Applications for Fall 2026 are Open! Apply Here

Proposed Projects:

Quantum Entanglement

Description: Readings on papers related to entanglement of quantum states. Quantum entanglement is a physical phenomena with important applications in information theory and computation. In this DRP we focus on the mathematical part of defining entanglement measures and try to see the more recent struggles in defining entanglement in large dimensinoal systems. 

Prerequisites: Students should have a background in linear algebra and basic quantum mechanics (e.g know Schrodinger's equation, Liouville–von Neumann equation, quantum entanglement, the basics of Hilbert spaces, etc.) for this.

Formal languages and Automata Theory

Description: What is a computer? Can we define problems that computers can't solve? Automata theory arose when mathematicians tried to define models of computation and understand the languages (or sets of strings) they accept. For this DRP, we will investigate what kinds of decision problems admit an algorithm that can compute the members of the set. We begin with deterministic finite automata and progressively "increase" the powers of the machine. If time permits, we will discuss problems which are undecidable even with a Turing machine, the first mathematical model of the modern-day computer.

References: Kozen, Automata and Computability, and a set of fantastic notes at https://people.cs.uchicago.edu/~timng/280/s26/notes/1.html

Prerequisites: Any experience with proofs (e.g. discrete math) suffices

Finite Representation Theory

Description: A representation of a finite group is a linear group action on a vector space. Studying the representations of a finite group allows us to use our knowledge of linear algebra to better understand the group itself. We'll read Serre's notes on finite rep theory and/or Fulton and Harris' Intro to Representation Theory.

Prerequisites: M373K and M373L, Algebraic Structures I.

Lie Groups and Lie Algebras

Description: Lie groups are groups that are also smooth manifolds, and a Lie algebra is an algebraic object associated with a Lie group that helps us study them. We will likely read Fulton & Harris' Intro to Representation Theory and/or Humphreys' Intro to Lie Algebras and Representation Theory.

Prerequisites: Algebraic Structures I and II, preferably also some exposure to representation theory

Mapping Class Groups

Description: The mapping class group is the group of diffeomorphisms of a manifold up to isotopy, and is an important object of study in geometric topology and geometric group theory. For this DRP project, we'll read Farb and Margalit's book "A Primer on Mapping Class Groups".

Prerequisites: We recommend at least one semester of undergraduate algebra and point-set topology.

Commutative Algebra

Description: Commutative algebra is a beautiful area of mathematics. For this project we will read Atiyah-Macdonald.

Prerequisites: Algebraic Structures II.

Advanced Linear Algebra

Description: We will read out of Advanced Linear Algebra by Roman. The goal would be to learn about applications of abstract algebra towards linear algebra (e.g. module theory and Jordan canonical form is something you could learn about; the description is vague, but we can look at the table of contents and see what's interesting)

Prerequisites: We recommend at least one semester of abstract algebra, preferably two (e.g. knowing some ring theory would be nice but not essential).

Calculus of Variations

Description: We will read out of Calculus of Variations by Gelfand and Fomin. The goal would be to to develop the basic theory and make sense of e.g., the Euler-Lagrange equation and the least action principle in physics. We should try to solve a few classic problems in the area; e.g., finding the brachistochrone curve.

Prerequisites: Strong background in undergraduate analysis

Introduction to Machine Learning: Theory and Applications

Description: This project provides an accessible but mathematically rigorous introduction to machine learning, with roughly equal emphasis on theory and applications. We will begin with a general introduction to AI and machine learning, then study topics including perceptrons, foundations of learning theory, linear and nonlinear regression/classification, logistic regression, support vector machines, Naive Bayes, decision trees, and ensemble learning. Our main reference will be Francis Bach’s Learning Theory from First Principles, supplemented by my notes, examples, and code. Students will work through problems and computational applications during the semester and complete a final project during the last few weeks.

Prerequisites: Calculus and linear algebra. Some familiarity with probability is helpful. No previous machine learning experience is required.

Gauge Theory

Description: Gauge theory is both a powerful tool for deriving invariants of certain smooth topological spaces and the framework that underlies much of modern particle physics. One of the primary goals in gauge theory is to study the space of solutions to a partial differential equation while being careful to ensure that the way in which you take measurements does not matter. We will follow "Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics" by Mark Hamilton.

Prerequisites: Ideally, it would be best to know what a smooth manifold is, but we can work around this. More importantly, a strong grasp of linear algebra, calculus, and basic group theory are recommended.

Smooth Manifolds

Description: In this project you will learn what a smooth manifold is, how we define vector fields and differential forms on them, and how doing so allows for calculus to be done on spaces which are not just a subset of Euclidean space. We will follow "Manifolds and Differential Geometry" by Jeffrey Lee.

Prerequisites: A solid understanding of linear algebra and calculus are recommended.

Analytic Methods in Enumeration

Description: In this project we can explore how analytic tools can be used to study combinatorial counting problems. We will focus on generating functions, how they encode combinatorial structures, and how the symbolic method allows us to translate counting problems into algebraic identities. Time and progress dependent we can introduce basic asymptotic methods, including singularity analysis, to understand the growth of sequences such as the Catalan numbers and related examples.

References: Flajolet and Sedgewick, Analytic Combinatorics, Chen and Koh, Principles and Techniques in Combinatorics.

Prerequisites: Some familiarity with proofs and basic combinatorics. Calculus I–II is sufficient for the necessary background on sequences and series.

Khovanov Homology

Description: We'll briefly introduce knots and links and then proceed to define an invariant of them that takes the form of a collection of abelian groups. We will attempt to get through the start of Melissa Zhang's notes but will detour to learn homological algebra as needed.

Prerequisites: Some knowledge of abstract algebra will be required, and knowledge of topology will be helpful but not necessary.

Introduction to Knot Theory

Description: This project will be a prerequisite-free and fun introduction to knot theory. We will read Adams’s The Knot Book and answer questions such as: how do we construct interesting knots? When are two knots the same? Different? There will be a lot of nice pictures!

Prerequisites: None.

Topos Theory and the Continuum Hypothesis

Description: The Continuum Hypothesis (CH) is the claim that there is no set S whose cardinality is strictly between that of the natural numbers and that of the real numbers (the continuum). In 1963, Cohen showed that the Continuum Hypothesis cannot proven from the currently accepted axioms of mathematics, establishing it as independent from these axioms. This project aims to present a modern version of Cohen’s proof through Topos Theory, a field of mathematics with applications to topology, logic, algebra, and geometry.

Prerequisites: At least 1 course each in abstract algebra and topology, basic understanding of category theory and universal properties.

Matroid Theory

Description: In this directed reading program, we will study matroid theory. Matroids generalize the notion of linear independence from linear algebra and its combinatorial analogue in graph theory. Depending on the student’s background and interests, we may explore connections to algebra, graph theory, polytopes, and geometry.

References: Our main reference will be James Oxley’s Matroid Theory.

Prerequisites: The only prerequisites are linear algebra and some previous exposure to combinatorics.

Classical Mechanics

Description: The aim of this project is to learn about the frame works for classical mechanics (the physics which tells us how objects move at medium size and energy) known as Lagrangian and Hamiltonian mechanics. We will start with an intro from Taylor's classical mechanics book and then learn how to work more rigorously using a math book like Marsden and Ratiu's intro to mechanics and symmetry.

Prerequisites: Vector calculus and linear algebra.

The Cobordism Hypothesis

Description: Lurie's work on extended TQFT beautifully packages maths of seemingly different flavors in a very concise statement (under appropriate setting of infinity categories). Examples vary from finite dimensional vector spaces (with or without inner product) to ribbon categories (which yield knot/link polynomials), Calabi-Yau structure, Serre duality, Hochschild homology and so on. I found it one of the most interesting connections between topology and algebra. Goals of the project is to have a sense of the statement (without going all the details of infinity categories) and see some interesting applications. For the first part, I think the following exposition by Freed is great: https://arxiv.org/pdf/1210.5100

Prerequisites: Ordinary category theory (1-categories), differential topology (manifolds), interests in topology (e.g. invariants of manifolds, knot polynomials) or interests in algebra/category (e.g. Frobenius algebras, Hopf algebras, monoidal categories). Knowing homotopy theory can be helpful but not required.

Principal Bundles and Characteristic Classes

Description: Principal bundles on a manifold M are geometric structures encoding interesting topological data of M. It simply generalizes covering spaces and vector bundles. Characteristic classes are invariants of principal bundles that measures how twisted the bundle is. In a way, it is similar to the characteristic polynomial of a square matrix which contains trace and determinant. It can be defined and treated purely homotopically (cohomology of classifying spaces) or differentially (Chern-Weil theory). The goal is to understand bundles with examples, define characteristic classes and see what kind of informations do they encode.

References: Characteristic Classes by Milnor–Stasheff, Differential Geometry: Connections, Curvature, and Characteristic Classes by Loring Tu, Vector Bundles and K-Theory (unfinished) by Hatcher.

Prerequisites: Basic algebraic topology and differential topology.

Geometric Group Theory

Description: Groups are usually introduced as algebraic objects, but many of the most interesting ones, such as free groups, surface groups, Coxeter groups, and mapping class groups, are best understood by turning them into geometric objects. Geometric group theory does exactly this: by having them act on metric spaces, we can ask how the large-scale geometry constrains the algebra. In this DRP project, we'll read "Office Hours with a Geometric Group Theorist" edited by Clay and Margalit, which is a collection of short, self-contained expository chapters by different authors that introduces the core ideas of geometric group theory such as Cayley graphs, quasi-isometries, free groups, hyperbolic groups, and their applications to the examples above and more. The book is written for exactly this kind of undegraduate reading project, with lots of pictures and exercises we'll work through together

Prerequisites: A first course in abstract algebra is all the background you need.