Applications Open

The application for the Spring 2026 semester is now open, apply here! Here are the proposed projects for this semester (note that you may apply even if you are not interested in any of the proposed projects):

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.

Knot Theory and the Alexander polynomial

Description: For more advanced students, an introduction to knot theory that emphasizes classic algebraic topology invariants of knots such as the Alexander polynomial. 

Prerequisites: Linear algebra and abstract algebra

Measure Theory

Description: When and how can we quantify the size of a set? For this DRP project, we'll follow a introduction to measure theory text and develop the abstract theory of measures. From there we can continue in two directions, either into probability or real analysis. 

Prerequisites: Basic real analysis.

Spin Geometry

Description: For this project we will mostly follow Lawson and Michelsohn's book "Spin Geometry." Spin geometry gives a very hands-on experience in the interplay of algebra, geometry, and topology. Through the semester we will hopefully touch on the basics of smooth manifolds, Lie groups, and topological invariants.

Prerequisites: A basic understanding of abstract algebra and a good grasp of multivariable calculus should be more than sufficient.

Semi-Riemannian Geometry

Description: For this project we will follow "Semi-Riemannian Geometry with Applications to Relativity" by Barrett O'Neill. This project is intended to give an introduction to the differential geometric formalism of relativity, as the title of the book suggests. Ideally, we would get to talk about smooth manifolds, tensor fields, and metrics.

Prerequisites: A strong background in linear algebra and multivariable calculus. A basic understanding of abstract algebra could be helpful but isn't necessary. 

Topos Theory and the Continuum Hypothesis

Description: A topos is a category which is similar to the category of sets in certain relevant respects. Drawing mainly from Mac Lane and Moerdijk’s “Sheaves in Geometry and Logic”, we will study how toposes provide models for set theory; the objective will be producing a model of set theory in which the continuum hypothesis fails. 

Prerequisites: Knowledge of basic category theory (limits, colimits, functors, natural transformations, Yoneda lemma, and adjunctions) is a mandatory prerequisite. Other nice-to-haves: experience with formal logic, set theory, and sheaves.

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 on Statistical Inference by Casella & Berger and work on some of the exercises. Our focus will be on rigorously developing and analyzing classic techniques used in statistical inference that are introduced in an introductory statistics course, such as maximum likelihood estimation, hypothesis testing, and confidence intervals. 

Prerequisites: An introductory statistics course and introductory probability.

Geometric Mechanics

Description: Classical mechanics seeks to describe the motion of objects by studying phase space, the space of all positions and "momenta" of a system. Geometric mechanics is a generalization of classical mechanics where one can use the tools of differential geometry and topology to study new questions. For this DRP project, we will read either part III of Arnold's "Mathematical Methods of Classical Mechanics" or select chapters from Marsden and Ratiu's "Introduction to Mechanics and Symmetry." 

Prerequisites: For this project you should have some experience with vector calculus, linear algebra, and group theory.

Introduction to topological manifolds

Description: Many people in the past (and some in the present) thought the Earth was flat. This make sense; from our point of view, the Earth looks like a flat plane. This is because the Earth is a manifold -- a space that looks locally like R^n (here n=2). In this project, we will build up a rigorous definition of a topological manifold and explore some of the nice properties they possess using John Lee's Introduction to Topological Manifolds. 

Prerequisites: Knowledge of point-set topology would be helpful but not strictly required; in its absence some real analysis, particularly knowledge of metric spaces, would be useful. 

Continued Fractions

Description: Continued fractions are a way of representing real numbers that have fascinating number theoretic properties. All real numbers have essentially unique continued fraction decompositions; a number's continued fraction decomposition is finite if and only if it is rational, it is periodic if and only if it is a quadratic irrational. Continued fractions are useful for solving certain Diophantine equations. Bizarrely, the geometric mean of the coefficients of almost all continued fraction decompositions is equal to a single constant (Khinchin's constant), even though this is not true for most well-known constants. 

Prerequisites: Basic knowledge of number theory.

The Math Behind Deep Learning

Description: We will explore the theoretical math underlying modern deep learning models/algorithms. Topics include optimization, stochastic processes, sampling, numerical approximation, and more. There is potential flexibility for the student’s own interests. We will use the following book: https://mathdl.github.io/.

Prerequisites: A solid handle on linear algebra and basic probability theory at a minimum. Real analysis or some experience with stochastic processes is a plus.

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.

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

Knots and 4-Manifolds

Description: An appropriately decorated knot diagram can be used to construct either 3 or 4-dimensional manifolds. Such diagrams are one of our principal tools for telling when two manifolds are the same, and manipulating them is more of an art than a science. In this project we will read 4-Manifolds and Kirby Calculus by Gompf and Stipsicz to learn how these diagrams represent manifolds, and how to transform them.

Prerequisites: Basic algebraic topology

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 (e.g. a height function on a sphere or torus).

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

Geometry of Surfaces

Description: Following Stillwell, Geometry of Surfaces or Bonahan, Low-dimensional Geometry, for this DRP project we will learn how to think about surfaces (2-dimensional shapes) with geometry (some notion of how the surface curves). In particular we will focus on constant-curvature surfaces: surfaces which everywhere locally look like either the Euclidean plane, the sphere, or the hyperbolic plane. We will see how all complete constant-curvature surfaces can be obtained from one of these three model surfaces using an operation called quotienting. Depending on time and interest we can also investigate similar topics for 3-dimensional manifolds.

Prerequisites: Multivariable calculus. Some exposure to basic group theory or topology would be helpful but is not necessary.

Classical Differential Geometry

Description: Investigate curves, surfaces, and their curvature using the language of calculus and linear algebra. We will follow texts by authors such as John Lee or Aleksey Penskoi, focusing on topics like Gauss-Bonnet theorem and geodesics.

Prerequisites: Multivariable calculus, basic topology would be helpful.

Geometric Relativity

Description: Study the geometric structure of spacetime and Einstein's equations through the lens of Riemannian and Lorentzian geometry. Under the guidance of Dan A. Lee, we'll explore concepts like initial data sets, black holes, and positive mass theorems.

Prerequisites: Multivariable calculus, basic topology would be helpful.

Hilbert Spaces

Description: Hilbert Spaces are infinite dimensional vector spaces used to solve Partial Differential Equations and model one-particle Quantum Physics. In this project, we will read parts of Brezi's book ""Functional Analysis, Sobolev Spaces, and Partial Differential Equations"" and solve a PDE with the Riesz Representation Theorem (and maybe some Spectral Theory if there is time). 

Prerequisites: Background in linear algebra, real analysis would be great for this DRP; however, students can still do well in this project without taking those courses.

Commutative Algebra

Description: Commutative algebra, while serving as the backbone of modern algebraic geometry, is an intrinsically beautiful subject that provides rich generalizations of many of the results from a first algebra class covering rings. For this DRP project, we will be reading through the classic, Atiyah and MacDonald's ""Introduction to Commutative Algebra.""

Prerequisites: Students should have a strong background in ring theory (i.e. have taken an undergraduate class on abstract algebra that covered rings or, better yet, a graduate course)

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

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.