Projects
Spring 2025
This list will be updated as more projects are confirmed.
- Measure Theory and Probability
- Text: Capinski and Kopp, Measure, Integral, and Probability
- Mentee: Andrew Kim
- Mentor: Jun Park
- Symplectic geometry
- Algebraic topology and the fundamental group
- Text: Crossley, Essential Topology and Lima, Fundamental Groups and Covering Spaces
- Mentee: Lucas McCulla
- Mentor: Mason Hart
- Homotopical Combinatorics
- Text: Bao, Hazel et al, Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games
- Mentee: Seth Bernstein
- Mentor: Ben Spitz
- Introduction to Elliptic Curves
- Text: Silverman and Tate, Rational Points on Elliptic Curves
- Mentee: Isaac Assink
- Mentor: Thomas Jaklitsch
- An Introduction to Geometric Group Theory
- Text: Clay and Margalit, Office Hours with a Geometric Group Theorist
- Mentee: Ismoil Nosirov
- Mentor: Beth Claire Branman
- The Hook Length Formula
- Text: Stanley, Enumerative Combinatorics
- Mentee: Nate Bryerton
- Mentor: Mikhail Tikhonov
Fall 2024
- Introduction to Algebraic Topology
- Text: Hatcher, Algebraic Topology
- Mentee: Wynn Smith
- Mentor: Sarah Blackwell
- Introduction to Functional Analysis
- Text: Eidelman, Milman and Tsolomitis, Functional Analysis: An Introduction
- Mentee: Kayla Denoo
- Mentor: Felipe Flores
- Differential Topology and the Generalized Stokes’ Theorem
- Text: Lee, Introduction to Smooth Manifolds
- Mentee: Alexander Kurtz
- Mentor: Nick Sweeney
- DNA, Associations Schemes, and the Linear Programming Bound
- Text: Bruower and Haemers, Spectra of Graphs
- Mentee: Giovanni Mazzeo
- Mentor: Kang Lu
- Quotients of Free Z-Modules of Finite Rank
- Text: David Tall and Ian Stewart, Algebraic Number Theory and Fermat’s Last Theorem
- Mentee: Josh Gao
- Mentor: Alejandro De Las Peñas Castaño
- p-adic Numbers
- Text: Gouvea, p-Adic Numbers
- Mentee: Alexander Seidenberg-Spitzer
- Mentor: Oliver Wang
- Differential geometry of curves and surfaces
- Text: Montiel, Curves and surfaces
- Mentee: Zubin Kaul
- Mentor: Pedro Brunialti Lima De Andrade
Spring 2024
- From calculus to de Rham cohomology
- Text: Madsen and Tornehave, From calculus to cohomology: de Rham cohomology and characteristic classes
- Mentee: Ethan Fu
- Mentor: Eleftherios Chatzitheodoridis
- Knots and Seifert surfaces
- Text: Livingston, Knot Theory
- Mentee: Claire Kehoe
- Mentor: Sarah Blackwell
- Knot Theory
- Text: Crowell and Fox, Introduction to Knot Theory
- Mentee: Wynn Smith
- Mentor: Beth Branman
- Algebraic Number Theory
- Text: Stewart and Tall, Algebraic Number Theory and Fermat’s Last Theorem
- Mentee: Brandon Werbel
- Mentor: Alejandro De Las Penas Castano
- Symmetric functions and representation theory
- Text: Prasad, Representation theory: A combinatorial viewpoint
- Mentee: Henghui Li
- Mentor: Kang Lu
- Category theory
- Texts: Lawvere and Rosebrugh, Sets for Mathematics; Riehl, Category Theory in Context; Leinster, Basic Category Theory
- Mentee: Luke Moyar
- Mentor: William Balderrama
- Matroids
- Text: Gordon and McNulty, Matroids: A geometric introduction
- Mentee: John Yan
- Mentor: Mark Pengitore
- Nonlinear Dynamics and Chaos
- Text: Strogatz, Nonlinear Dynamics and Chaos
- Mentee: Mary Edith Plunkket
- Mentor: Pedro Brunialti
- An Overview of Combinatorial and Geometric Group Theory
- Text: Löh, Geometric Group Theory: An Introduction
- Mentee: Grady Hollar
- Mentor: Darien Farnham
- The RSK Correspondence and its applications
- Texts: Schensted, Longest increasing and decreasing subsequences; Stanley, Increasing and decreasing subsequences of permutations and their variants
- Mentee: Brendan Malaugh
- Mentor: Petch Chueluecha
Fall 2023
- Choosing the optimal number of clusters for the k-means algorithm
- Text: Murphey, Machine learning: a probabalistic perspective
- Mentee: Michelle Giulajan
- Mentor: Beth Branman
- Elliptic curves and the congruent number problem
- Texts: Silverman and Tate, Rational points on elliptic curves; Koblitz, Introduction to elliptic curves and modular forms
- Mentee: Owen Guttilla
- Mentor: Alejandro De Las Peñas Castaño
- Brouwer’s fixed point theorem and the fundamental group
- Text: Munkres, Topology
- Mentee: Alan Hale
- Mentor: William Balderrama
- Random knot theory
- Texts: Livingston, Knot theory; Even-Zohar, Models of random knots
- Mentee: Samuel Lichtman
- Mentor: Sarah Blackwell
- Hamiltonian mechanics and Poisson brackets
- Texts: Axler, Linear algebra done right; Hall, Quantum theory for mathematicians
- Mentee: Brian Meiser
- Mentor: Pedro Brunialti Lima de Andrade
- Convex optimization and support vector machines
- Texts: Boyd and Vandenberghe, Convex optimization; Faisal, Ong, and Deisenroth, Mathematics for machine learning
- Mentee: Alex Ning
- Mentor: Kang Lu
- Musical chord transformations as groups
- Texts: Rotman, An introduction to the theory of groups; Zhang, The framework of music theory as represented with groups
- Mentee: Peter Tessier
- Mentor: Maximiliano Sánchez Garza
Spring 2023
- Topological data analysis
- Text: Carlsson and Vejdemo-Johansson, Topological data analysis with applications
- Mentee: Lior Ben-David
- Mentor: Miika Tuominen
- Geometry of groups
- Mentee: Anthony Doll
- Mentor: Mark Pengitore
- From spaces and continuity to categories and functoriality
- Text: Bradley, Bryson, and Terilla, Topology: a categorical approach
- Mentee: Yejun Kim
- Mentor: Eleftherios Chatzitheodoridis
- On primes and irreducibles: aren’t they the same?
- Text: Ireland and Rosen, A classical introduction to modern number theory
- Mentee: Jake Fulford
- Mentor: Alejandro de las Peñas Castaño
- The central limit theorem
- Text: Lesigne, Heads or tails: an introduction to limit theorems in probability
- Mentee: Charlie Wang
- Mentor: Yaolong Shen
Fall 2022
- Naturality of the double dual evaluation map
- Text: Serge Lang, Linear algebra & Harold Simmons, An introduction to category theory
- Mentee: John Layne
- Mentor: Eleftherios Chatzitheodoridis
- Measure theory: the foundations of probability
- Text: Sydney Resnick, A probability path
- Mentee: Zoe Zerui Wang
- Mentor: Daniel Slonim
- Hilbert’s 10th problem
- Text: Murty and Fodden, Hilbert’s tenth problem: an introduction to logic, number theory, and computability
- Mentee: Trevor Millard
- Mentor: Michael Wills
- The fast Fourier transform
- Text: M. C. Pereyra and L. A. Ward, Harmonic analysis: from Fourier to wavelets
- Mentee: Zihan Mei
- Mentor: Joe Webster
- Foundations of machine learning
- Text: Shai Shalev-Shwartz and Shai Ben-David, Understanding machine learning: from theory to algorithms
- Mentee: Qingyang Wang
- Mentor: Mojdeh Tarighat Feller
- Fractal geometry
- Text: Falconer, Fractal Geometry
- Mentee: Maahi Vyas
- Mentor: Miika Tuominen
- Quadratic reciprocity
- Mentee: Bliss Bodawala
- Mentor: Kian Cheong Aik
- Partitions and representation theory of Sn
- Text: Howe, An invitation to representation theory
- Mentee: Grace Guinan
- Mentor: Eleanor McSpirit
- Lattice-based cryptography
- Text: Hoffstein, Pipher, and Silverman, An introduction to mathematical cryptography
- Mentee: Jack Liu
- Mentor: Alejandro de las Peñas Castaño
Spring 2022
- Introduction to Knot Theory
- Text: The Knot Book, Adams
- Mentee: Faith Ellis
- Mentor: Louisa Liles
- Reasoning and Mathematical Elegance
- Text: Proofs from THE BOOK, Aigner
- Mentee: Blair Stilley
- Mentor: Walker Stern
- Categories and Arrows
- Text: Category Theory in Context, Riehl
- Mentee: Declan Stacy
- Mentor: Walker Stern
- Complex Analysis
- Text: Functions of One Complex Variable, Conway
- Mentee: David Bass
- Mentor: Shunyu Wan
- Elliptic Curves and Cryptography
- Rational Points on Elliptic Curves, Silverman
- Mentee: Said Kaili
- Mentor: Will Craig
- Introduction to Machine Learning
- Text: Machine Learning: A Probabilistic Perspective, Murphy
- Mentee: Mina Kim
- Mentor: Valentina Zapata Castro
- General Topology and an Application
- Text: Topology, Munkres
- Mentee: Zacary Abusmaies
- Mentor: Tanner Carawan
- p-adic Numbers and Interpolation of Special Zeta Values
- Text: p-adic Numbers, p-adic Analysis, and Zeta-Functions. Koblitz
- Mentee: Tanner Leonard
- Mentor: Joe Webster
- Fractals and Hausdorff Dimension
- Text: Fractal Geometry: Mathematical Foundations and Applications, Falconer
- Mentee: Ivory Tang
- Mentor: Gennady Uraltsev
- Introduction to Fourier Analysis
- Text: Fourier Analysis: An Introduction, Stein, Shakarchi
- Mentee: Ningxin Zhang
- Mentor: Gennady Uraltsev
Fall 2021
- Geometric Group Theory
- Text: Office Hours with a Geometric Group Theorist, Clay and Margalit
- Mentee: Olivia Goodrich
- Mentor: Alec Traaseth
- The Congruent Number Problem
- Text: Introduction to Elliptic Curves and Modular Forms, Koblitz
- Mentee: Catherine Cossaboom
- Mentor: Will Craig
- Brownian Motion and Stochastic Differential Equations
- Text: Probability: Theory and Examples, Durrett
- Mentee: Jeremy Nachison
- Mentor: Don Phillips
- Point Set Topology
- Text: Introduction to Topology, Baker
- Mentee: Liran Li
- Mentor: Shunyu Wan
- A Survey of Galois Theory
- Text: A Book of Abstract Algebra, Pinter
- Mentee: Yunsheng Lu
- Mentor: Matt McCarthy
- Finite Dimensional Lie algebras and their Root Systems
- Text: Lie Algebras of Finite and Affine Type, Carter
- Mentee: Alex Emmert
- Mentor: Weinan Zhang
- Möbius Inversion and Cyclotomic Polynomials
- Text: Generating Functionology, Wilf
- Mentee: Michael Walker
- Mentor: Joe Webster
- The Fundamental Group of a Knot Complement
- Texts: The Knot Book, Adams
- Mentee: Donovan Cates
- Mentor: Louisa Liles
- Cryptographic Hash Functions from Expander Graphs
- Text: Introduction to Modern Cryptography, Katz and Lindell
- Mentee: Eli Kin
- Mentor: Neelav Dutta
Spring 2021
- Hilbert’s Nullstellensatz
- Text: Abstract Algebra, Dummit and Foote
- Mentee: Omar Alsalihi
- Mentor: Alec Traaseth
- Isometries of the Hyperbolic Plane
- Text: Notes of Caroline Series
- Mentee: William MacMillan
- Mentor: Ian Runnels
- Exploring the Robinson-Schensted-Knuth Algorithm
- Text: Young Tableaux With Applications to Representation Theory and Geometry, William Fulton
- Mentee: Katherine Cadzow
- Mentor: Mojdeh Tarighat
- Grid Homology
- Text: Grid Homology for Knots and Links, Ozsváth, Stipsicz, and Szabó
- Mentee: Nick Cooney
- Mentor: Peter Johnson
Fall 2020
- Galois Theory and its Applications
- Text: A Book of Abstract Algebra, Charles Pinter
- Mentee: Bilal Khan
- Mentee: Casia Siegel
- Mentor: Will Craig
- Group Theory and Rubik’s Cube
- Text: Notes of Janet Chen
- Mentee: Sabrina Fuller
- Mentor: Alec Traaseth
- Classification of Surfaces
- Text: A Guide to the Classification Theorem for Compact Surfaces, Gallier and Xu
- Mentee: Harsh Padhye
- Mentee: Zhijun Cao
- Mentor: Alec Traaseth
- Analyzing Spaces
- Text: Functional Analysis, Bachman and Narici
- Mentee: Emma Graham
- Mentor: Bruno Braga
- Synthetic Topology
- Text: Synthetic Topology of Data Types and Classical Spaces, Martin Escardo
- Mentee: Joey Rudek
- Mentor: Neelav Dutta
Spring 2020
- Homological proof of the Brouwer fixed point theorem
- Text: Algebraic Topology, Hatcher
- Mentee: Thomas Blue
- Mentor: Chris Lloyd
Fall 2019
- The Geometries of Surfaces
- Text: Low-Dimensional Geometry, Bonahon
- Mentee: Haoyu Li
- Mentor: Ian Runnels
- Point-set Topology
- Text: Topology, Munkres
- Mentee: Dean Sublett
- Mentor: Shunyu Wan
- Fibered Categories
- Text: Categorical Logic and Type Theory, Jacobs
- Mentee: Nijat Khanbabayev
- Mentor: John Harnois
- Character Theory
- Text: Linear Representations of Finite Groups, Serre
- Mentee: Greg Conneen
- Mentee: Spencer Martin
- Mentor: Chris Lloyd
- Alexander’s Theorem
- Text: The Knot Book, Adams
- Mentee: Noah Brenny
- Mentor: Rostislav Akhmechet
Spring 2019
- Gröbner bases and applications in characteristic p
- Text: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Cox, O’Shea, and Little
- Mentee: Abrar Majidi Idrissi
- Mentor: Huy Dang
- Classification of the Simple Lie Algebras
- Text: Introduction to Lie Algebras, Erdmann and Wildon
- Mentee: Hisham M Assana
- Mentor: Arun Kannan
- The Geometry of Free Groups
- Text: Geometric Group Theory: An Introduction, Loeh
- Mentee: Katherine Andrews
- Mentor: Ian Runnels
- Hamiltonian ODEs: Deterministic and probabilistic studies
- Text: ODEs and Dynamical Systems, Teschl
- Mentee: Yunlu Li
- Mentor: Mouhamadou Sy
- The Yoneda Embedding and its Applications
- Text: Categories for the Working Computer Scientist, Barr and Wells
- Mentee: Joseph Snitzer
- Mentor: Chris Chung
- Real Analysis and Probability Theory
- Text: Real Analysis, Folland
- Mentee: Chris Pufko
- Mentor: Ethan Zell
- Representation Theory of Finite Groups
- Text: Representations and Characters of Groups, James and Liebeck
- Mentee: Jonah Newman
- Mentor: Liron Speyer
- Exploring Schottky Groups
- Text: Indra’s Pearls: The Vision of Felix Klein, Mumford, Series, Wright
- Mentee: Aaron Christian
- Mentor: Jiajun Yan
Fall 2018
- Support vector machines and kernel methods
- Text: Foundations of Machine Learning, Mohri, Rostamizadeh, and Talwalkar
- Mentee: Dylan Hunt
- Mentor: George Seelinger
- All Groups can be Realized as a Fundamental Group
- Text: Algebraic Topology, Hatcher
- Mentee: Kyle Hoffmann
- Mentor: John Harnois
- Topological Proof of the Nielsen-Schreier Theorem (a subgroup of a
free group is free)
- Text: Algebraic Topology, Hatcher
- Mentee: Mia Shaker
- Mentor: Chris Lloyd
- Difference Between Algebraic Bases and Schauder Bases
- Text: Real Analysis: Modern Techniques and Their Applications, Folland
- Mentee: Yifei Yang
- Mentor: Joseph Eisner
- Wirtinger Presentations of Fundamental Groups of Knot Complements
- Text: Algebraic Topology, Hatcher
- Mentee: Zach Baugher
- Mentor: Rostislav Akhmechet
- Theorem: Let G be a finitely generated group with two different
finite generating sets S and S’. Then G with the word metric induced
by S is quasi-isometric to G with the word metric induced by S’
- Text: Office Hours with a Geometric Group Theorist, Clay and Margalit
- Mentee: Sam Dulin
- Mentor: Ian Runnels
- Constructing Hyperbolic Structures on Closed Surfaces (of genus g)
- Text: Low-dimensional geometry: from euclidean surfaces to hyperbolic knots, Bonahon
- Mentee: Andrew Zazzera
- Mentor: Jiajun Yan
- Brownian Motion and its Applications
- Text: Brownian Motion, Peter Morters and Yuval Peres
- Mentee: Xinru Cheng
- Mentor: Ethan Zell
Summer 2018
- An Introduction to Computability Theory and ‘Universal Programs’
- Text: Computability: An introduction to recursive function theory, N.J. Cutland
- Mentee: Parker Lazear
- Mentor: John Harnois
- A Counterintuitive Subset of the Baer Space
- Text: Notes On Set Theory, Yiannis N. Moschovakis
- Mentee: Henry Carscadden
- Mentor: John Harnois
A great place to see more DRP projects is at other DRP program pages.