- 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
- 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

- Homological proof of the Brouwer fixed point theorem
- Text: Algebraic Topology, Hatcher
- Mentee: Thomas Blue
- Mentor: Chris Lloyd

- 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

- Gröbner bases and applications in characteristic p
- Text: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, David A.Cox, Donal O’Shea, and John B. 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, Clara Loeh
- Mentee: Katherine Andrews
- Mentor: Ian Runnels

- Hamiltonian ODEs: Deterministic and probabilistic studies
- Text: ODEs and Dynamical Systems, Gerald Teschl
- Mentee: Yunlu Li
- Mentor: Mouhamadou

- 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

- Support vector machines and kernel methods
- Text: Foundations of Machine Learning by 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 by Peter Morters and Yuval Peres
- Mentee: Xinru Cheng
- Mentor: Ethan Zell

- 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.