Directed Reading Program

The DRP will run again in Winter 2026!

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Differential Geometry

Mentor: Friedrich Bauermeister

We will discuss semi-Riemannian geometry, which generalizes both Riemannian geometry ("the geometry of space") and Lorentzian geometry ("the geometry of spacetime"), from the ground up. The book used will be Barret O'Neill's Semi-Riemannian geometry with applications to relativity, which is a really good book. This DRP is an excellent chance to (1) first learn about smooth manifolds or to strengthen ones familiarity with them, (2) gain insights into how to "properly" generalize notions of straight lines, distances, and angles to the setting of smooth manifolds, and (3) and to learn about the mathematics underlying general relativity in a rigorous way.

Suggested References: Semi-Riemannian geometry with applications to relativity

Prerequisites: (proof-based) experience with linear algebra and with at least one of analysis or point-set topology. Experience with smooth manifolds is not required but very advantageous.

Fourier Optics

Mentor: Patrick Addona

This reading course will serve as an introduction to the principles and mathematical foundations of Fourier optics. Students will learn how Fourier analysis applies to optical imaging systems, diffraction, and filtering. In particular, we will develop an intuition for the relationship between spatial and frequency domains. Ultimately, we will see how this applies to signal processing, which can be used to transform digital images!

Suggested References: Introduction to Fourier Optics - Joseph W Goodman

Prerequisites: Multivariable calculate (MATH 13) and linear algebra (MATH 22). Introductory knowledge of optical physics (diffraction, wave propagation, interference) is recommended but not required! We will discuss this as background in the first session.

Fractal Geometry and Measure Theory

Mentor: Rohan Kapoor

We will study the construction of the Hausdorff measure and fractal construction, and use it to analyze the 'size' of fractal sets. We will then compute the fractal dimension of some basic examples, and learn different techniques to do so (known as potential theoretic methods). After this, we can either further build out the theory of fractals and work up to problems in current geometric measure theory research (like the recently solved Kakeya conjecture), or learn about the applications of this theory to number theory, iterated function systems, or Brownian motion.

Suggested References: Fractal Geometry: Mathematical Foundations and Applications (Falconer)

Prerequisites: Honors Real Analysis (Math 63) or Math 103, basic programming experience

Galois Cohomology

Mentor: Ben Singer

Understanding the structure of the absolute Galois group of a given field has long been the culminating question of number theory. Many of number theory�s greatest advances (e.g. class field theory, modularity for elliptic curves) and biggest question marks (e.g. the Langlands correspondences, inverse Galois problem, Fontaine-Mazur conjecture) surround progress on this particular question. In this DRP, we will study the behavior of the absolute Galois group of a field through its group cohomology, the homological study of its actions on abelian groups (usually called Galois modules). This will include preliminary coverage of profinite groups, a refresher on derived functors, and an exploration of the main theorems of group cohomology. From there, we will discuss the relationship between Galois cohomology and central simple algebras, covering applications such as the Hasse-Minkowski theorem, computation of the Brauer group of the p-adic numbers, and the Hasse-Brauer-Noether-Albert exact sequence. Depending on time and interests, we may also cover some more advanced aspects of Galois cohomology and Galois actions, such as cohomological aspects of elliptic curves, an introduction to Galois representations and modularity, or a discussion of Milnor K-Theory and the Merkurjev-Suslin Theorem.

Suggested References: Gille-Szamuely, "Central Simple Algebras and Galois Cohomology"

Prerequisites: Math 81/111. Math 101 and Math 54 (or equivalent) are recommended, but a student willing to read extra material will have no problem getting by without them.

Representation theory of finite groups

Mentor: Allison Tsypin

Representation theory is a fascinating subject that allows us to link abstract algebraic structures (groups) to concrete, more easily understandable objects (matrices). A "representation" of a group is a vector space together with a linear action of this group, which lets us associate each group element with a matrix. It turns out that we can build an entire theory about these representations. What are their building blocks? What properties must they have? What does it mean for two representations to be "the same", and how can we check this property? We'll see how concepts of group theory and linear algebra work together to give us answers to these questions. Depending on student interest, we could also look at connections to Fourier analysis on finite groups or links to combinatorics through representations of the symmetric group.

Suggested References: Representation theory of finite groups - Benjamin Steinberg (https://link.springer.com/book/10.1007/978-1-4614-0776-8)

Prerequisites: Math 31 or Math 71

Quantum Information Theory

Mentor: Gage Hoefer

In the classical communication paradigm, we are concerned with the transmission of information between two parties. Oftentimes, the channels we use to transmit information are imperfect, and inherent defects (or noise) of the channel can affect the messages we send. Basic methods for addressing these defects, and the possibility of correcting random errors in transmission, were first introduced by Shannon in the '50's; these methods established a link between asymptotic combinatorics and classical information theory which is still utilized today. With the advent of quantum computing and communication, we are interested in addressing the same fundamental issue in this setting--- namely, how to correct or protect against random errors introduced by noisy quantum channels. In this project, we will learn the basics of quantum information theory: what quantum information and quantum channels are, how noise affects quantum information, and different ways to address this problem (through zero-error transmission or quantum error correction). Time permitting, we might look into the use of different classes of resources in zero-error channel coding, and connections to non-local games.

Suggested References: "The Theory of Quantum Information" by John Watrous, or "Quantum Computation and Quantum Information" by Nielsen and Chuang

Prerequisites: MATH 22, MATH 24 (MATH 40 might also be helpful but is not necessary. No physics background is required)

Linear Algebra Methods in Combinatorics

Mentor: Alex Moon

Combinatorics, or the study of enumeration of finite objects, is one of the oldest and most wide reaching branches of mathematics. One classic combinatorial bound is that of dimension - the size of a linearly independent subset of a vector space cannot exceed its dimension. Thus, a powerful combinatorial technique is to "linearize" a problem so that the objects of interest form a linearly independent set in a finite dimensional vector space, then use the dimension bound to get a cheap and powerful limit on the number of objects. In this book, we will study applications and variations of this technique to a wide range of combinatorial problems, including finding bounds for Ramsey numbers, generalizing Helly's theorem to other discrete settings, several versions of Bollob�s' theorem, and more.

Suggested References: Linear algebra methods in combinatorics (Babai and Frankl)

Prerequisites: Linear algebra, some flavor of proofs

The Math of SET

Mentor: Beth Anne Castellano

SET, also known as "the family game of visual perception," is a seemingly simple card game that has deep mathematical connections and has even inspired recent mathematics research. Together we'll read The Joy of SET and ask/answer some of the many mathematical questions that the game invites in areas such as finite affine geometry, combinatorics, and probability. Depending on student interests, students can also work on one of the projects suggested in the book, or we can explore related research papers on the "maximal cap" problem or other variations of SET. This DRP does not require any prerequisites -- any mathematically curious students are welcome, and I hope it will serve as an introduction to some fun math you haven't seen before! (Yes, we will play SET a lot too :))

Suggested References: The Joy of SET: The Many Mathematical Dimensions of a Seemingly Simple Card Game (McMahon, Gordon^3)

Prerequisites: Math 22 may be helpful for particular topics, but not necessary!

(Arithmetic of) Elliptic Curves

Mentor: Aidan Hennessey

Number theory is all about understanding the whole number and rational number solutions to polynomial equations. For 2-variable polynomials of degree 1 or 2, this is well understood. Once you hit degree 3, things become much more interesting. These degree 3 polynomials, called elliptic curves, have a ton of structure, tying into some of the biggest questions in math (such as Fermat's last theorem). They also underpin the most common encryption algorithm used in all modern communications, so they are essential for computer scientists. This DRP will study elliptic curves and their group laws, diving into specific topics in accordance with student interest.

Suggested References: Rational Points on Elliptic Curves, by Silverman and Tate

Prerequisites: Math 31 or 71 is preferable (taking them concurrently is totally fine).

Ergodic Theory and Dynamical Systems

Mentor: Rohan Kapoor

We will cover the pure-mathematical aspects of ergodic theory: the study of chaotic systems. Based on the students' background, we will either highlight the topological or the analytical/measure-theoretic aspects of dynamical systems theory. Highlights include some applications of this formalism to classical mechanics, chaos theory, and number theory.

Suggested References: Walters: Introduction to Ergodic Theory

Prerequisites: Math 22/24, and either Honors Real Analysis (Math 63) or Point-Set Topology (Math 54)

Introduction to Operator Algebras

Mentor: Rohan Kapoor

This would be a graduate-level reading course in operator algebras, which requires a strong background in functional analysis. Operator algebras are an extremely powerful and abstract tool that allow us to systemize the study of quantum mechanics, functional analysis, and partial differential equations using algebraic tools. We will cover the basics of both C* and von-Neumann algebras, and work through the theory of Gelfand duality and the Gelfand-Naimark_Segal construction. Based on the students' interests, we can then explore applications to geometric group theory, representation theory, ergodic theory, or quantum information theory.

Suggested References: An Invitation to C*-Algebras - Averson + Notes of Prof. Dana Williams

Prerequisites: Functional Analysis (Math 113)

Learning theory and Kernel Machines

Mentor: Rohan Kapoor

This course will use the framework of functional analysis to study problems in machine learning and learning theory. We will prove the celebrated representer theorem and see how kernel algorithms are useful to solve nonlinear problems in a linear framework. We will also see many perspectives on kernel integral operators, as regularizing operators, sources of orthonormal bases, and as feature maps for a kernel machine. This would be useful for people interested in research in optimization, statistical learning theory, or machine learning from a mathematical perspective. Some programming background in Python is necessary.

Suggested References: Learning with Kernels: Scholkopf and Smola

Prerequisites: Honors Real Analysis (Math 63), Linear Algebra (22/24)

History of Mathematics

Mentor: Stoyan Dimitrov, David Shuster and Alejandro Galvan

"Students who enroll in this DRP will be able to pick their specific topic of interest, from the list of intriguing and diverse topics within History of Mathematics below: 1. Tracing the history of the major ideas in calculus: from its global origins to its modern form. Sources: the excellent book by D. Bressoud, together with his online course. 2. The leading statistician, Ronald Fisher - life, work, and motivation coming from eugenics? Sources: the book by J. Fisher, together with two articles on the topic. 3. The man who outperformed Ramanujan in calculations: The life and work of Major Percy MacMahon. Sources: P. Garcia's doctoral thesis. 4. Studying the history of the Central Limit Theorem. Sources: the book by H. Fischer. 5. Five great mathematical journeys, told through the original texts of the mathematicians who paved the way. Sources: the book by R. Laubenbacher and D. Pengelley."

Suggested References: 1. Calculus Reordered: A History of the Big Ideas - David M. Bressoud. 2. R. A. Fisher, the Life of a Scientist - Joan Fisher Box. 3. The life and work of Major Percy Alexander MacMahon - Paul Garcia. 4. A History of the Central Limit Theorem: From Classical to Modern Probability Theory - Hans Fischer. 5. Mathematical Expeditions: Chronicles by the Explorers - Reinhard C. Laubenbacher, David Pengelley.

Prerequisites: None

Inverse Problems in Quantum Information

Mentor: Casey Dowdle

We will review the principles of quantum information and quantum computing. Then we will discuss various applications of inverse problems such at quantum state tomography and error mitigation on devices. Finally we will interact with real quantum computers through the cloud to solve real world inverse problems.

Suggested References: �Quantum Computation and Quantum Information � Michael A. Nielsen & Isaac L. Chuang

Prerequisites: Comfortable with calculus and linear algebra.

C*-algebras and K-theory

Mentor: Ahmadreza Hajsaeedisadegh

C*-algebras arise from the algebra of observables in quantum mechanics, which carry not only an algebraic structure but a topology as well. We will discuss many examples of C*-algebras. We also discuss the K-theory of these algebras and their connection to the index theory of Fredholm operators. This topic is closely related to the topological K-theory of Atiyah and Hirzebruch. If time allows we will explore the latter.

Suggested References: An Introduction to K-theory for the C*-algebras by Rordam, Larsen, and Lausten

Prerequisites: Math 31, familiarity with groups and rings

Hyperbolic geometry

Mentor: Ahmadreza Hajsaeedisadegh

We study one of the most well-known non-Euclidean geometries. We initially learn about complex numbers and then move on to experiment with models of the hyperbolic plane, such as Poincar�'s half-plane and disk models. We learn the correct notion of line, length, and distance, and compare results in Euclidean geometry with those in hyperbolic geometry. We discuss M�bius transformations and their conformity. If time allows, we will cover Gauss-Bonnet's theorem.

Suggested References: Hyperbolic Geometry by James Anderson

Prerequisites: Math 31 or equivalent.