guth made some impressive progress on this problem using polynomial partitioning, a divide and conquer technique developed by Guth and Katz for some problems in incidence geometry.<br />
In this talk, I will introduce the restriction problem and the polynomial partitioning method. In addition, I will present some sharp L^p to L^q estimates for the Fourier extension operator that use an estimate of Guth as a black box.<br />
<br />
=== Roger Temam (Colloquium) ===<br />
''On the mathematical modeling of the humid atmosphere''<br />
<br />
The humid atmosphere is a multi-phase system, made of air, water vapor, cloud-condensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, non-continuous (and non-monotone) in the framework of nonlinear partial differential equations.<br />
We will discuss some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasi-variational inequalities.<br />
<br />
=== Roger Temam (Seminar) ===<br />
''Weak solutions of the Shigesada-Kawasaki-Teramoto system''<br />
<br />
We will present a result of existence of weak solutions to the Shigesada-Kawasaki-Teramoto system, in all dimensions. The method is based on new a priori estimates, the construction of approximate solutions and passage to the limit. The proof of existence is completely self-contained and does not rely on any earlier result.<br />
Based on an article with Du Pham, to appear in Nonlinear Analysis.<br />
<br />
=== Xianghong Chen ===<br />
''Restricting the Fourier transform to some oscillating curves''<br />
<br />
I will talk about Fourier restriction to some compact smooth curves. The problem is relatively well understood for curves with nonvanishing torsion due to work of Drury from the 80's, but is less so for curves that contain 'flat' points (i.e. vanishing torsion). Sharp results are known for some monomial-like or finite type curves by work of Bak-Oberlin-Seeger, Dendrinos-Mueller, and Stovall, where a geometric inequality (among others) plays an important role. Such an inequality fails to hold if the torsion demonstrates strong sign-changing behavior, in which case endpoint restriction bounds may fail. In this talk I will present how one could obtain sharp non-endpoint results for certain space curves of this kind. Our approach uses a covering lemma for smooth functions that strengthens a variation bound of Sjolin, who used it to obtain a similar result for plane curves. This is joint work with Dashan Fan and Lifeng Wang.<br />
<br />
===Sylvia Serfaty ===<br />
<br />
''Mean Field Limits for Ginzburg Landau Vortices''<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
<br />
=== Brian Cook ===<br />
''Twists on the twisted ergodic theorems''<br />
<br />
The classical pointwise ergodic theorem has been adapted to include averages twisted by a phase polynomial, primary examples being the ergodic theorems of Wiener-Wintner and Lesigne. Certain uniform versions of these results are also known. Here uniformity refers to the collection of polynomials of degree less than some prescribed number. In this talk we wish to consider weakening the hypothesis in these latter results by considering uniformity over a smaller class of polynomials, which is naturally motivated when considering certain applications related to the circle method.<br />
<br />
<br />
=== Laura Cladek ===<br />
''Endpoint bounds for the lacunary spherical maximal operator''<br />
<br />
Define the lacunary spherical maximal operator as the maximal operator corresponding to averages over spheres of radius 2^k for k an integer. This operator may be viewed as a model case for studying more general classes of singular maximal operators and Radon transforms. It is a classical result in harmonic analysis that this operator is bounded on L^p for p>1, but the question of weak-type (1, 1) boundedness (which would correspond to pointwise convergence of lacunary spherical averages for functions in L^1 has remained open. Although this question still remains open, we discuss some new endpoint bounds for the operator near L^1 that allows us to conclude almost everywhere pointwise convergence of lacunary spherical means for functions in a slightly smaller space than L\log\log\log L. This is based on joint work with Ben Krause.<br />
<br />
<br />
=== Francesco di Plinio ===<br />
''Sparse domination of singular integral operators''<br />
<br />
Singular integral operators, which are a priori signed and non-local, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means, and singular integrals along submanifolds with curvature. Collaborators: Amalia Culiuc, Laura Cladek, Jose Manuel Conde-Alonso, Yen Do, Yumeng Ou and Gennady Uraltsev.<br />
<br />
<br />
===Xianghong Gong===<br />
''Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary''<br />
<br />
Abstract: We derive a new homotopy formula for a bounded strictly pseudoconvex domain of C^2 boundary by using a method of Lieb and Range, and we obtain estimates for its homotopy operator. We show that the d-bar equation on the domain admits a solution gaining half-derivative in the Hoelder-Zygmund spaces. The estimates are also applied to obtain a boundary regularity for D-solutions on a suitable product domain in the Levi-flat Euclidean spaces.<br />
<br />
===Chris Henderson===<br />
''A local-in-time Harnack inequality and applications to reaction-diffusion equations''<br />
<br />
Abstract: The classical Harnack inequality requires one to look back in time to obtain a uniform lower bound on the solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Fall_2021_and_Spring_2022_Analysis_Seminars&diff=13664Fall 2021 and Spring 2022 Analysis Seminars2017-04-14T18:09:42Z<p>Jessica: /* Abstracts */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
[http://www.math.wisc.edu/~seeger/curr.html Current Semester]<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Andreas at seeger(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule Spring 2017 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 17, Math Department Colloquium<br />
| Fabio Pusateri (Princeton) <br />
|[[#Fabio Pusateri | The Water Waves Problem ]]<br />
| Sigurd Angenent<br />
|<br />
|-<br />
|January 24, Joint Analysis/Geometry Seminar<br />
| Tamás Darvas (Maryland) <br />
|[[#Tamás Darvas | Existence of constant scalar curvature Kähler metrics and properness of the K-energy ]]<br />
| Jeff Viaclovsky<br />
|<br />
|-<br />
|Monday, January 30, 3:30, VV901 (PDE Seminar)<br />
| Serguei Denissov (UW Madison)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid ]]<br />
| <br />
|-<br />
|February 7<br />
| Andreas Seeger (UW Madison)<br />
|[[#Andreas Seeger| The Haar system in Sobolev spaces]]<br />
|<br />
|-<br />
|February 21<br />
| Jongchon Kim (UW Madison)<br />
|[[#Jongchon Kim | Some remarks on Fourier restriction estimates ]]<br />
| Andreas Seeger<br />
|-<br />
|March 7, Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana) <br />
|[[#Roger Temam (Colloquium) | On the mathematical modeling of the humid atmosphere ]]<br />
| Leslie Smith<br />
|-<br />
|Wednesday, March 8, Joint Applied Math/PDE/Analysis Seminar<br />
| Roger Temam (Indiana) <br />
|[[#Roger Temam (Seminar) | Weak solutions of the Shigesada-Kawasaki-Teramoto system]]<br />
| Leslie Smith<br />
|-<br />
|March 14<br />
| Xianghong Chen (UW Milwaukee)<br />
|[[#Xianghong Chen | Restricting the Fourier transform to some oscillating curves ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|March 21<br />
| SPRING BREAK<br />
|[[#linktoabstract | ]]<br />
<br />
<br />
|-<br />
|Monday, March 27 (joint PDE/Analysis Seminar), 3:30, VV901<br />
| Sylvia Serfaty (NYU)<br />
|[[#Sylvia Serfaty |Mean Field Limits for Ginzburg Landau Vortices ]]<br />
| Hung Tran<br />
|<br />
|-<br />
|March 28<br />
| Brian Cook (Fields Institute) <br />
|[[#Brian Cook |Twists on the twisted ergodic theorems ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|Friday, March 31, 4:00 p.m., B139<br />
| Laura Cladek (UBC) <br />
|[[#Laura Cladek | Endpoint bounds for the lacunary spherical maximal operator ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|April 4<br />
| Francesco Di Plinio (Virginia)<br />
|[[#Francesco di Plinio | Sparse domination of singular integral operators ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|April 11<br />
| Xianghong Gong (UW Madison)<br />
|[[#lXianghong Gong | Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary ]]<br />
| <br />
|<br />
|-<br />
|April 25 (joint PDE/Analysis Seminar)<br />
| Chris Henderson (University of Chicago)<br />
|[[#Chris Henderson | A local-in-time Harnack inequality and applications to reaction-diffusion equations]<br />
| Jessica Lin<br />
|}<br />
<br />
=Abstracts=<br />
<br />
=== Fabio Pusateri ===<br />
''The Water Waves problem''<br />
<br />
We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas.<br />
<br />
=== Tamás Darvas ===<br />
''Existence of constant scalar curvature Kähler metrics and properness of the K-energy''<br />
<br />
Given a compact Kähler manifold $(X,\omega)$, we show that if there exists a constant<br />
scalar curvature Kähler metric cohomologous to $\omega$ then Mabuchi's K-energy is J-proper in an<br />
appropriate sense, confirming a conjecture of Tian from the nineties. The proof involves a careful<br />
study of weak minimizers of the K-energy, and involves a surprising amount of analysis. This is<br />
joint work with Robert Berman and Chinh H. Lu.<br />
<br />
=== Serguei Denissov ===<br />
''Instability in 2D Euler equation of incompressible inviscid fluid''<br />
<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
=== Andreas Seeger ===<br />
''The Haar system in Sobolev spaces''<br />
<br />
We consider the Haar system on Sobolev spaces and ask:<br />
When is it a Schauder basis?<br />
When is it an unconditional basis?<br />
Some answers are given in recent joint work Tino Ullrich and Gustavo Garrigós.<br />
<br />
=== Jongchon Kim ===<br />
''Some remarks on Fourier restriction estimates''<br />
<br />
The Fourier restriction problem, raised by Stein in the 1960’s, is a hard open problem in harmonic analysis. Recently, Guth made some impressive progress on this problem using polynomial partitioning, a divide and conquer technique developed by Guth and Katz for some problems in incidence geometry.<br />
In this talk, I will introduce the restriction problem and the polynomial partitioning method. In addition, I will present some sharp L^p to L^q estimates for the Fourier extension operator that use an estimate of Guth as a black box.<br />
<br />
=== Roger Temam (Colloquium) ===<br />
''On the mathematical modeling of the humid atmosphere''<br />
<br />
The humid atmosphere is a multi-phase system, made of air, water vapor, cloud-condensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, non-continuous (and non-monotone) in the framework of nonlinear partial differential equations.<br />
We will discuss some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasi-variational inequalities.<br />
<br />
=== Roger Temam (Seminar) ===<br />
''Weak solutions of the Shigesada-Kawasaki-Teramoto system''<br />
<br />
We will present a result of existence of weak solutions to the Shigesada-Kawasaki-Teramoto system, in all dimensions. The method is based on new a priori estimates, the construction of approximate solutions and passage to the limit. The proof of existence is completely self-contained and does not rely on any earlier result.<br />
Based on an article with Du Pham, to appear in Nonlinear Analysis.<br />
<br />
=== Xianghong Chen ===<br />
''Restricting the Fourier transform to some oscillating curves''<br />
<br />
I will talk about Fourier restriction to some compact smooth curves. The problem is relatively well understood for curves with nonvanishing torsion due to work of Drury from the 80's, but is less so for curves that contain 'flat' points (i.e. vanishing torsion). Sharp results are known for some monomial-like or finite type curves by work of Bak-Oberlin-Seeger, Dendrinos-Mueller, and Stovall, where a geometric inequality (among others) plays an important role. Such an inequality fails to hold if the torsion demonstrates strong sign-changing behavior, in which case endpoint restriction bounds may fail. In this talk I will present how one could obtain sharp non-endpoint results for certain space curves of this kind. Our approach uses a covering lemma for smooth functions that strengthens a variation bound of Sjolin, who used it to obtain a similar result for plane curves. This is joint work with Dashan Fan and Lifeng Wang.<br />
<br />
===Sylvia Serfaty ===<br />
<br />
''Mean Field Limits for Ginzburg Landau Vortices''<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
<br />
=== Brian Cook ===<br />
''Twists on the twisted ergodic theorems''<br />
<br />
The classical pointwise ergodic theorem has been adapted to include averages twisted by a phase polynomial, primary examples being the ergodic theorems of Wiener-Wintner and Lesigne. Certain uniform versions of these results are also known. Here uniformity refers to the collection of polynomials of degree less than some prescribed number. In this talk we wish to consider weakening the hypothesis in these latter results by considering uniformity over a smaller class of polynomials, which is naturally motivated when considering certain applications related to the circle method.<br />
<br />
<br />
=== Laura Cladek ===<br />
''Endpoint bounds for the lacunary spherical maximal operator''<br />
<br />
Define the lacunary spherical maximal operator as the maximal operator corresponding to averages over spheres of radius 2^k for k an integer. This operator may be viewed as a model case for studying more general classes of singular maximal operators and Radon transforms. It is a classical result in harmonic analysis that this operator is bounded on L^p for p>1, but the question of weak-type (1, 1) boundedness (which would correspond to pointwise convergence of lacunary spherical averages for functions in L^1 has remained open. Although this question still remains open, we discuss some new endpoint bounds for the operator near L^1 that allows us to conclude almost everywhere pointwise convergence of lacunary spherical means for functions in a slightly smaller space than L\log\log\log L. This is based on joint work with Ben Krause.<br />
<br />
<br />
=== Francesco di Plinio ===<br />
''Sparse domination of singular integral operators''<br />
<br />
Singular integral operators, which are a priori signed and non-local, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means, and singular integrals along submanifolds with curvature. Collaborators: Amalia Culiuc, Laura Cladek, Jose Manuel Conde-Alonso, Yen Do, Yumeng Ou and Gennady Uraltsev.<br />
<br />
<br />
===Xianghong Gong===<br />
''Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary''<br />
<br />
Abstract: We derive a new homotopy formula for a bounded strictly pseudoconvex domain of C^2 boundary by using a method of Lieb and Range, and we obtain estimates for its homotopy operator. We show that the d-bar equation on the domain admits a solution gaining half-derivative in the Hoelder-Zygmund spaces. The estimates are also applied to obtain a boundary regularity for D-solutions on a suitable product domain in the Levi-flat Euclidean spaces.<br />
<br />
===Chris Henderson===<br />
''A local-in-time Harnack inequality and applications to reaction-diffusion equations''<br />
<br />
Abstract: The classical Harnack inequality requires one to look back in time to obtain a uniform lower bound on the solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Fall_2021_and_Spring_2022_Analysis_Seminars&diff=13663Fall 2021 and Spring 2022 Analysis Seminars2017-04-14T18:08:54Z<p>Jessica: /* Analysis Seminar Schedule Spring 2017 */</p>
<hr />
<div>'''Analysis Seminar<br />
'''<br />
[http://www.math.wisc.edu/~seeger/curr.html Current Semester]<br />
<br />
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.<br />
<br />
If you wish to invite a speaker please contact Andreas at seeger(at)math<br />
<br />
===[[Previous Analysis seminars]]===<br />
<br />
= Analysis Seminar Schedule Spring 2017 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 17, Math Department Colloquium<br />
| Fabio Pusateri (Princeton) <br />
|[[#Fabio Pusateri | The Water Waves Problem ]]<br />
| Sigurd Angenent<br />
|<br />
|-<br />
|January 24, Joint Analysis/Geometry Seminar<br />
| Tamás Darvas (Maryland) <br />
|[[#Tamás Darvas | Existence of constant scalar curvature Kähler metrics and properness of the K-energy ]]<br />
| Jeff Viaclovsky<br />
|<br />
|-<br />
|Monday, January 30, 3:30, VV901 (PDE Seminar)<br />
| Serguei Denissov (UW Madison)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid ]]<br />
| <br />
|-<br />
|February 7<br />
| Andreas Seeger (UW Madison)<br />
|[[#Andreas Seeger| The Haar system in Sobolev spaces]]<br />
|<br />
|-<br />
|February 21<br />
| Jongchon Kim (UW Madison)<br />
|[[#Jongchon Kim | Some remarks on Fourier restriction estimates ]]<br />
| Andreas Seeger<br />
|-<br />
|March 7, Mathematics Department Distinguished Lecture<br />
| Roger Temam (Indiana) <br />
|[[#Roger Temam (Colloquium) | On the mathematical modeling of the humid atmosphere ]]<br />
| Leslie Smith<br />
|-<br />
|Wednesday, March 8, Joint Applied Math/PDE/Analysis Seminar<br />
| Roger Temam (Indiana) <br />
|[[#Roger Temam (Seminar) | Weak solutions of the Shigesada-Kawasaki-Teramoto system]]<br />
| Leslie Smith<br />
|-<br />
|March 14<br />
| Xianghong Chen (UW Milwaukee)<br />
|[[#Xianghong Chen | Restricting the Fourier transform to some oscillating curves ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|March 21<br />
| SPRING BREAK<br />
|[[#linktoabstract | ]]<br />
<br />
<br />
|-<br />
|Monday, March 27 (joint PDE/Analysis Seminar), 3:30, VV901<br />
| Sylvia Serfaty (NYU)<br />
|[[#Sylvia Serfaty |Mean Field Limits for Ginzburg Landau Vortices ]]<br />
| Hung Tran<br />
|<br />
|-<br />
|March 28<br />
| Brian Cook (Fields Institute) <br />
|[[#Brian Cook |Twists on the twisted ergodic theorems ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|Friday, March 31, 4:00 p.m., B139<br />
| Laura Cladek (UBC) <br />
|[[#Laura Cladek | Endpoint bounds for the lacunary spherical maximal operator ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|April 4<br />
| Francesco Di Plinio (Virginia)<br />
|[[#Francesco di Plinio | Sparse domination of singular integral operators ]]<br />
| Andreas Seeger<br />
|<br />
|-<br />
|April 11<br />
| Xianghong Gong (UW Madison)<br />
|[[#lXianghong Gong | Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary ]]<br />
| <br />
|<br />
|-<br />
|April 25 (joint PDE/Analysis Seminar)<br />
| Chris Henderson (University of Chicago)<br />
|[[#Chris Henderson | A local-in-time Harnack inequality and applications to reaction-diffusion equations]<br />
| Jessica Lin<br />
|}<br />
<br />
=Abstracts=<br />
<br />
=== Fabio Pusateri ===<br />
''The Water Waves problem''<br />
<br />
We will begin by introducing the free boundary Euler equations which are a system of nonlinear PDEs modeling the motion of fluids, such as waves on the surface of the ocean. We will discuss several works done on this system in recent years, and how they fit into the broader context of the study of nonlinear evolution problems. We will then focus on the question of global regularity for water waves, present some of our main results - obtained in collaboration with Ionescu and Deng-Ionescu-Pausader - and sketch some of the main ideas.<br />
<br />
=== Tamás Darvas ===<br />
''Existence of constant scalar curvature Kähler metrics and properness of the K-energy''<br />
<br />
Given a compact Kähler manifold $(X,\omega)$, we show that if there exists a constant<br />
scalar curvature Kähler metric cohomologous to $\omega$ then Mabuchi's K-energy is J-proper in an<br />
appropriate sense, confirming a conjecture of Tian from the nineties. The proof involves a careful<br />
study of weak minimizers of the K-energy, and involves a surprising amount of analysis. This is<br />
joint work with Robert Berman and Chinh H. Lu.<br />
<br />
=== Serguei Denissov ===<br />
''Instability in 2D Euler equation of incompressible inviscid fluid''<br />
<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
=== Andreas Seeger ===<br />
''The Haar system in Sobolev spaces''<br />
<br />
We consider the Haar system on Sobolev spaces and ask:<br />
When is it a Schauder basis?<br />
When is it an unconditional basis?<br />
Some answers are given in recent joint work Tino Ullrich and Gustavo Garrigós.<br />
<br />
=== Jongchon Kim ===<br />
''Some remarks on Fourier restriction estimates''<br />
<br />
The Fourier restriction problem, raised by Stein in the 1960’s, is a hard open problem in harmonic analysis. Recently, Guth made some impressive progress on this problem using polynomial partitioning, a divide and conquer technique developed by Guth and Katz for some problems in incidence geometry.<br />
In this talk, I will introduce the restriction problem and the polynomial partitioning method. In addition, I will present some sharp L^p to L^q estimates for the Fourier extension operator that use an estimate of Guth as a black box.<br />
<br />
=== Roger Temam (Colloquium) ===<br />
''On the mathematical modeling of the humid atmosphere''<br />
<br />
The humid atmosphere is a multi-phase system, made of air, water vapor, cloud-condensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, non-continuous (and non-monotone) in the framework of nonlinear partial differential equations.<br />
We will discuss some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasi-variational inequalities.<br />
<br />
=== Roger Temam (Seminar) ===<br />
''Weak solutions of the Shigesada-Kawasaki-Teramoto system''<br />
<br />
We will present a result of existence of weak solutions to the Shigesada-Kawasaki-Teramoto system, in all dimensions. The method is based on new a priori estimates, the construction of approximate solutions and passage to the limit. The proof of existence is completely self-contained and does not rely on any earlier result.<br />
Based on an article with Du Pham, to appear in Nonlinear Analysis.<br />
<br />
=== Xianghong Chen ===<br />
''Restricting the Fourier transform to some oscillating curves''<br />
<br />
I will talk about Fourier restriction to some compact smooth curves. The problem is relatively well understood for curves with nonvanishing torsion due to work of Drury from the 80's, but is less so for curves that contain 'flat' points (i.e. vanishing torsion). Sharp results are known for some monomial-like or finite type curves by work of Bak-Oberlin-Seeger, Dendrinos-Mueller, and Stovall, where a geometric inequality (among others) plays an important role. Such an inequality fails to hold if the torsion demonstrates strong sign-changing behavior, in which case endpoint restriction bounds may fail. In this talk I will present how one could obtain sharp non-endpoint results for certain space curves of this kind. Our approach uses a covering lemma for smooth functions that strengthens a variation bound of Sjolin, who used it to obtain a similar result for plane curves. This is joint work with Dashan Fan and Lifeng Wang.<br />
<br />
===Sylvia Serfaty ===<br />
<br />
''Mean Field Limits for Ginzburg Landau Vortices''<br />
<br />
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.<br />
<br />
<br />
<br />
=== Brian Cook ===<br />
''Twists on the twisted ergodic theorems''<br />
<br />
The classical pointwise ergodic theorem has been adapted to include averages twisted by a phase polynomial, primary examples being the ergodic theorems of Wiener-Wintner and Lesigne. Certain uniform versions of these results are also known. Here uniformity refers to the collection of polynomials of degree less than some prescribed number. In this talk we wish to consider weakening the hypothesis in these latter results by considering uniformity over a smaller class of polynomials, which is naturally motivated when considering certain applications related to the circle method.<br />
<br />
<br />
=== Laura Cladek ===<br />
''Endpoint bounds for the lacunary spherical maximal operator''<br />
<br />
Define the lacunary spherical maximal operator as the maximal operator corresponding to averages over spheres of radius 2^k for k an integer. This operator may be viewed as a model case for studying more general classes of singular maximal operators and Radon transforms. It is a classical result in harmonic analysis that this operator is bounded on L^p for p>1, but the question of weak-type (1, 1) boundedness (which would correspond to pointwise convergence of lacunary spherical averages for functions in L^1 has remained open. Although this question still remains open, we discuss some new endpoint bounds for the operator near L^1 that allows us to conclude almost everywhere pointwise convergence of lacunary spherical means for functions in a slightly smaller space than L\log\log\log L. This is based on joint work with Ben Krause.<br />
<br />
<br />
=== Francesco di Plinio ===<br />
''Sparse domination of singular integral operators''<br />
<br />
Singular integral operators, which are a priori signed and non-local, can be dominated in norm, pointwise, or dually, by sparse averaging operators, which are in contrast positive and localized. The most striking consequence is that weighted norm inequalities for the singular integral follow from the corresponding, rather immediate estimates for the averaging operators. In this talk, we present several positive sparse domination results of singular integrals falling beyond the scope of classical Calderón-Zygmund theory; notably, modulation invariant multilinear singular integrals including the bilinear Hilbert transforms, variation norm Carleson operators, matrix-valued kernels, rough homogeneous singular integrals and critical Bochner-Riesz means, and singular integrals along submanifolds with curvature. Collaborators: Amalia Culiuc, Laura Cladek, Jose Manuel Conde-Alonso, Yen Do, Yumeng Ou and Gennady Uraltsev.<br />
<br />
<br />
===Xianghong Gong===<br />
''Hoelder estimates for homotopy operators on strictly pseudoconvex domains with C^2 boundary''<br />
<br />
Abstract: We derive a new homotopy formula for a bounded strictly pseudoconvex domain of C^2 boundary by using a method of Lieb and Range, and we obtain estimates for its homotopy operator. We show that the d-bar equation on the domain admits a solution gaining half-derivative in the Hoelder-Zygmund spaces. The estimates are also applied to obtain a boundary regularity for D-solutions on a suitable product domain in the Levi-flat Euclidean spaces.<br />
<br />
=Extras=<br />
[[Blank Analysis Seminar Template]]</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=13290PDE Geometric Analysis seminar2017-02-06T20:44:24Z<p>Jessica: /* PDE GA Seminar Schedule Spring 2017 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2017]]===<br />
<br />
= PDE GA Seminar Schedule Spring 2017 =<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|-<br />
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck<br />
| Sigurd Angenent (UW)<br />
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|January 30<br />
| Serguei Denissov (UW)<br />
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]<br />
| Kim & Tran<br />
|- <br />
<br />
<br />
|-<br />
|February 6<br />
| Benoit Perthame (University of Paris VI)<br />
|[[#| ]]<br />
| Wasow lecture<br />
|- <br />
<br />
<br />
|-<br />
|February 13<br />
| Bing Wang (UW)<br />
|[[#Bing Wang | The extension problem of the mean curvature flow]]<br />
| Kim & Tran<br />
|- <br />
<br />
|-<br />
|February 20<br />
| <br />
|<br />
| <br />
|- <br />
<br />
|-<br />
|February 27<br />
| Ben Seeger (University of Chicago)<br />
|[[#Ben Seeger | ]]<br />
| Tran<br />
|- <br />
<br />
|-<br />
|March 7 - Applied math/PDE/Analysis seminar<br />
| Roger Temam (Indiana University) <br />
|[[#| ]]<br />
| Mathematics Department Distinguished Lecture <br />
|- <br />
<br />
<br />
|-<br />
|March 8 - Applied math/PDE/Analysis seminar<br />
| Roger Temam (Indiana University) <br />
|[[#| ]]<br />
| Mathematics Department Distinguished Lecture <br />
|-<br />
<br />
|-<br />
|March 13<br />
| Sona Akopian (UT-Austin)<br />
|[[#Sona Akopian | ]]<br />
| Kim<br />
<br />
|-<br />
|March 27 - Analysis/PDE seminar<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | ]]<br />
| Tran<br />
<br />
|-<br />
|March 29<br />
| Sylvia Serfaty (Courant)<br />
|[[#Sylvia Serfaty | ]]<br />
| Wasow lecture<br />
<br />
|-<br />
|April 3<br />
| Zhenfu Wang (Maryland)<br />
|[[#Zhenfu Wang | ]]<br />
| Kim<br />
<br />
|-<br />
|April 10<br />
| Andrei Tarfulea (Chicago)<br />
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]<br />
| Baer<br />
<br />
<br />
|-<br />
|April 24<br />
| Chris Henderson (Chicago)<br />
|[[#Chris Henderson | TBA]]<br />
| Lin<br />
<br />
|-<br />
|May 1st<br />
| Jeffrey Streets (UC-Irvine)<br />
|[[#Jeffrey Streets | ]]<br />
| Bing Wang<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Sigurd Angenent===<br />
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo; In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.<br />
<br />
<br />
===Serguei Denissov===<br />
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.<br />
<br />
<br />
===Andrei Tarfulea===<br />
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and<br />
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.<br />
<br />
===Bing Wang===<br />
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11732PDE Geometric Analysis seminar2016-04-03T23:41:02Z<p>Jessica: /* Seminar Schedule Spring 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
|Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
|Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
|Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
|Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| No seminar<br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
|Misha Feldman (UW)<br />
|[[#Misha Feldman | Shock reflection, free boundary problems and degenerate elliptic equations ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
|Sergey Bolotin (UW)<br />
|[[#Sergey Bolotin | Degenerate billiards in celestial mechanics]]<br />
|<br />
|-<br />
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations<br />
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 3 (Joint Analysis-PDE seminar )<br />
|Stanley Snelson (University of Chicago) <br />
|[[# | ]]<br />
| Seeger & Tran.<br />
|-<br />
|May 16-20, Conference in Harmonic Analysis in Honor of Michael Christ<br />
|Link: https://www.math.wisc.edu/ha_2016/<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.<br />
<br />
===Nestor Guillen===<br />
<br />
Min-max formulas for integro-differential equations and applications<br />
<br />
We show under minimal assumptions that a nonlinear operator satisfying what is known as a "global comparison principle" can be represented by a min-max formula in terms of very special linear operators (Levy operators, which involve drift-diffusion and integro-differential terms). Such type of formulas have been very useful in the theory of second order equations -for instance, by allowing the representation of solutions as value functions for differential games. Applications include results on the structure of Dirichlet-to-Neumann mappings for fully nonlinear second order elliptic equations.<br />
<br />
===Ryan Denlinger===<br />
<br />
The propagation of chaos for a rarefied gas of hard spheres in vacuum<br />
<br />
We are interested in the rigorous mathematical justification of<br />
Boltzmann's equation starting from the deterministic evolution of<br />
many-particle systems. O. E. Lanford was able to derive Boltzmann's<br />
equation for hard spheres, in the Boltzmann-Grad scaling, on a short<br />
time interval. Improvements to the time in Lanford's theorem have so far<br />
either relied on a small data hypothesis, or have been restricted to<br />
linear regimes. We revisit the small data regime, i.e. a sufficiently<br />
dilute gas of hard spheres dispersing into vacuum; this is a regime<br />
where strong bounds are available globally in time. Subject to the<br />
existence of such bounds, we give a rigorous proof for the propagation<br />
of Boltzmann's ``one-sided'' molecular chaos.<br />
<br />
===Misha Feldman===<br />
<br />
Shock reflection, free boundary problems and degenerate elliptic equations.<br />
<br />
Abstract: We will discuss shock reflection problem for compressible gas dynamics, and von Neumann<br />
conjectures on transition between regular and Mach reflections. We will discuss existence of solutions <br />
of regular reflection structure for potential flow equation, and also regularity of solutions, and<br />
properties of the shock curve (free boundary). Our approach is to reduce the shock reflection problem<br />
to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems <br />
will also be discussed, including uniqueness.<br />
The talk is based on the joint works with Gui-Qiang Chen, Myoungjean Bae and Wei Xiang.<br />
<br />
===Jessica Lin===<br />
<br />
Optimal Quantitative Error Estimates in Stochastic<br />
Homogenization for Elliptic Equations in Nondivergence Form<br />
<br />
Abstract: I will present optimal quantitative error estimates in the<br />
stochastic homogenization for uniformly elliptic equations in<br />
nondivergence form. From the point of view of probability theory,<br />
stochastic homogenization is equivalent to identifying a quenched<br />
invariance principle for random walks in a balanced random<br />
environment. Under strong independence assumptions on the environment,<br />
the main argument relies on establishing an exponential version of the<br />
Efron-Stein inequality. As an artifact of the optimal error estimates,<br />
we obtain a regularity theory down to microscopic scale, which implies<br />
estimates on the local integrability of the invariant measure<br />
associated to the process. This talk is based on joint work with Scott<br />
Armstrong.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11731PDE Geometric Analysis seminar2016-04-03T23:40:42Z<p>Jessica: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
|Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
|Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
|Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
|Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| No seminar<br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
|Misha Feldman (UW)<br />
|[[#Misha Feldman | Shock reflection, free boundary problems and degenerate elliptic equations ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
|Sergey Bolotin (UW)<br />
|[[#Sergey Bolotin | Degenerate billiards in celestial mechanics]]<br />
|<br />
|-<br />
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations<br />
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 3 (Joint Analysis-PDE seminar )<br />
|Stanley Snelson (University of Chicago) <br />
|[[# | ]]<br />
| Seeger & Tran.<br />
|-<br />
|May 16-20, Conference in Harmonic Analysis in Honor of Michael Christ<br />
|Link: https://www.math.wisc.edu/ha_2016/<br />
|-<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.<br />
<br />
===Nestor Guillen===<br />
<br />
Min-max formulas for integro-differential equations and applications<br />
<br />
We show under minimal assumptions that a nonlinear operator satisfying what is known as a "global comparison principle" can be represented by a min-max formula in terms of very special linear operators (Levy operators, which involve drift-diffusion and integro-differential terms). Such type of formulas have been very useful in the theory of second order equations -for instance, by allowing the representation of solutions as value functions for differential games. Applications include results on the structure of Dirichlet-to-Neumann mappings for fully nonlinear second order elliptic equations.<br />
<br />
===Ryan Denlinger===<br />
<br />
The propagation of chaos for a rarefied gas of hard spheres in vacuum<br />
<br />
We are interested in the rigorous mathematical justification of<br />
Boltzmann's equation starting from the deterministic evolution of<br />
many-particle systems. O. E. Lanford was able to derive Boltzmann's<br />
equation for hard spheres, in the Boltzmann-Grad scaling, on a short<br />
time interval. Improvements to the time in Lanford's theorem have so far<br />
either relied on a small data hypothesis, or have been restricted to<br />
linear regimes. We revisit the small data regime, i.e. a sufficiently<br />
dilute gas of hard spheres dispersing into vacuum; this is a regime<br />
where strong bounds are available globally in time. Subject to the<br />
existence of such bounds, we give a rigorous proof for the propagation<br />
of Boltzmann's ``one-sided'' molecular chaos.<br />
<br />
===Misha Feldman===<br />
<br />
Shock reflection, free boundary problems and degenerate elliptic equations.<br />
<br />
Abstract: We will discuss shock reflection problem for compressible gas dynamics, and von Neumann<br />
conjectures on transition between regular and Mach reflections. We will discuss existence of solutions <br />
of regular reflection structure for potential flow equation, and also regularity of solutions, and<br />
properties of the shock curve (free boundary). Our approach is to reduce the shock reflection problem<br />
to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems <br />
will also be discussed, including uniqueness.<br />
The talk is based on the joint works with Gui-Qiang Chen, Myoungjean Bae and Wei Xiang.<br />
<br />
===Jessica Lin===<br />
<br />
Optimal Quantitative Error Estimates in Stochastic<br />
Homogenization for Elliptic Equations in Nondivergence Form<br />
<br />
Abstract: I will present optimal quantitative error estimates in the<br />
stochastic homogenization for uniformly elliptic equations in<br />
nondivergence form. From the point of view of probability theory,<br />
stochastic homogenization is equivalent to identifying a quenched<br />
invariance principle for random walks in a balanced random<br />
environment. Under strong independence assumptions on the environment,<br />
the main argument relies on establishing an exponential version of the<br />
Efron-Stein inequality. As an artifact of the optimal error estimates,<br />
we obtain a regularity theory down to microscopic scale, which implies<br />
estimates on the local integrability of the invariant measure<br />
associated to the process. This talk is based on joint work with Scott<br />
Armstrong.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11604PDE Geometric Analysis seminar2016-03-07T17:05:28Z<p>Jessica: /* Nestor Guillen */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
|Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
|Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
|Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
|Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations<br />
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.<br />
<br />
===Nestor Guillen===<br />
<br />
Min-max formulas for integro-differential equations and applications<br />
<br />
We show under minimal assumptions that a nonlinear operator satisfying what is known as a "global comparison principle" can be represented by a min-max formula in terms of very special linear operators (Levy operators, which involve drift-diffusion and integro-differential terms). Such type of formulas have been very useful in the theory of second order equations -for instance, by allowing the representation of solutions as value functions for differential games. Applications include results on the structure of Dirichlet-to-Neumann mappings for fully nonlinear second order elliptic equations.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11603PDE Geometric Analysis seminar2016-03-07T17:05:09Z<p>Jessica: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
|Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
|Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
|Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
|Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations<br />
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.<br />
<br />
===Nestor Guillen===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
We show under minimal assumptions that a nonlinear operator satisfying what is known as a "global comparison principle" can be represented by a min-max formula in terms of very special linear operators (Levy operators, which involve drift-diffusion and integro-differential terms). Such type of formulas have been very useful in the theory of second order equations -for instance, by allowing the representation of solutions as value functions for differential games. Applications include results on the structure of Dirichlet-to-Neumann mappings for fully nonlinear second order elliptic equations.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11602PDE Geometric Analysis seminar2016-03-07T17:04:33Z<p>Jessica: /* Seminar Schedule Spring 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
|Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
|Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
|Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
|Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations<br />
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11601PDE Geometric Analysis seminar2016-03-07T17:04:12Z<p>Jessica: /* Seminar Schedule Spring 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
|Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
|Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
|Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | Min-max formulas for integro-differential equations and applications. ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
|Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations<br />
|Link: http://www.ki-net.umd.edu/content/conf?event_id=493<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11532PDE Geometric Analysis seminar2016-02-22T22:31:33Z<p>Jessica: /* Seminar Schedule Spring 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]<br />
| Tran & Kim<br />
|-<br />
|February 15 <br />
| Paul Rabinowitz (UW Madison)<br />
|[[#Paul Rabinowitz | On A Double Well Potential System ]]<br />
| Tran & Kim<br />
|-<br />
|February 22 <br />
| Hong Zhang (Brown)<br />
|[[#Hong Zhang | On an elliptic equation arising from composite material ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[#Aaron Yip | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]<br />
| Tran<br />
|-<br />
|March 15<br />
| Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | TBA ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
| Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 14: 2:25 PM in VV 901-Joint with Probability Seminar<br />
|Jessica Lin (UW-Madison)<br />
|[[#Jessica Lin | Optimal Quantitative Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form ]]<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.<br />
<br />
===Jingrui Cheng===<br />
<br />
Semi-geostrophic system with variable Coriolis parameter.<br />
<br />
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.<br />
<br />
<br />
===Paul Rabinowitz===<br />
<br />
On A Double Well Potential System<br />
<br />
We will discuss an elliptic system of partial differential equations of the form<br />
\[<br />
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}<br />
\]<br />
\[<br />
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,<br />
\]<br />
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. <br />
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. <br />
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will <br />
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,<br />
i.e. solutions that are of phase transition type.<br />
<br />
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).<br />
<br />
===Hong Zhang===<br />
<br />
On an elliptic equation arising from composite material<br />
<br />
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.<br />
<br />
===Aaron Yip===<br />
<br />
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media<br />
<br />
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.<br />
<br />
<br />
===Hiroyoshi Mitake===<br />
<br />
Selection problem for fully nonlinear equations<br />
<br />
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Colloquia/Fall18&diff=11364Colloquia/Fall182016-01-29T13:59:28Z<p>Jessica: /* Abstracts */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2016 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
| '''January 22''' <br />
|<!--[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)--><br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!--Host--><br />
|-<br />
| '''January 28 (Th 4pm VV901)''' <br />
| [https://web.math.princeton.edu/~ssivek/ Steven Sivek] (Princeton)<br />
| [[Colloquia#September 11: Speaker (University) | The augmentation category of a Legendrian knot]] <br />
| Ellenberg<br />
|-<br />
| '''January 29''' <br />
|[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)<br />
| [[Colloquia#September 11: Ana Caraiani (Princeton) | Locally symmetric spaces, torsion classes, and the geometry of period domains]] <br />
| Ellenberg<br />
|-<br />
| '''February 5''' <br />
|[http://math.uchicago.edu/~souganidis/ Takis Souganidis] (University of Chicago)<br />
| [[Colloquia#September 11: Takis Souganidis (University of Chicago) | Scalar Conservation Laws with Rough Dependence]]<br />
| Lin<br />
|-<br />
| '''February 12''' <br />
|[http://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Jean-Luc<br />
|-<br />
| '''February 19''' <br />
| [https://people.math.osu.edu/lafont.1/ Jean-François Lafont] (Ohio State) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Dymarz<br />
|-<br />
| '''February 26''' <br />
|Hiroyoshi Mitake (Hiroshima university) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Tran<br />
|-<br />
| '''March 4''' <br />
| [http://www.columbia.edu/~gb2030/ Guillaume Bal] (Columbia University)<br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Li, Jin<br />
|-<br />
| '''March 11''' <br />
| [http://math.umn.edu/~luskin Mitchell Luskin] (University of Minnesota)<br />
| [[Colloquia#March 11: Mitchell Luskin (UMN) | Mathematical Modeling of Incommensurate 2D Materials]]<br />
| Li<br />
|-<br />
| '''March 18''' <br />
| [http://www.math.lsa.umich.edu/~spatzier/ Ralf Spatzier] (University of Michigan) <br />
| [[Colloquia#March 18: Ralf Spatzier (University of Michigan) | TBA]]<br />
| Dymarz<br />
|-<br />
| '''March 25''' <br />
| Spring Break<!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 1''' <br />
| [http://www.math.uci.edu/~cterng/ Chuu-Lian Terng] (UC Irvine) --> <br />
| [[Colloquia#April 1: Chuu-Lian Terng (UC Irvine) | TBA]] --><br />
| Mari-Beffa<br />
|-<br />
| '''April 8''' <br />
| [https://web.math.princeton.edu/~aionescu/ Alexandru Ionescu] (Princeton) <br />
| [[Colloquia#April 8: Alexandru Ionescu (Princeton) | TBA]] <br />
| Wainger/Seeger<br />
|-<br />
| '''April 15''' <br />
| [https://www.kcl.ac.uk/nms/depts/mathematics/people/atoz/wigmani.aspx Igor Wigman] (King's College - London) <br />
| [[Colloquia#September 11: Speaker (University) |Nodal Domains of Eigenfunctions]]<br />
| Gurevich/Marshall<br />
|-<br />
| '''April 22''' <br />
| [http://www.cims.nyu.edu/~bourgade/ Paul Bourgade] (NYU)<br />
| [[Colloquia#April 22: Paul Bourgade (NYU) | TBA]]<br />
| Seppalainen/Valko<br />
|-<br />
| '''April 29''' <br />
| [http://www.physics.upenn.edu/~kamien/kamiengroup/ Randall Kamien] (U Penn) <br />
| [[Colloquia#April 29: Randall Kamien (U Penn) | TBA]] <br />
| Spagnolie<br />
|-<br />
| '''May 6''' <br />
| [https://www.math.upenn.edu/~shaneson/ Julius Shaneson] (University of Pennsylvania) <br />
| [[Colloquia#September 11: Julius Shaneson (University of Pennsylvania) | TBA]] <br />
| Maxim/Kjuchukova<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
=== January 28: Steven Sivek (Princeton) === <br />
Title: The augmentation category of a Legendrian knot<br />
<br />
Abstract: A well-known principle in symplectic geometry says that information about the smooth structure on a manifold should be captured by the symplectic geometry of its cotangent bundle. One prominent example of this is Nadler and Zaslow's microlocalization correspondence, an equivalence between a category of constructible sheaves on a manifold and a symplectic invariant of its cotangent bundle called the Fukaya category.<br />
<br />
The goal of this talk is to describe a model for a relative version of this story in the simplest case, corresponding to Legendrian knots in the standard contact 3-space. This construction, called the augmentation category, is a powerful invariant which is defined in terms of holomorphic curves but can also be described combinatorially. I will describe some interesting properties of this category and relate it to a category of sheaves on the plane. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Maslow.<br />
<br />
=== January 29: Ana Caraiani (Princeton) === <br />
Title: Locally symmetric spaces, torsion classes, and the geometry of period domains<br />
<br />
Abstract: The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.<br />
<br />
I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.<br />
<br />
<br />
=== February 5: Takis Souganidis (University of Chicago) === <br />
Title: Scalar Conservation Laws with Rough Dependence<br />
<br />
I will present a recently developed theory for scalar conservation laws with nonlinear multiplicative rough signal dependence. I will describe the difficulties, introduce the notion of pathwise entropy/kinetic solution and its well-posedness. I will also talk about the long time behavior of the solutions as well as some regularization by noise type results.<br />
<br />
<br />
===March 11: Mitchell Luskin (UMN) ===<br />
Title: Mathematical Modeling of Incommensurate 2D Materials<br />
<br />
Abstract: Incommensurate materials are found in crystals, liquid crystals, and quasi-crystals. Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the elastic, electronic, and optical properties of these materials. One of the main issues encountered in the mathematical modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. This leads to complex commensurate-incommensurate transitions and pattern formation.<br />
<br />
Even basic concepts like the Cauchy-Born strain energy density, the electronic density of states, and the Kubo-Greenwood formulas for transport properties have not been given a rigorous analysis in the incommensurate setting. New approximate approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.<br />
==== ====<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Colloquia/Fall18&diff=11363Colloquia/Fall182016-01-29T13:58:01Z<p>Jessica: /* Spring 2016 */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2016 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
| '''January 22''' <br />
|<!--[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)--><br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!--Host--><br />
|-<br />
| '''January 28 (Th 4pm VV901)''' <br />
| [https://web.math.princeton.edu/~ssivek/ Steven Sivek] (Princeton)<br />
| [[Colloquia#September 11: Speaker (University) | The augmentation category of a Legendrian knot]] <br />
| Ellenberg<br />
|-<br />
| '''January 29''' <br />
|[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)<br />
| [[Colloquia#September 11: Ana Caraiani (Princeton) | Locally symmetric spaces, torsion classes, and the geometry of period domains]] <br />
| Ellenberg<br />
|-<br />
| '''February 5''' <br />
|[http://math.uchicago.edu/~souganidis/ Takis Souganidis] (University of Chicago)<br />
| [[Colloquia#September 11: Takis Souganidis (University of Chicago) | Scalar Conservation Laws with Rough Dependence]]<br />
| Lin<br />
|-<br />
| '''February 12''' <br />
|[http://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Jean-Luc<br />
|-<br />
| '''February 19''' <br />
| [https://people.math.osu.edu/lafont.1/ Jean-François Lafont] (Ohio State) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Dymarz<br />
|-<br />
| '''February 26''' <br />
|Hiroyoshi Mitake (Hiroshima university) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Tran<br />
|-<br />
| '''March 4''' <br />
| [http://www.columbia.edu/~gb2030/ Guillaume Bal] (Columbia University)<br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Li, Jin<br />
|-<br />
| '''March 11''' <br />
| [http://math.umn.edu/~luskin Mitchell Luskin] (University of Minnesota)<br />
| [[Colloquia#March 11: Mitchell Luskin (UMN) | Mathematical Modeling of Incommensurate 2D Materials]]<br />
| Li<br />
|-<br />
| '''March 18''' <br />
| [http://www.math.lsa.umich.edu/~spatzier/ Ralf Spatzier] (University of Michigan) <br />
| [[Colloquia#March 18: Ralf Spatzier (University of Michigan) | TBA]]<br />
| Dymarz<br />
|-<br />
| '''March 25''' <br />
| Spring Break<!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 1''' <br />
| [http://www.math.uci.edu/~cterng/ Chuu-Lian Terng] (UC Irvine) --> <br />
| [[Colloquia#April 1: Chuu-Lian Terng (UC Irvine) | TBA]] --><br />
| Mari-Beffa<br />
|-<br />
| '''April 8''' <br />
| [https://web.math.princeton.edu/~aionescu/ Alexandru Ionescu] (Princeton) <br />
| [[Colloquia#April 8: Alexandru Ionescu (Princeton) | TBA]] <br />
| Wainger/Seeger<br />
|-<br />
| '''April 15''' <br />
| [https://www.kcl.ac.uk/nms/depts/mathematics/people/atoz/wigmani.aspx Igor Wigman] (King's College - London) <br />
| [[Colloquia#September 11: Speaker (University) |Nodal Domains of Eigenfunctions]]<br />
| Gurevich/Marshall<br />
|-<br />
| '''April 22''' <br />
| [http://www.cims.nyu.edu/~bourgade/ Paul Bourgade] (NYU)<br />
| [[Colloquia#April 22: Paul Bourgade (NYU) | TBA]]<br />
| Seppalainen/Valko<br />
|-<br />
| '''April 29''' <br />
| [http://www.physics.upenn.edu/~kamien/kamiengroup/ Randall Kamien] (U Penn) <br />
| [[Colloquia#April 29: Randall Kamien (U Penn) | TBA]] <br />
| Spagnolie<br />
|-<br />
| '''May 6''' <br />
| [https://www.math.upenn.edu/~shaneson/ Julius Shaneson] (University of Pennsylvania) <br />
| [[Colloquia#September 11: Julius Shaneson (University of Pennsylvania) | TBA]] <br />
| Maxim/Kjuchukova<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
=== January 28: Steven Sivek (Princeton) === <br />
Title: The augmentation category of a Legendrian knot<br />
<br />
Abstract: A well-known principle in symplectic geometry says that information about the smooth structure on a manifold should be captured by the symplectic geometry of its cotangent bundle. One prominent example of this is Nadler and Zaslow's microlocalization correspondence, an equivalence between a category of constructible sheaves on a manifold and a symplectic invariant of its cotangent bundle called the Fukaya category.<br />
<br />
The goal of this talk is to describe a model for a relative version of this story in the simplest case, corresponding to Legendrian knots in the standard contact 3-space. This construction, called the augmentation category, is a powerful invariant which is defined in terms of holomorphic curves but can also be described combinatorially. I will describe some interesting properties of this category and relate it to a category of sheaves on the plane. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Maslow.<br />
<br />
=== January 29: Ana Caraiani (Princeton) === <br />
Title: Locally symmetric spaces, torsion classes, and the geometry of period domains<br />
<br />
Abstract: The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.<br />
<br />
I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.<br />
<br />
<br />
===March 11: Mitchell Luskin (UMN) ===<br />
Title: Mathematical Modeling of Incommensurate 2D Materials<br />
<br />
Abstract: Incommensurate materials are found in crystals, liquid crystals, and quasi-crystals. Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the elastic, electronic, and optical properties of these materials. One of the main issues encountered in the mathematical modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. This leads to complex commensurate-incommensurate transitions and pattern formation.<br />
<br />
Even basic concepts like the Cauchy-Born strain energy density, the electronic density of states, and the Kubo-Greenwood formulas for transport properties have not been given a rigorous analysis in the incommensurate setting. New approximate approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.<br />
==== ====<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Colloquia/Fall18&diff=11362Colloquia/Fall182016-01-29T13:57:09Z<p>Jessica: /* Spring 2016 */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2016 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
| '''January 22''' <br />
|<!--[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)--><br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!--Host--><br />
|-<br />
| '''January 28 (Th 4pm VV901)''' <br />
| [https://web.math.princeton.edu/~ssivek/ Steven Sivek] (Princeton)<br />
| [[Colloquia#September 11: Speaker (University) | The augmentation category of a Legendrian knot]] <br />
| Ellenberg<br />
|-<br />
| '''January 29''' <br />
|[https://web.math.princeton.edu/~caraiani/ Ana Caraiani] (Princeton)<br />
| [[Colloquia#September 11: Ana Caraiani (Princeton) | Locally symmetric spaces, torsion classes, and the geometry of period domains]] <br />
| Ellenberg<br />
|-<br />
| '''February 5''' <br />
|[http://math.uchicago.edu/~souganidis/ Takis Souganidis] (University of Chicago)<br />
| <!-- [[Colloquia#September 11: Takis Souganidis (University of Chicago) | Scalar Conservation Laws with Rough Dependence<br />
] --><br />
| Lin<br />
|-<br />
| '''February 12''' <br />
|[http://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Jean-Luc<br />
|-<br />
| '''February 19''' <br />
| [https://people.math.osu.edu/lafont.1/ Jean-François Lafont] (Ohio State) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Dymarz<br />
|-<br />
| '''February 26''' <br />
|Hiroyoshi Mitake (Hiroshima university) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Tran<br />
|-<br />
| '''March 4''' <br />
| [http://www.columbia.edu/~gb2030/ Guillaume Bal] (Columbia University)<br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Li, Jin<br />
|-<br />
| '''March 11''' <br />
| [http://math.umn.edu/~luskin Mitchell Luskin] (University of Minnesota)<br />
| [[Colloquia#March 11: Mitchell Luskin (UMN) | Mathematical Modeling of Incommensurate 2D Materials]]<br />
| Li<br />
|-<br />
| '''March 18''' <br />
| [http://www.math.lsa.umich.edu/~spatzier/ Ralf Spatzier] (University of Michigan) <br />
| [[Colloquia#March 18: Ralf Spatzier (University of Michigan) | TBA]]<br />
| Dymarz<br />
|-<br />
| '''March 25''' <br />
| Spring Break<!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 1''' <br />
| [http://www.math.uci.edu/~cterng/ Chuu-Lian Terng] (UC Irvine) --> <br />
| [[Colloquia#April 1: Chuu-Lian Terng (UC Irvine) | TBA]] --><br />
| Mari-Beffa<br />
|-<br />
| '''April 8''' <br />
| [https://web.math.princeton.edu/~aionescu/ Alexandru Ionescu] (Princeton) <br />
| [[Colloquia#April 8: Alexandru Ionescu (Princeton) | TBA]] <br />
| Wainger/Seeger<br />
|-<br />
| '''April 15''' <br />
| [https://www.kcl.ac.uk/nms/depts/mathematics/people/atoz/wigmani.aspx Igor Wigman] (King's College - London) <br />
| [[Colloquia#September 11: Speaker (University) |Nodal Domains of Eigenfunctions]]<br />
| Gurevich/Marshall<br />
|-<br />
| '''April 22''' <br />
| [http://www.cims.nyu.edu/~bourgade/ Paul Bourgade] (NYU)<br />
| [[Colloquia#April 22: Paul Bourgade (NYU) | TBA]]<br />
| Seppalainen/Valko<br />
|-<br />
| '''April 29''' <br />
| [http://www.physics.upenn.edu/~kamien/kamiengroup/ Randall Kamien] (U Penn) <br />
| [[Colloquia#April 29: Randall Kamien (U Penn) | TBA]] <br />
| Spagnolie<br />
|-<br />
| '''May 6''' <br />
| [https://www.math.upenn.edu/~shaneson/ Julius Shaneson] (University of Pennsylvania) <br />
| [[Colloquia#September 11: Julius Shaneson (University of Pennsylvania) | TBA]] <br />
| Maxim/Kjuchukova<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
=== January 28: Steven Sivek (Princeton) === <br />
Title: The augmentation category of a Legendrian knot<br />
<br />
Abstract: A well-known principle in symplectic geometry says that information about the smooth structure on a manifold should be captured by the symplectic geometry of its cotangent bundle. One prominent example of this is Nadler and Zaslow's microlocalization correspondence, an equivalence between a category of constructible sheaves on a manifold and a symplectic invariant of its cotangent bundle called the Fukaya category.<br />
<br />
The goal of this talk is to describe a model for a relative version of this story in the simplest case, corresponding to Legendrian knots in the standard contact 3-space. This construction, called the augmentation category, is a powerful invariant which is defined in terms of holomorphic curves but can also be described combinatorially. I will describe some interesting properties of this category and relate it to a category of sheaves on the plane. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende, and Eric Maslow.<br />
<br />
=== January 29: Ana Caraiani (Princeton) === <br />
Title: Locally symmetric spaces, torsion classes, and the geometry of period domains<br />
<br />
Abstract: The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.<br />
<br />
I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.<br />
<br />
<br />
===March 11: Mitchell Luskin (UMN) ===<br />
Title: Mathematical Modeling of Incommensurate 2D Materials<br />
<br />
Abstract: Incommensurate materials are found in crystals, liquid crystals, and quasi-crystals. Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the elastic, electronic, and optical properties of these materials. One of the main issues encountered in the mathematical modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. This leads to complex commensurate-incommensurate transitions and pattern formation.<br />
<br />
Even basic concepts like the Cauchy-Born strain energy density, the electronic density of states, and the Kubo-Greenwood formulas for transport properties have not been given a rigorous analysis in the incommensurate setting. New approximate approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.<br />
==== ====<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11084PDE Geometric Analysis seminar2016-01-20T17:14:00Z<p>Jessica: /* Russell Schwab */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[# Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[# Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[# Jingrui Cheng | ]]<br />
|<br />
|-<br />
|February 15 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 22 <br />
| Hong Zhang (Brown)<br />
|[[# Hong Zhang | ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[# Aaron Yip | TBD ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[# Hiroyoshi Mitake | TBD ]]<br />
| Tran<br />
|-<br />
|March 15<br />
| Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | TBA ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
| Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11079PDE Geometric Analysis seminar2016-01-20T15:25:40Z<p>Jessica: /* Tianling Jin */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[# Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[# Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[# Jingrui Cheng | ]]<br />
|<br />
|-<br />
|February 15 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 22 <br />
| Hong Zhang (Brown)<br />
|[[# Hong Zhang | ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[# Aaron Yip | TBD ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[# Hiroyoshi Mitake | TBD ]]<br />
| Tran<br />
|-<br />
|March 15<br />
| Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | TBA ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
| Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.<br />
<br />
===Russell Schwab===<br />
<br />
Neumann homogenization via integro-differential methods<br />
<br />
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with \emph{non-co-normal} oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=11078PDE Geometric Analysis seminar2016-01-20T15:25:05Z<p>Jessica: /* Seminar Schedule Spring 2016 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2016 | Tentative schedule for Fall 2016]]===<br />
<br />
= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
||Tianling Jin (HKUST and Caltech)<br />
|[[# Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations ]]<br />
| Zlatos<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[# Russell Schwab | Neumann homogenization via integro-differential methods ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|Jingrui Cheng (UW Madison)<br />
|[[# Jingrui Cheng | ]]<br />
|<br />
|-<br />
|February 15 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 22 <br />
| Hong Zhang (Brown)<br />
|[[# Hong Zhang | ]]<br />
| Kim<br />
|-<br />
|February 29<br />
|Aaron Yip (Purdue university) <br />
|[[# Aaron Yip | TBD ]]<br />
| Tran<br />
|-<br />
|March 7<br />
|Hiroyoshi Mitake (Hiroshima university) <br />
||[[# Hiroyoshi Mitake | TBD ]]<br />
| Tran<br />
|-<br />
|March 15<br />
| Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | TBA ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
| Ryan Denlinger (Courant Institute)<br />
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]<br />
| Lee<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 25<br />
| Moon-Jin Kang (UT-Austin)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
=Abstracts=<br />
<br />
===Tianling Jin===<br />
<br />
Holder gradient estimates for parabolic homogeneous p-Laplacian equations<br />
<br />
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation <br />
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),<br />
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=10475PDE Geometric Analysis seminar2015-10-16T01:30:03Z<p>Jessica: /* Connor Mooney */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Spring 2016 | Tentative schedule for Spring 2016]]===<br />
<br />
<br />
<br />
= Seminar Schedule Fall 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 7 (Labor Day)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|September 14 (special room: B115)<br />
| Hung Tran (Madison)<br />
|[[#Hung Tran | Some inverse problems in periodic homogenization of Hamilton--Jacobi equations ]]<br />
| <br />
|- <br />
|September 21 (special room: B115) <br />
| Eric Baer (Madison)<br />
||[[#Eric Baer | Optimal function spaces for continuity of the Hessian determinant as a distribution ]]<br />
| <br />
|-<br />
|September 28<br />
| Donghyun Lee (Madison)<br />
|[[#Donghyun Lee | FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT]]<br />
| <br />
|-<br />
|October 5 <br />
|Hyung-Ju Hwang (Postech & Brown Univ)<br />
|[[#Hyung-Ju Hwang | The Fokker-Planck equation in bounded domains ]]<br />
| Kim<br />
|-<br />
|October 12<br />
| Minh-Binh Tran (Madison)<br />
|[[#Minh-Binh Tran | Nonlinear approximation theory for kinetic equations ]]<br />
| <br />
|-<br />
|October 19<br />
| Bob Jensen (Loyola University Chicago)<br />
||[[#Bob Jensen | Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs ]]<br />
| Tran<br />
|-<br />
|October 26<br />
|Luis Silvestre (Chicago)<br />
|[[# Luis Silvestre | TBA ]]<br />
|Kim<br />
|-<br />
|November 2<br />
| Connor Mooney (UT Austin)<br />
|[[# Connor Mooney | Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations ]]<br />
|Lin<br />
|-<br />
|November 9<br />
| Javier Gomez-Serrano (Princeton)<br />
||[[# Javier Gomez-Serrano | TBA ]]<br />
|Zlatos<br />
|-<br />
|November 16<br />
| Yifeng Yu (UC Irvine)<br />
|[[# Yifeng Yu | TBA ]]<br />
| Tran<br />
|-<br />
|November 23<br />
| Nam Le (Indiana)<br />
|[[# Nam Le | TBA ]]<br />
|Tran<br />
|-<br />
|November 30<br />
| Qin Li (Madison)<br />
|[[# Qin Li | TBA ]]<br />
|<br />
|-<br />
|December 7<br />
| Lu Wang (Madison)<br />
||[[# Lu Wang | TBA ]]<br />
|<br />
|-<br />
|December 14<br />
| Christophe Lacave (Paris 7)<br />
|[[# Christophe Lacave | TBA ]]<br />
| Zlatos<br />
|}<br />
<br />
=Abstract=<br />
<br />
===Hung Tran===<br />
<br />
Some inverse problems in periodic homogenization of Hamilton--Jacobi equations.<br />
<br />
Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.<br />
<br />
<br />
===Eric Baer===<br />
<br />
Optimal function spaces for continuity of the Hessian determinant as a distribution.<br />
<br />
Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(2-2/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(2-2/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2). The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-2/N,p)$ with $p>N$. Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.<br />
<br />
===Donghyun Lee===<br />
<br />
FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT.<br />
<br />
Abstract : Free-boundary problems of incompressible fluids have been studied for several decades. In the viscous case, it is basically solved by Stokes regularity. However, the inviscid case problem is generally much harder, because the problem is purely hyperbolic. In this talk, we approach the problem via vanishing viscosity limit, which is a central problem of fluid mechanics. To correct boundary layer behavior, conormal Sobolev space will be introduced. In the spirit of the recent work by N.Masmoudi and F.Rousset (2012, non-surface tension), we will see how to get local regularity of incompressible free-boundary Euler, taking surface tension into account. This is joint work with Tarek Elgindi.<br />
If possible, we also talk about applying the similar technique to the free-boundary MHD(Magnetohydrodynamics). Especially, we will see that strong zero initial boundary condition is still valid for this coupled PDE. For the general boundary condition (for perfect conductor), however, the problem is still open.<br />
<br />
=== Hyung-Ju Hwang===<br />
<br />
The Fokker-Planck equation in bounded domains<br />
<br />
abstract: In this talk, we consider the initial-boundary value problem for the Fokker-Planck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of well-posedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang, J. Jung, and J. Velazquez.<br />
<br />
=== Minh-Binh Tran ===<br />
<br />
Nonlinear approximation theory for kinetic equations<br />
<br />
Abstract: Numerical resolution methods for the Boltzmann equation plays a very important role in the practical a theoretical study of the theory of rarefied gas. The main difficulty in the approximation of the Boltzmann equation is due to the multidimensional structure of the Boltzmann collision operator. The major problem with deterministic numerical methods using to solve Boltzmann equation is that we have to truncate the domain or to impose nonphysical conditions to keep the supports of the solutions in the velocity space uniformly compact. I<br />
n this talk, we will introduce our new way to make the connection between nonlinear approximation theory and kinetic theory. Our nonlinear wavelet approximation is nontruncated and based on an adaptive spectral method associated with a new wavelet filtering technique. The approximation is proved to converge and preserve many properties of the homogeneous Boltzmann equation. The nonlinear approximation solves the equation without having to impose non-physics conditions on the equation.<br />
<br />
=== Bob Jensen ===<br />
<br />
Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs<br />
<br />
Abstract: I will discuss C-L viscosity solutions of uniformly elliptic partial differential equations for operators with only measurable spatial regularity. E.g., $L[u] = \sum a_{i\,j}(x)\,D_{i\,j}u(x)$ where $a_{i\,j}(x)$ is bounded, uniformly elliptic, and measurable in $x$. In general there isn't a meaningful extension of the C-L viscosity solution definition to operators with measurable spatial dependence. But under uniform ellipticity there is a natural extension. Though there isn't a general comparison principle in this context, we will see that the extended definition is robust and uniquely characterizes the ``right" solutions for such problems.<br />
<br />
=== Connor Mooney ===<br />
<br />
Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations<br />
<br />
Abstract: W^{2,1} estimates for the Monge-Ampere equation \det D^2u = f in R^n were first obtained by De Philippis and Figalli in the case that f is bounded between positive constants. Motivated by applications to the semigeostrophic equation, we consider the case that f is bounded but allowed to be zero on some set. In this case there are simple counterexamples to W^{2,1} regularity in dimension n \geq 3 that have a Lipschitz singularity. In contrast, if n = 2 a classical theorem of Alexandrov on the propagation of Lipschitz singularities shows that solutions are C^1. We will discuss a counterexample to W^{2,1} regularity in two dimensions whose second derivatives have nontrivial Cantor part, and also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=10473PDE Geometric Analysis seminar2015-10-15T18:37:03Z<p>Jessica: /* Abstract */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Spring 2016 | Tentative schedule for Spring 2016]]===<br />
<br />
<br />
<br />
= Seminar Schedule Fall 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 7 (Labor Day)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|September 14 (special room: B115)<br />
| Hung Tran (Madison)<br />
|[[#Hung Tran | Some inverse problems in periodic homogenization of Hamilton--Jacobi equations ]]<br />
| <br />
|- <br />
|September 21 (special room: B115) <br />
| Eric Baer (Madison)<br />
||[[#Eric Baer | Optimal function spaces for continuity of the Hessian determinant as a distribution ]]<br />
| <br />
|-<br />
|September 28<br />
| Donghyun Lee (Madison)<br />
|[[#Donghyun Lee | FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT]]<br />
| <br />
|-<br />
|October 5 <br />
|Hyung-Ju Hwang (Postech & Brown Univ)<br />
|[[#Hyung-Ju Hwang | The Fokker-Planck equation in bounded domains ]]<br />
| Kim<br />
|-<br />
|October 12<br />
| Minh-Binh Tran (Madison)<br />
|[[#Minh-Binh Tran | Nonlinear approximation theory for kinetic equations ]]<br />
| <br />
|-<br />
|October 19<br />
| Bob Jensen (Loyola University Chicago)<br />
||[[#Bob Jensen | Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs ]]<br />
| Tran<br />
|-<br />
|October 26<br />
|Luis Silvestre (Chicago)<br />
|[[# Luis Silvestre | TBA ]]<br />
|Kim<br />
|-<br />
|November 2<br />
| Connor Mooney (UT Austin)<br />
|[[# Connor Mooney | Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations ]]<br />
|Lin<br />
|-<br />
|November 9<br />
| Javier Gomez-Serrano (Princeton)<br />
||[[# Javier Gomez-Serrano | TBA ]]<br />
|Zlatos<br />
|-<br />
|November 16<br />
| Yifeng Yu (UC Irvine)<br />
|[[# Yifeng Yu | TBA ]]<br />
| Tran<br />
|-<br />
|November 23<br />
| Nam Le (Indiana)<br />
|[[# Nam Le | TBA ]]<br />
|Tran<br />
|-<br />
|November 30<br />
| Qin Li (Madison)<br />
|[[# Qin Li | TBA ]]<br />
|<br />
|-<br />
|December 7<br />
| Lu Wang (Madison)<br />
||[[# Lu Wang | TBA ]]<br />
|<br />
|-<br />
|December 14<br />
| Christophe Lacave (Paris 7)<br />
|[[# Christophe Lacave | TBA ]]<br />
| Zlatos<br />
|}<br />
<br />
=Abstract=<br />
<br />
===Hung Tran===<br />
<br />
Some inverse problems in periodic homogenization of Hamilton--Jacobi equations.<br />
<br />
Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.<br />
<br />
<br />
===Eric Baer===<br />
<br />
Optimal function spaces for continuity of the Hessian determinant as a distribution.<br />
<br />
Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(2-2/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(2-2/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2). The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-2/N,p)$ with $p>N$. Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.<br />
<br />
===Donghyun Lee===<br />
<br />
FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT.<br />
<br />
Abstract : Free-boundary problems of incompressible fluids have been studied for several decades. In the viscous case, it is basically solved by Stokes regularity. However, the inviscid case problem is generally much harder, because the problem is purely hyperbolic. In this talk, we approach the problem via vanishing viscosity limit, which is a central problem of fluid mechanics. To correct boundary layer behavior, conormal Sobolev space will be introduced. In the spirit of the recent work by N.Masmoudi and F.Rousset (2012, non-surface tension), we will see how to get local regularity of incompressible free-boundary Euler, taking surface tension into account. This is joint work with Tarek Elgindi.<br />
If possible, we also talk about applying the similar technique to the free-boundary MHD(Magnetohydrodynamics). Especially, we will see that strong zero initial boundary condition is still valid for this coupled PDE. For the general boundary condition (for perfect conductor), however, the problem is still open.<br />
<br />
=== Hyung-Ju Hwang===<br />
<br />
The Fokker-Planck equation in bounded domains<br />
<br />
abstract: In this talk, we consider the initial-boundary value problem for the Fokker-Planck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of well-posedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang, J. Jung, and J. Velazquez.<br />
<br />
=== Minh-Binh Tran ===<br />
<br />
Nonlinear approximation theory for kinetic equations<br />
<br />
Abstract: Numerical resolution methods for the Boltzmann equation plays a very important role in the practical a theoretical study of the theory of rarefied gas. The main difficulty in the approximation of the Boltzmann equation is due to the multidimensional structure of the Boltzmann collision operator. The major problem with deterministic numerical methods using to solve Boltzmann equation is that we have to truncate the domain or to impose nonphysical conditions to keep the supports of the solutions in the velocity space uniformly compact. I<br />
n this talk, we will introduce our new way to make the connection between nonlinear approximation theory and kinetic theory. Our nonlinear wavelet approximation is nontruncated and based on an adaptive spectral method associated with a new wavelet filtering technique. The approximation is proved to converge and preserve many properties of the homogeneous Boltzmann equation. The nonlinear approximation solves the equation without having to impose non-physics conditions on the equation.<br />
<br />
=== Bob Jensen ===<br />
<br />
Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs<br />
<br />
Abstract: I will discuss C-L viscosity solutions of uniformly elliptic partial differential equations for operators with only measurable spatial regularity. E.g., $L[u] = \sum a_{i\,j}(x)\,D_{i\,j}u(x)$ where $a_{i\,j}(x)$ is bounded, uniformly elliptic, and measurable in $x$. In general there isn't a meaningful extension of the C-L viscosity solution definition to operators with measurable spatial dependence. But under uniform ellipticity there is a natural extension. Though there isn't a general comparison principle in this context, we will see that the extended definition is robust and uniquely characterizes the ``right" solutions for such problems.<br />
<br />
=== Connor Mooney ===<br />
<br />
Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations<br />
<br />
Abstract: W^{2,1} estimates for the Monge-Ampere equation \det D^2u = f in R^n were first obtained by De Philippis and Figalli in the case that f is bounded between positive constants. We consider the case that f is bounded but allowed to be zero on some set. In this case there are simple counterexamples to W^{2,1} regularity in dimension n \geq 3 that have a Lipschitz singularity. In contrast, if n = 2 a classical result of Alexandrov on the propagation of Lipschitz singularities shows that solutions are C^1. We will discuss a counterexample to W^{2,1} regularity in two dimensions whose second derivatives have nontrivial Cantor part, and also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=10472PDE Geometric Analysis seminar2015-10-15T18:36:03Z<p>Jessica: /* Seminar Schedule Fall 2015 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Spring 2016 | Tentative schedule for Spring 2016]]===<br />
<br />
<br />
<br />
= Seminar Schedule Fall 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 7 (Labor Day)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|September 14 (special room: B115)<br />
| Hung Tran (Madison)<br />
|[[#Hung Tran | Some inverse problems in periodic homogenization of Hamilton--Jacobi equations ]]<br />
| <br />
|- <br />
|September 21 (special room: B115) <br />
| Eric Baer (Madison)<br />
||[[#Eric Baer | Optimal function spaces for continuity of the Hessian determinant as a distribution ]]<br />
| <br />
|-<br />
|September 28<br />
| Donghyun Lee (Madison)<br />
|[[#Donghyun Lee | FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT]]<br />
| <br />
|-<br />
|October 5 <br />
|Hyung-Ju Hwang (Postech & Brown Univ)<br />
|[[#Hyung-Ju Hwang | The Fokker-Planck equation in bounded domains ]]<br />
| Kim<br />
|-<br />
|October 12<br />
| Minh-Binh Tran (Madison)<br />
|[[#Minh-Binh Tran | Nonlinear approximation theory for kinetic equations ]]<br />
| <br />
|-<br />
|October 19<br />
| Bob Jensen (Loyola University Chicago)<br />
||[[#Bob Jensen | Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs ]]<br />
| Tran<br />
|-<br />
|October 26<br />
|Luis Silvestre (Chicago)<br />
|[[# Luis Silvestre | TBA ]]<br />
|Kim<br />
|-<br />
|November 2<br />
| Connor Mooney (UT Austin)<br />
|[[# Connor Mooney | Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations ]]<br />
|Lin<br />
|-<br />
|November 9<br />
| Javier Gomez-Serrano (Princeton)<br />
||[[# Javier Gomez-Serrano | TBA ]]<br />
|Zlatos<br />
|-<br />
|November 16<br />
| Yifeng Yu (UC Irvine)<br />
|[[# Yifeng Yu | TBA ]]<br />
| Tran<br />
|-<br />
|November 23<br />
| Nam Le (Indiana)<br />
|[[# Nam Le | TBA ]]<br />
|Tran<br />
|-<br />
|November 30<br />
| Qin Li (Madison)<br />
|[[# Qin Li | TBA ]]<br />
|<br />
|-<br />
|December 7<br />
| Lu Wang (Madison)<br />
||[[# Lu Wang | TBA ]]<br />
|<br />
|-<br />
|December 14<br />
| Christophe Lacave (Paris 7)<br />
|[[# Christophe Lacave | TBA ]]<br />
| Zlatos<br />
|}<br />
<br />
=Abstract=<br />
<br />
===Hung Tran===<br />
<br />
Some inverse problems in periodic homogenization of Hamilton--Jacobi equations.<br />
<br />
Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.<br />
<br />
<br />
===Eric Baer===<br />
<br />
Optimal function spaces for continuity of the Hessian determinant as a distribution.<br />
<br />
Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(2-2/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(2-2/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2). The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-2/N,p)$ with $p>N$. Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.<br />
<br />
===Donghyun Lee===<br />
<br />
FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT.<br />
<br />
Abstract : Free-boundary problems of incompressible fluids have been studied for several decades. In the viscous case, it is basically solved by Stokes regularity. However, the inviscid case problem is generally much harder, because the problem is purely hyperbolic. In this talk, we approach the problem via vanishing viscosity limit, which is a central problem of fluid mechanics. To correct boundary layer behavior, conormal Sobolev space will be introduced. In the spirit of the recent work by N.Masmoudi and F.Rousset (2012, non-surface tension), we will see how to get local regularity of incompressible free-boundary Euler, taking surface tension into account. This is joint work with Tarek Elgindi.<br />
If possible, we also talk about applying the similar technique to the free-boundary MHD(Magnetohydrodynamics). Especially, we will see that strong zero initial boundary condition is still valid for this coupled PDE. For the general boundary condition (for perfect conductor), however, the problem is still open.<br />
<br />
=== Hyung-Ju Hwang===<br />
<br />
The Fokker-Planck equation in bounded domains<br />
<br />
abstract: In this talk, we consider the initial-boundary value problem for the Fokker-Planck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of well-posedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang, J. Jung, and J. Velazquez.<br />
<br />
=== Minh-Binh Tran ===<br />
<br />
Nonlinear approximation theory for kinetic equations<br />
<br />
Abstract: Numerical resolution methods for the Boltzmann equation plays a very important role in the practical a theoretical study of the theory of rarefied gas. The main difficulty in the approximation of the Boltzmann equation is due to the multidimensional structure of the Boltzmann collision operator. The major problem with deterministic numerical methods using to solve Boltzmann equation is that we have to truncate the domain or to impose nonphysical conditions to keep the supports of the solutions in the velocity space uniformly compact. I<br />
n this talk, we will introduce our new way to make the connection between nonlinear approximation theory and kinetic theory. Our nonlinear wavelet approximation is nontruncated and based on an adaptive spectral method associated with a new wavelet filtering technique. The approximation is proved to converge and preserve many properties of the homogeneous Boltzmann equation. The nonlinear approximation solves the equation without having to impose non-physics conditions on the equation.<br />
<br />
=== Bob Jensen ===<br />
<br />
Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs<br />
<br />
Abstract: I will discuss C-L viscosity solutions of uniformly elliptic partial differential equations for operators with only measurable spatial regularity. E.g., $L[u] = \sum a_{i\,j}(x)\,D_{i\,j}u(x)$ where $a_{i\,j}(x)$ is bounded, uniformly elliptic, and measurable in $x$. In general there isn't a meaningful extension of the C-L viscosity solution definition to operators with measurable spatial dependence. But under uniform ellipticity there is a natural extension. Though there isn't a general comparison principle in this context, we will see that the extended definition is robust and uniquely characterizes the ``right" solutions for such problems.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Graduate_student_reading_seminar&diff=10442Graduate student reading seminar2015-10-11T22:31:00Z<p>Jessica: /* 2015 Fall */</p>
<hr />
<div>==2015 Fall==<br />
<br />
Tuesday 2:25pm, Social Sciences 6101.<br />
<br />
This semester we will focus on tools and methods.<br />
<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 Dae Han<br />
<br />
10/13, 10/20: 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 />
10/27, 11/3: Hao Kai<br />
<br />
11/10, 11/17: Chris<br />
<br />
11/24, 12/1: Louis<br />
<br />
12/8, 12/15: Jinsu<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
1/26, 2/2: Hans<br />
<br />
2/9, 2/16: Fan<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 />
==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>Jessicahttps://wiki.math.wisc.edu/index.php?title=Graduate_student_reading_seminar&diff=10441Graduate student reading seminar2015-10-11T16:49:33Z<p>Jessica: /* 2015 Fall */</p>
<hr />
<div>==2015 Fall==<br />
<br />
Tuesday 2:25pm, Social Sciences 6101.<br />
<br />
This semester we will focus on tools and methods.<br />
<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 Dae Han<br />
<br />
10/13, 10/20: 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 the Efron-Stein inequality and touch upon its connection with Log-Sobolev inequalities. <br />
<br />
10/27, 11/3: Hao Kai<br />
<br />
11/10, 11/17: Chris<br />
<br />
11/24, 12/1: Louis<br />
<br />
12/8, 12/15: Jinsu<br />
<br />
<br />
<br />
2016 Spring:<br />
<br />
1/26, 2/2: Hans<br />
<br />
2/9, 2/16: Fan<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 />
==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>Jessicahttps://wiki.math.wisc.edu/index.php?title=Spring_2016&diff=10056Spring 20162015-08-30T09:10:50Z<p>Jessica: /* Seminar Schedule Spring 2016 */</p>
<hr />
<div>= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[# Russell Schwab | TBA ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|<br />
|[[# | ]]<br />
|<br />
|-<br />
|February 15<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|February 22 <br />
|<br />
|[[# | ]]<br />
|<br />
|-<br />
|February 29<br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|March 7<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|March 15<br />
| Nestor Guillen (UMass Amherst)<br />
|[[#Nestor Guillen | TBA ]]<br />
| Lin<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 25<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Probability_group_timetable&diff=9970Probability group timetable2015-08-25T12:53:57Z<p>Jessica: </p>
<hr />
<div>{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| || Kurt 221, Chiara 221 || || Kurt 221, Chiara 221 || <br />
|-<br />
| 10-11|| Kurt 721, Chiara 721 || Kurt 221 || Kurt 721, Chiara 721 || Kurt 221 || Kurt 721, Chiara 721<br />
|-<br />
| 11-12|| Chiara 221 || Chiara 703 || Chiara 221 || Chiara 703 || Chiara 221<br />
|-<br />
| 12-1|| Dave 605, Sebastien 632, Kurt 714 || || Dave 605, Sebastien 632, Kurt 714 || || Dave 605, Sebastien 632, Kurt 721 <br />
|-<br />
| 1-2|| || Kurt 733, Chiara 733 || || Kurt 733, Chiara 733 || <br />
|-<br />
| 2-3|| Benedek 431 (2:25) || reading seminar (2:25pm) || Benedek 431 (2:25) || probability seminar (2:25pm) || Benedek 431 (2:25)<br />
|-<br />
| 3-4|| Benedek OH (3:30), Kurt 221 (3:30), PDE Seminar (3:30) || Benedek OH (3:30) || Benedek OH (3:30), Kurt 221 (3:30) || || Kurt 221 (3:30)<br />
|-<br />
| 4-5|| Kurt 221 (4:20) || Analysis Seminar || Kurt 221 (4:20) || || Colloquium, Kurt 221 (4:20)<br />
|-<br />
| 5-6|| || || || ||<br />
|}</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=9900PDE Geometric Analysis seminar2015-08-08T22:52:49Z<p>Jessica: /* Seminar Schedule Fall 2015 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Spring 2016 | Tentative schedule for Spring 2016]]===<br />
<br />
<br />
<br />
= Seminar Schedule Fall 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 7 (Labor Day)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|September 14<br />
|<br />
|[[# | ]]<br />
| <br />
|-<br />
|September 21 <br />
|Luis Silvestre (Chicago)<br />
||[[# Luis Silvestre | TBA ]]<br />
| Kim<br />
|-<br />
|September 28<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|October 5 <br />
|Hyung-Ju Hwang (Postech)<br />
|[[# Hyung-Ju Hwang | TBA ]]<br />
| Kim<br />
|-<br />
|October 12<br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|October 19<br />
| Bob Jensen (Loyola University Chicago)<br />
||[[# Bob Jensen | ]]<br />
| Tran<br />
|-<br />
|October 26<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|November 2<br />
| Connor Mooney (UT Austin)<br />
|[[# Connor Mooney | TBA ]]<br />
|Lin<br />
|-<br />
|November 9<br />
| Yifeng Yu (UC Irvine)<br />
|[[# Yifeng Yu | TBA ]]<br />
| Tran<br />
|-<br />
|November 16<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|November 23<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|November 30<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|December 7<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|December 14<br />
| reserved<br />
|[[# | ]]<br />
| Zlatos<br />
|}<br />
<br />
= Abstracts =</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Spring_2016&diff=9887Spring 20162015-08-06T15:10:39Z<p>Jessica: /* Seminar Schedule Spring 2016 */</p>
<hr />
<div>= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 25<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[# Russell Schwab | TBA ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|<br />
|[[# | ]]<br />
|<br />
|-<br />
|February 15<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|February 22 <br />
|<br />
|[[# | ]]<br />
|<br />
|-<br />
|February 29<br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|March 7<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|March 15<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 25<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Spring_2016&diff=9886Spring 20162015-08-06T15:09:32Z<p>Jessica: /* Seminar Schedule Spring 2016 */</p>
<hr />
<div>= Seminar Schedule Spring 2016 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
| <br />
| <br />
|[[# | ]]<br />
|<br />
|}<br />
<br />
|-<br />
|January 25<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|February 1<br />
|Russell Schwab (Michigan State University)<br />
|[[# Russell Schwab | TBA ]]<br />
| Lin<br />
|-<br />
|February 8 <br />
|<br />
|[[# | ]]<br />
|<br />
|-<br />
|February 15<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|February 22 <br />
|<br />
|[[# | ]]<br />
|<br />
|-<br />
|February 29<br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|March 7<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|March 15<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 21 (Spring Break)<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|March 28<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 4<br />
| <br />
||[[# | ]]<br />
| <br />
|-<br />
|April 11<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 18<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|April 25<br />
| <br />
|[[# | ]]<br />
|<br />
|-<br />
|May 2<br />
| <br />
|[[# | ]]<br />
|</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Colloquia/Spring_2016&diff=9869Colloquia/Spring 20162015-08-04T03:15:16Z<p>Jessica: /* Spring 2016 */</p>
<hr />
<div>== Spring 2016 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
| '''January 22''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''January 29''' <br />
| Amir Mohammadi (Texas-Austin) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Marshall<br />
|-<br />
| '''February 5''' <br />
|Takis Souganidis (University of Chicago)<br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Lin<br />
|-<br />
| '''February 12''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''February 19''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''February 26''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''March 4''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''March 11''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''March 18''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''March 25''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 1''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 8''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 15''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 22''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 29''' <br />
| [http://www.physics.upenn.edu/~kamien/kamiengroup/ Randall Kamien] (U Penn) <br />
| [[Colloquia#April 29: Randall Kamien (U Penn) | TBA]] <br />
| Spagnolie<br />
|-<br />
| '''May 6''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|}</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=Colloquia/Spring_2016&diff=9868Colloquia/Spring 20162015-08-04T03:14:07Z<p>Jessica: /* Spring 2016 */</p>
<hr />
<div>== Spring 2016 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
| '''January 22''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''January 29''' <br />
| Amir Mohammadi (Texas-Austin) <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| Marshall<br />
|-<br />
| '''February 5''' <br />
| <!-- [Takis Souganidis] (University of Chicago) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- Lin--><br />
|-<br />
| '''February 12''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''February 19''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''February 26''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''March 4''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''March 11''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''March 18''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''March 25''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 1''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 8''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 15''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 22''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|-<br />
| '''April 29''' <br />
| [http://www.physics.upenn.edu/~kamien/kamiengroup/ Randall Kamien] (U Penn) <br />
| [[Colloquia#April 29: Randall Kamien (U Penn) | TBA]] <br />
| Spagnolie<br />
|-<br />
| '''May 6''' <br />
| <!-- [webpage Speaker Name] (University) --> <br />
| <!-- [[Colloquia#September 11: Speaker (University) | title]] --><br />
| <!-- host --><br />
|}</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=9480PDE Geometric Analysis seminar2015-03-06T21:52:58Z<p>Jessica: </p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2015 | Tentative schedule for Fall 2015]]===<br />
<br />
= Seminar Schedule Spring 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 21 (Departmental Colloquium: 4PM, B239) <br />
|Jun Kitagawa (Toronto) <br />
|[[#Jun Kitagawa (Toronto) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics ]]<br />
|Feldman <br />
|-<br />
|February 9 <br />
|Jessica Lin (Madison)<br />
|[[#Jessica Lin (Madison) | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations ]]<br />
|Kim<br />
|-<br />
|February 17 (Tuesday) (joint with Analysis Seminar: 4PM, B139)<br />
|Chanwoo Kim (Madison) <br />
|[[#Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]<br />
|Seeger <br />
|-<br />
|February 23 (special time*, '''3PM, B119''') <br />
| Yaguang Wang (Shanghai Jiao Tong)<br />
|[[ #Yaguang Wang | Stability of Three-dimensional Prandtl Boundary Layers ]]<br />
|Jin<br />
|-<br />
|March 2 <br />
|Benoit Pausader (Princeton)<br />
|[[#Benoit Pausader (Princeton) | Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions]]<br />
|Kim<br />
|-<br />
|March 9 <br />
|Haozhao Li (University of Science and Technology of China) <br />
|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]]<br />
|Wang <br />
|-<br />
|March 16 <br />
| Jennifer Beichman (Madison) <br />
|[[#Jennifer Beichman (Madison) | ]]<br />
| Kim<br />
|-<br />
|March 23 <br />
| Ben Fehrman (University of Chicago)<br />
|[[#Ben Fehrman (University of Chicago | On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments ]]<br />
| Lin<br />
|-<br />
|March 30 <br />
| Spring recess Mar 28-Apr 5 (S-N)<br />
|[[# | ]]<br />
| <br />
|-<br />
|April 6 <br />
| Vera Hur (UIUC)<br />
|[[# | ]]<br />
| Yao<br />
|-<br />
|April 13 <br />
| Sung-Jin Oh (Berkeley)<br />
|[[# | ]]<br />
| Kim<br />
|-<br />
|April 20 <br />
|Yuan Lou (Ohio State)<br />
|[[#Yuan Lou (Ohio State) | TBA]]<br />
|Zlatos<br />
|-<br />
|April 27 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|May 4 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|}<br />
<br />
<br />
== Abstracts ==<br />
<br />
===Jun Kitagawa (Toronto)===<br />
<br />
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics<br />
<br />
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.<br />
<br />
===Jessica Lin (Madison)===<br />
<br />
Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations<br />
<br />
We establish error estimates for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. Based on the approach of Armstrong and Smart in the elliptic setting, we construct a quantity which captures the geometric behavior of solutions to parabolic equations. The error estimates are shown to be of algebraic order. This talk is based on joint work with Charles Smart.<br />
<br />
<br />
===Yaguang Wang (Shanghai Jiao Tong)===<br />
<br />
Stability of Three-dimensional Prandtl Boundary Layers<br />
<br />
In this talk, we shall study the stability of the Prandtl boundary layer<br />
equations in three space variables. First, we obtain a well-posedness<br />
result of the three-dimensional Prandtl equations under some constraint on<br />
its flow structure. It reveals that the classical Burgers equation plays an<br />
important role in determining this type of flow with special structure,<br />
that avoids the appearance of the complicated secondary flow in the<br />
three-dimensional Prandtl boundary layers. Second, we give an instability<br />
criterion for the Prandtl equations in three space variables. Both of<br />
linear and nonlinear stability are considered. This criterion shows that<br />
the monotonic shear flow is linearly stable for the three dimensional<br />
Prandtl equations if and only if the tangential velocity field direction is<br />
invariant with respect to the normal variable, which is an exact complement<br />
to the above well-posedness result for a special flow. This is a joint work<br />
with Chengjie Liu and Tong Yang.<br />
<br />
<br />
===Benoit Pausader (Princeton)===<br />
<br />
Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions<br />
<br />
It is well known that pure compressible fluids tend to develop shocks, even from small perturbation. We study how self consistent electromagnetic fields can stabilize these fluids. In a joint work with A. Ionescu and Y. Deng, we consider a compressible fluid of electrons in 2D, subject to its own electromagnetic field and to a field created by a uniform background of positively charged ions. We show that small smooth and irrotational perturbations of a uniform background at rest lead to solutions that remain globally smooth, in contrast with neutral fluids. This amounts to proving small data global existence for a system of quasilinear Klein-Gordon equations with different speeds.<br />
<br />
<br />
===Haozhao Li (University of Science and Technology of China)===<br />
<br />
Regularity scales and convergence of the Calabi flow<br />
<br />
We define regularity scales to study the behavior of the Calabi flow. <br />
Based on estimates of the regularity scales, we obtain convergence theorems<br />
of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence<br />
of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for<br />
the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra<br />
assumption that the scalar curvature is uniformly bounded.<br />
<br />
===Ben Fehrman (University of Chicago)===<br />
<br />
On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments<br />
<br />
I will discuss the existence of a unique mutually absolutely continuous invariant measure for isotropic diffusions in random environment, of dimension at least three, which are small perturbations of Brownian motion satisfying a finite range dependence. This framework was first considered in the continuous setting by Sznitman and Zeitouni and in the discrete setting by Bricmont and Kupiainen. The results of this talk should be seen as an extension of their work.<br />
<br />
I will furthermore mention applications of this analysis to the stochastic homogenization of the related elliptic and parabolic equations with random oscillatory boundary data and, explain how the existence of an invariant measure can be used to prove a Liouville property for the environment. In the latter case, the methods were motivated by work in the discrete setting by Benjamini, Duminil-Copin, Kozma and Yadin.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=9231PDE Geometric Analysis seminar2015-01-26T18:15:18Z<p>Jessica: /* Jessica Lin (Madison) */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2015 | Tentative schedule for Fall 2015]]===<br />
<br />
= Seminar Schedule Spring 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 21 (Departmental Colloquium: 4PM, B239) <br />
|Jun Kitagawa (Toronto) <br />
|[[#Jun Kitagawa (Toronto) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics ]]<br />
|Feldman <br />
|-<br />
|February 2 <br />
|Jessica Lin (Madison)<br />
|[[#Jessica Lin (Madison) | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations ]]<br />
|Kim<br />
|-<br />
|February 9<br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 16 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 17 (joint with Analysis Seminar: 4PM, B139)<br />
|Chanwoo Kim (Madison) <br />
|[[#Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]<br />
|Seeger <br />
|-<br />
|February 23 <br />
|Jennifer Beichman (Madison) <br />
|[[#Jennifer Beichman (Madison) | TBA ]]<br />
| Kim<br />
|-<br />
|March 2 <br />
|Benoit Pausader (Princeton)<br />
|[[#Benoit Pausader (Princeton) | TBA]]<br />
|Kim<br />
|-<br />
|March 9 <br />
|Haozhao Li (University of Science and Technology of China) <br />
|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]]<br />
|Wang <br />
|-<br />
|March 16 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|March 23 <br />
| Ben Fehrman (University of Chicago)<br />
|[[#Ben Fehrman (University of Chicago | TBA ]]<br />
| Lin<br />
|-<br />
|March 30 <br />
| Spring recess Mar 28-Apr 5 (S-N)<br />
|[[# | ]]<br />
| <br />
|-<br />
|April 6 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|April 13 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|April 20 <br />
|Yuan Lou (Ohio State)<br />
|[[#Yuan Lou (Ohio State) | TBA]]<br />
|Zlatos<br />
|-<br />
|April 27 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|May 4 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|}<br />
<br />
<br />
== Abstracts ==<br />
<br />
===Jun Kitagawa (Toronto)===<br />
<br />
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics<br />
<br />
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.<br />
<br />
===Jessica Lin (Madison)===<br />
<br />
Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations<br />
<br />
We establish error estimates for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. Based on the approach of Armstrong and Smart in the elliptic setting, we construct a quantity which captures the geometric behavior of solutions to parabolic equations. The error estimates are shown to be of algebraic order. This talk is based on joint work with Charles Smart.<br />
<br />
===Haozhao Li (University of Science and Technology of China)===<br />
Regularity scales and convergence of the Calabi flow<br />
<br />
We define regularity scales to study the behavior of the Calabi flow. <br />
Based on estimates of the regularity scales, we obtain convergence theorems<br />
of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence<br />
of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for<br />
the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra<br />
assumption that the scalar curvature is uniformly bounded.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=9230PDE Geometric Analysis seminar2015-01-26T18:14:40Z<p>Jessica: /* Seminar Schedule Spring 2015 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2015 | Tentative schedule for Fall 2015]]===<br />
<br />
= Seminar Schedule Spring 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 21 (Departmental Colloquium: 4PM, B239) <br />
|Jun Kitagawa (Toronto) <br />
|[[#Jun Kitagawa (Toronto) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics ]]<br />
|Feldman <br />
|-<br />
|February 2 <br />
|Jessica Lin (Madison)<br />
|[[#Jessica Lin (Madison) | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations ]]<br />
|Kim<br />
|-<br />
|February 9<br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 16 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 17 (joint with Analysis Seminar: 4PM, B139)<br />
|Chanwoo Kim (Madison) <br />
|[[#Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]<br />
|Seeger <br />
|-<br />
|February 23 <br />
|Jennifer Beichman (Madison) <br />
|[[#Jennifer Beichman (Madison) | TBA ]]<br />
| Kim<br />
|-<br />
|March 2 <br />
|Benoit Pausader (Princeton)<br />
|[[#Benoit Pausader (Princeton) | TBA]]<br />
|Kim<br />
|-<br />
|March 9 <br />
|Haozhao Li (University of Science and Technology of China) <br />
|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]]<br />
|Wang <br />
|-<br />
|March 16 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|March 23 <br />
| Ben Fehrman (University of Chicago)<br />
|[[#Ben Fehrman (University of Chicago | TBA ]]<br />
| Lin<br />
|-<br />
|March 30 <br />
| Spring recess Mar 28-Apr 5 (S-N)<br />
|[[# | ]]<br />
| <br />
|-<br />
|April 6 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|April 13 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|April 20 <br />
|Yuan Lou (Ohio State)<br />
|[[#Yuan Lou (Ohio State) | TBA]]<br />
|Zlatos<br />
|-<br />
|April 27 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|May 4 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|}<br />
<br />
<br />
== Abstracts ==<br />
<br />
===Jun Kitagawa (Toronto)===<br />
<br />
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics<br />
<br />
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.<br />
<br />
===Jessica Lin (Madison)===<br />
<br />
Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations<br />
<br />
We establish error estimates for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. Based on an approach of Armstrong and Smart in the elliptic setting, we construct a quantity which captures the geometric behavior of solutions to parabolic equations. The error estimates are shown to be of algebraic order. This talk is based on joint work with Charles Smart. <br />
<br />
<br />
===Haozhao Li (University of Science and Technology of China)===<br />
Regularity scales and convergence of the Calabi flow<br />
<br />
We define regularity scales to study the behavior of the Calabi flow. <br />
Based on estimates of the regularity scales, we obtain convergence theorems<br />
of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence<br />
of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for<br />
the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra<br />
assumption that the scalar curvature is uniformly bounded.</div>Jessicahttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=9161PDE Geometric Analysis seminar2015-01-21T15:53:52Z<p>Jessica: /* Seminar Schedule Spring 2015 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2015 | Tentative schedule for Fall 2015]]===<br />
<br />
= Seminar Schedule Spring 2015 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 21 (Departmental Colloquium: 4PM, B239) <br />
|Jun Kitagawa (Toronto) <br />
|[[#Jun Kitagawa (Toronto) | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics ]]<br />
|Feldman <br />
|-<br />
|January 26 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 2 <br />
|Jessica Lin (Madison)<br />
|[[#Jessica Lin (Madison) | TBA ]]<br />
|Kim<br />
|-<br />
|February 9<br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 16 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|February 17 (joint with Analysis Seminar: 4PM, B139)<br />
|Chanwoo Kim (Madison) <br />
|[[#Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]<br />
|Seeger <br />
|-<br />
|February 23 <br />
|Jennifer Beichman (Madison) <br />
|[[#Jennifer Beichman (Madison) | TBA ]]<br />
| Kim<br />
|-<br />
|March 2 <br />
|Benoit Pausader (Princeton)<br />
|[[#Benoit Pausader (Princeton) | TBA]]<br />
|Kim<br />
|-<br />
|March 9 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|March 16 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|March 23 <br />
| Ben Fehrman (University of Chicago)<br />
|[[#Ben Fehrman (University of Chicago | TBA ]]<br />
| Lin<br />
|-<br />
|March 30 <br />
| Spring recess Mar 28-Apr 5 (S-N)<br />
|[[# | ]]<br />
| <br />
|-<br />
|April 6 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|April 13 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|April 20 <br />
|Yuan Lou (Ohio State)<br />
|[[#Yuan Lou (Ohio State) | TBA]]<br />
|Zlatos<br />
|-<br />
|April 27 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|May 4 <br />
| <br />
|[[# | ]]<br />
| <br />
|-<br />
|}<br />
<br />
<br />
== Abstracts ==<br />
<br />
===Jun Kitagawa (Toronto)===<br />
<br />
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics<br />
<br />
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.</div>Jessica