Projects

Here are some of the topics and books studied by past DRP participants. If you are looking for project ideas, you might also find it helpful to look at this list of books and their prerequisites.

Spring 2019

Project Resources
An Introduction to the p-adic Numbers P-adic numbers: An Introduction by Gouvea
Line Bundles on Elliptic Curves (presentation) Vector Bundles Over an Elliptic Curve by Atiyah
Stochastic Processes: Martingales
Algorithmic/High-Frequency Trading
Automated Market Making
Eigenvalues for Odd-dimensional Real Vector Spaces Elementary Number Theory, Group Theory, and Ramanujan Graphs by Davinoff, Sarnak, and Valette
Representations of p-adic Groups Berstein's lecture notes
Convex Optimization
Optimal Control for PDE Systems/Linear Systems
Winning Strategies for Nim (presentation) On Numbers and Games by Conway
Four Coin Welter's Game On Numbers and Games by Conway
Jordan Canonical Form
Commutative Algebra and Point Set Topology Towards Category Theory A Term of Commutative Algebra by Altman and Kleinman, Topology by Munkres, Foundations in Algebraic Geometry by Vakil
Stochastic Gradient Descent
Sheaves and Algebraic Curves Geometry of Algebraic Curves by Griffiths, et. al.
Fiber Sequences in Topology
Introduction to Quantum Mechanics and the Kochen-Specker Theorem Incompleteness, Non-localism, and Reality by Michael Redhead
Number Theory/Prime Number Theorem
Groups and Rubik's Cubes
Using Knot Theory to Untangle a Lightbulb Cord The Knot Book by Colin Adams
Random Matrix Theory and Wigner's Semicircle Law
The Projective Plane Rational Points on Elliptic Curves by Silverman
Elliptic Curves and Fruit Algebra Rational Points on Elliptic Curves by Silverman
Generalized Cantor Functions

Spring 2018

Project Resources
Basic Topology An Introduction to Topology and Homotopy by Allan Sieradski
Introduction to Logic
Modular Arithmetic and Quadratic Reciprocity Elementary Number Theory by Jones and Jones
An Introduction to Computational Algebra and Geometry Ideals, Varieties, Algorithms by Cox, Little and O'Shea
Applications of Monte Carlo Methods
Monte Carlo Methods for Finance Monte Carlo Methods in Financial Engineering by Paul Glasserman
p-adic numbers p-adic Numbers: an Introduction by Gouvea
An Introduction to the Theory of Numbers An Introduction to the Theory of Numbers by Hardy and Wright
Combinatorics and Problem Solving Combinatorics Through Guided Discovery by Kenneth Bogart
Number Theory and Ramanujan Graphs Elementary Number Theory, Group Theory and Ramanujan Graphs by Davidoff, Sarnak, Valette
Finite Volume Methods for Linear and Non-linear Conservation Laws Finite Volume Methods for Hyperbolic Problems by Leveque
Tangent Spaces of Algebraic Curves
Group theory and Burnside's Lemma

Fall 2018

Project Resources
Coherent Cohomology of Sheaves
Computational Algebra Ideals, Varieties, Algorithms by Cox, Little and O'Shea
Random Sampling
An Introduction to Number theory and Algebraic Geometry
Linear Algebra Linear Algebra Done Right by Sheldon Axler
Fundamental Groups and Covering Spaces Algebraic Topology by Hatcher
Singular Functions Principles of Mathematical Analysis by Rudin
Differential Equations Finite Difference Methods for Ordinary and Partial Differential Equations by Leveque
How to read a terse book on Commutative Algebra A Term of Commutative Algebra by Altman and Kleiman
Introduction to Topology Shape of Space by Jeff Weeks
Algorithmic Information Theory An introduction to Kolmogorov complexity and its applications by Li and Vitanyi
Analytic Number Theory Introduction to Analytic Number Theory by Apostol
Representation Theory Representations and Characters of Groups by James and Liebeck
Combinatorics A Walk Through Combinatorics by Bona
TheRiemann Hurwitz Theorem Riemann Surfaces and Algebraic Curves by Cavalieri and Miles
Introduction toLogic Foundations of Mathematics by Kunen
Random Walk Random Walk and the Heat Equation by Lawler
Category Theory
Reading Tate's Thesis
Mathematics for Finance Essentials of Stochastic Processes by Rick Durrett

Spring 2017

Project Resources
History of Mathematical Ideas Mathematics and it's History by Stillwell
Stochastic Math Biology
The Hairy Ball Theorem Topology from a Differentiable Viewpoint by John Milnor
Quantitative Investment Strategies
Dirichlet's Theorem on Arithmetic Progressions Introduction to Analytic Number Theory by Apostol
Chaotic Dynamics
Category Theory Abstract and Concrete Categories - The Joy of Cats by J.Adámek, H.Herrlich, G. Strecker
Young Tableaux Young Tableaux by Fulton
Combinatorics and Graph Theory
Square the Circle
Probabilistic Methods in Combinatorics
Matrix Factorization
Nonstandard Analysis
Braid Groups

Fall 2017

Project Resources
The fundamental group and covering spaces
Introduction to probabilistic methods and heuristics
Introduction to stochastic calculus and stochastic differential equations
Quantum computation
Orthogonal polynomials
Set theory Elements of Set Theory by Herbert Enderton
Bezout's Theorem Undergraduate Algebraic Geometry by Miles Reid
Introduction to group theory Groups and Symmetry by M. A. Armstrong
p-adic numbers p-adic Numbers, p-adic Analysis, and Zeta-Functions by Koblitz