The DRP will run again in Spring 2026!
Applications are closed and students have been paired with their mentors. The projects which are running are listed blow.
Take a look at our Past ProjectsIntroduction to Frame Theory
Mentor: Trevor Camper
This reading program introduces frame theory, a mathematical framework for representing signals using collections of vectors that allow helpful redundancy. Unlike traditional bases, frames do not require every element to be unique or strictly necessary.
Suggested References: An Introduction to Frames and Riesz Bases by Ole Christensen
Prerequisites: Linear algebra, real analysis
Category Theory
Mentor: Ben Singer
Category theory is the unifying backbone of all of modern mathematics. To make an analogy, if different areas of mathematics are languages, then the foundations of category theory can be thought of as rigorous phonetics and phonology. In this project, students will become familiar with the foundations of category theory, such as functors, natural transformations, (co)limits, adjoint functors, and (potentially) monads. The goal will be to see how these concepts can fundamentally deepen our understanding of mathematics we have seen before, as well as develop new intuition for problem solving with powerful new tools. The culminating results of this project will be the Yoneda Lemma, the explicit expression of (co)limits in (co)complete categories, the interactions between adjoint functors and (co)limits, (potentially) the Beck monadicity theorem, and all of their deep philosophical consequences. Furthermore, we will also use category theory to give simplified proofs of classical results throughout mathematics (e.g. the Brouwer fixed point theorem, fundamental groups of topological groups are abelian, etc.). If we have time at the end, we may discuss some more advanced topics, such as beginning applications to homological algebra via abelian categories and derived functors, Grothendieck topologies and sheaves, some aspects of simplicial homotopy theory and infinity-categories, or the foundations of category theory's rich relationship with modern homotopy theory and higher algebra through Quillen's model categories.
Suggested References: Category Theory in Context by Riehl
Prerequisites: Math 71 and Math 74/114 (taking it concurrently is fine too!), or equivalent
Combinatorial Game Theory
Mentor: Michaela Polley
You've probably heard of games like Go and chess in which there is no element of randomness and no information is hidden. Games that satisfy these two properties are called "combinatorial games," and there is a lot of rich math to learn by studying them. We will explore lots of examples and discuss if there is always a winning strategy for one of the players. Depending on the background of the students we will be either more or less formal in our explanations and proofs.
Suggested References: Lessons in Play by Albert, Nowakowsi, and Wolfe
Prerequisites: None!
Introduction to Cryptography
Mentor: Allison Tsypin
ryptography has been used since ancient times to protect sensitive information. We will begin by studying classical examples such as substitution and shift ciphers, as well as basic ideas about encoding and breaking simple systems (with lots of examples along the way). From there, we will learn about the mathematical structures behind the basics of modern cryptography, including modular arithmetic, number-theoretic ideas, and groups and rings. Depending on student interest, we can read about RSA and Diffie–Hellman key exchange, which are used to secure communication over the internet today.
Suggested References: An Introduction to Mathematical Cryptography by Hoffstein, Pipher, and Silverman
Prerequisites: Basic linear algebra, e.g. math 22 or 24
Metric Geometry & Length Spaces
Mentor: Friedrich Bauermeister
Let $M$ be a smooth connected compact manifold and $(g_n)_{n\in \N}$ a sequence of Riemannian metrics. Let $(d_n)_{n\in \N}$ be the sequence of distance functions induced by the Riemannian metrics. Even if the sequence of distance functions (d_n) converges to a distance function d (in a certain sense), it will not generally be one induced by a Riemannian metric. Still, such limit metrics make $M$ into what is called a length space. A length space is a metric space in which the distance of two points is the infimum of the length of all paths connecting them. The geometry of length spaces is rich. For example, we still have a notion of geodesics and even curvature. This DRP will work through (parts of) the textbook "A course in Metric Geometry". I will also be working through the textbook while giving the DRP as I have been meaning to learn more about this subject.
Suggested References: A Course in Metric Geometry by Burago, Burago & Ivanov
Prerequisites: Real Analysis + Metric Spaces required. Point set topology would be very helpful. Some Riemannian Geometry would be very beneficial so that the book is well-motivated.
Mathematics of Card Shuffling
Mentor: GaYee Park
Card shuffling is a classic problem that can be studied using various mathematical tools. Among the many shuffling techniques ranging from the naive method of moving one card at a time to the dexterous riffle shuffle which are the most effective? How can this knowledge help us understand card games, magic tricks, or even traffic flow? While probability theory provides a natural framework for exploring these questions, shuffling also connects to other mathematical fields, including combinatorics, representation theory, and geometry. In this program, we will first study the application of linear algebra and probability to analyze shuffling, then study algebraic structures of shuffles and its connection to combinatorics and geometry (and more given enough time!).
Suggested References: The Mathematics of Shuffling Cards by Persi Diaconis and Jason Fulman
Prerequisites: Math 31 (or equivalent courses on abstract algebra)