Projects

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.