Applications Open!

The application for the Fall 2025 semester is now open! It will close on Monday, September 8. 

Proposed Projects:

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.

Advanced Knot Theory

Description: We will study knot theory from the 3-manifold perspective, using Rolfsen's text Knots and Links. Topics include classical invariants such as the Alexander polynomial, as well as 3-manifold techniques like Dehn surgery.

Prerequisites: Upper level proof based math and abstract algebra.

p-adic Numbers

Description: We will read from a subset of a number of sources, including an REU paper by Pomerantz and a book by Gouvêa. The p-adic numbers are a peculiar field that arises from taking the completion of the rationals with respect to an unusual absolute value. This results in what is called a local field, which has interesting topological properties.

Prerequisites: M373K, Algebraic Structures I.

Measure Theory and Lebesgue Integration

Description: Measure theory provides a rigorous way to assign a notion of size to sets, even highly irregular ones like the rationals or the Cantor set. Building on this, the Lebesgue integral extends the classical Riemann integral and serves as a cornerstone of modern analysis. In this DRP project, we’ll develop the theory of sigma-algebras, measurable sets, and measures, then construct the Lebesgue integral and study its fundamental convergence theorems (such as dominated and monotone convergence). Along the way, we’ll encounter examples of non-measurable sets and explore applications depending on your interests. Possible references include Axler’s Measure, Integration & Real Analysis and Royden & Fitzpatrick’s Real Analysis.

Prerequisites: Basic point-set topology, undergraduate real analysis would be good too.

Intro to smooth topology via Morse Theory

Description: For this DRP project, we will read Matsumoto's An Introduction to Morse Theory. Morse Theory is a technique used to study surfaces (and higher dimensional analogs) through examining the gradient vector fields of differentiable functions defined on them (eg. a height function on a sphere or torus).

Prerequisites: Comfortability working with multivariable calculus (ie, you did well in and enjoyed 408D/408M). Some experience with topology or geometry is helpful but not required!

Geometric Group Theory

Description: We can learn a lot about groups by studying how they arise as symmetries of various geometric objects. As a starting point, every group can be thought of as the group of symmetries of its own Cayley graph. For this DRP project, we will work through chapters from the book Office Hours with a Geometric Group Theorist by Clay and Margalit.

Prerequisites: Some basic familiarity with groups, e.g. one semester of abstract algebra

Statistical Inference

Description: Statistical inference allows us to make sense of the world when we only have access to limited data. In this DRP project, we will read the legendary textbook Statistical Inference by Casella & Berger and work on some of the exercises. Our focus will be on formally developing and analyzing classic techniques used in statistical inference that are introduced (but never motivated!) in an introductory statistics course, namely, maximum likelihood estimation, hypothesis testing, and confidence intervals.

Prerequisites: An introductory statistics course and introductory probability (one that covers random variables and, more importantly, functions of random variables).

Algebraic Geometry

Description: Algebraic geometry is a subject to study the vanishing locus of polynomials. The modern and systematic way to study this subject is to use the notion of scheme developed by Grothendieck. In this DRP project, we will make our first step in learning scheme theory.

Prerequisites: Abstract Algebra (Groups, Rings and Modules), Point Set Topology. Some understanding of smooth manifolds and commutative algebra might help.

Smooth Manifolds

Description: Smooth manifolds are a type of topological space that are in some sense locally equivalent to Euclidean space. For this project, we would explore their definition, some topological invariants of them, and what it means to do calculus on such a space. To do this we will read John Lee's Introduction to Smooth Manifolds.

Prerequisites: Basic point-set topology and vector calculus.

Intro to Algebraic Topology

Description: Algebraic invariants of topological spaces and their construction are one of the first places people may encounter functors. Our goal for the semester will largely be to introduce the concept of homotopy and to construct the fundamental group functor. For this project, we will be reading Allen Hatcher's Algebraic Topology.

Prerequisites: Solid understanding of groups.

The Topology and Geometry of Surfaces via Curves

Description: In this project, we will explore aspects of the geometry and topology of surfaces using a wide variety of techniques. You'll learn some algebraic topology, differential topology/geometry, and hyperbolic geometry in the course of this project. Depending on the student's background, possible texts include Thurston's Work on Surfaces or A Primer on Mapping Class Groups.

Prerequisites: Multivariable calculus and linear algebra at a minimum. Ideally, some topology and abstract algebra.

Algebraic Combinatorics

Description: Some counting problems look simple but turn out to hide deep structure. For example:

– How many ways can you tile an m by n rectangle with 2 by 1 dominoes?

– How many spanning trees does a given graph have?

– How many different ways are there to place non-attacking rooks on a chessboard with holes?

All of these are tough to approach by direct counting, but algebraic tools—linear algebra, determinants, generating functions—reveal surprisingly elegant answers. This project is an introduction to those tools and their use in solving otherwise intractable combinatorial problems.

Prerequisites: Command of linear algebra and elementary combinatorics, mathematical maturity. We will follow Stanley's Enumerative Combinatorics Vol. 1.

Geometric Foundations of Data Science

Description: Many classical machine learning algorithms can be understood from a geometric perspective. We will learn about the underlying geometry of supervised and unsupervised algorithms and discuss how these insights can help us apply these algorithms to data. This project is entirely theoretical; no coding is required or expected.

Prerequisites: Multivariable Calculus, Probability, Linear Algebra

Qualitative Behavior of Solutions to Differential Equations

Description: Differential equations are used to model a great variety of systems in physics, engineering, chemistry, biology, ... Unfortunately, most equations have solutions that cannot be written down explicitly. However, even if that is the case, there are many tools that can be used to prove that solutions will behave in a particular way. For example, in a model for two populations competing for resources in an ecosystem, it is possible to predict when one of them will go extinct.

Prerequisites: The minimal prerequisite would be knowing one variable calculus (derivation and integration). Exposure to differential equations would be very useful although it is not completely necessary. If the student knows multivariable calculus then we could study more complex models based on partial differential equations.

AI and Topology

Description: How can ideas from machine learning help us study geometric or topological objects? One direction is to use AI techniques to learn self-diffeomorphisms of manifolds in order to sample complicated probability distributions that live on them. For instance, what happens if we try this on a 2-sphere or other simple surfaces? More broadly, the project could explore other ways topology and AI interact, depending on the student’s interests.

Prerequisites: Solid background in linear algebra (vectors and matrices) and multivariable calculus (through Calc III). Some familiarity with Python is helpful; willingness to learn PyTorch or similar frameworks is important. Basic exposure to parametrizations of 2-manifolds is a plus but not required.

Introduction to p-adic L-functions

Description: Roughly follow part 1 of https://arxiv.org/abs/2309.15692. The goal is to read a proof of the construction of the Kubota-Leopoldt p-adic L-function.

Prerequisites: Know what p-adic and L-function mean (separately). Working understanding of algebraic number theory and complex analysis.

Modular Forms

Description: Follow Diamond–Shurman, focusing on the Hecke operators theory. We focus mainly on ch. 5 (apart from the basics), and use results from ch. 1–4 only as required. The goal is to read a proof of the weak multiplicity one theorem for newforms.

Prerequisites: Working understanding of algebraic number theory and complex analysis.

Topology of Tiling Spaces

Description: Aperiodic tilings have been of interest recently, especially with the Einstein tile making the headlines in 2022. We will read Sadun’s book Topology of Tiling Spaces to see what kind of topological properties aperiodic tilings have (namely cohomology in this book).

Prerequisites: Point set topology, linear algebra; algebraic topology would be nice but not required.

Intro to Algebraic Geometry

Description: Algebraic geometry is the study of the solutions to polynomial equations. Possible reading would include Shafarevich’s Intro to Algebraic Geometry and Reid’s Undergraduate Algebraic Geometry. For some background in commutative algebra, we could read Atiyah–Macdonald’s Introduction to Commutative Algebra.

Prerequisites: M373K and M373L (Algebraic Structures I and II). M380C Graduate Algebra I would be helpful but not necessary.

Hilbert Spaces and Compact Operators

Description: This DRP deals with infinite dimensional vector spaces which are useful when studying classes of continuous functions. We will study these “Hilbert spaces” and the linear mappings between them which are called “operators.” The compact operators enjoy several properties that mirror finite dimensional matrices, and this leads to applications in differential equations and PDE. Topics include Hilbert Spaces, the Open Mapping Theorem, the Spectral Theorem, and the Fredholm Alternative. We will follow Functional Analysis for the Applied Mathematician by Arbogast and Bona.
Prerequisites: Linear algebra, Real analysis, point-set topology.