Mentor profiles
Fall 2026
Fall 2026
Applications for Fall 2026 DRP are now open! Link here!
Mathematical Interests: My mathematical interests are broadly in mathematical finance, applied probability, optimization, and machine learning.
Some possible projects: We can have three possible project directions. The first is diffusion models and their mathematical foundations, where we can study the ideas behind score-based generative modeling, stochastic differential equations, and how diffusion models learn to generate data.
The second is high-frequency trading and optimal market making, where we can study stochastic optimal control models for quoting and inventory risk, and then ask whether modern generative models can make these classical frameworks a little smarter.
Ever wondered how a pile of mortgages turned into a global financial headache in 2008? :) That leads to the third project: mortgage credit risk and prepayment. We can use Fannie Mae and Freddie Mac loan-level data to study default, prepayment, and credit risk using tools from probability, statistics, and machine learning.
Mathematical Interests: I like thinking about interactions between algebraic geometry and homotopy theory. Problems I think about usually revolve around derived algebraic geometry, algebraic K-theory, moduli geometry, motivic homotopy theory, and chromatic homotopy theory. I have also recently taken a keen interest in the philosophy and history of mathematics.
Some possible projects: One intermediate level project would be to learn about cobordism and its fascinating properties that have made it a central object in algebraic topology. We could start by reading early references on the Pontryagin-Thom construction and Thom's theorem (like Thom's original paper on the subject) that investigate the relationship between complex cobordism and fiber bundles.
An advanced project on the same subject would be to learn the relationship between complex cobordism and formal groups. One could start by reading Milnor and Novikov's computation of the cobordism ring using the Adams spectral sequence. And then follow up with Quillen's theorem and Landweber's exact functor theorem which establish a deep connection between stable homotopy theory and formal group laws.
More algebro-geometric projects include reading (part of) any one of these: SGA 4.5, Mumford's Picard Groups of Moduli Problems, GAGA, Quillen's Higher Algebraic K-Theory I.
Mathematical Interests: I am interested in probability (high-dimensional, integrable, random matrices) as well as some connections to combinatorics, statistical physics, and machine learning.
Some possible projects: I would be interested in doing a guided reading on one of two topics:
1) High-dimensional probability. Books by Ramon van Handel, Vershynin, Wainwright, etc.
2) Introductory spin glass models (focusing on SK and p-spin). Books by Bovier, Panchenko, Talagrand. etc.
Goal:
Read and meet (roughly once per week) to discuss material and solve problems. Deliverables are (1) summary presentations delivered to me each time we meet, (2) write-ups of the problems we solve, and (3) a final presentation highlighting core concepts learned.
Other:
1. I plan to spend 1-3 hours per week on this.
2. I expect the mentee to spend 6-8 hours per week on this.
Mathematical Interests: I am broadly interested in geometry. Some interests include complex geometry, metric geometry and convex geometry.
Some possible projects: Geometric valuation theory (Alesker's "Integral Geometry and Valuations") or convex geometry (Schneider's "Convex Bodies: The Brunn-Minkowski Theory).
Mathematical Interests: I'm interested broadly in topology and geometry, usually of a complex flavor. I've also got a soft spot for connections between combinatorics and topology.
Mathematical Interests: I'm primarily interested in physics, particularly statistical and plasma physics. Most of my current work is computational, but I have also worked with and am very interested in analytical methods as well. I also enjoy learning about algorithms used for common software tasks (image/video compression/encoding, pseudorandom number generators, fast methods for elementary/special functions, etc).
Some possible projects: I'm open to a pretty wide range of topics in applied math. Here's a few ideas:
- Geometric algebra and its applications in physics/computer science (see Doran's book)
- Numerical PDE methods, with special focus on inhomogeneous transport equations and semi-Lagrangian/DG/SLDG methods
- Any physics topic of interest -- I'm perfectly happy to hijack this into a physics DRP
Mathematical Interests: I'm of the belief that while pure math is beautiful, applied math is more fun. My interests predominantly lie in applications of linear algebra and differential equations, with a focus on fast algorithms.
Some possible projects: I've lead two successful DRP's in the past:
- Quantum algorithms (based on Lin Lin's lecture notes https://lin.caltech.edu/qasc/live_notes_0429.pdf)
- Time series models and high dimensional probability (see table of contents here https://acme.byu.edu/00000179-a1fc-d475-a57d-f5fe0dbf0000/v3toc-pdf)
I would be happy to lead a DRP on:
- Approximation theory (following Trefethen)
- Reinforcement learning (following Barto and Sutton)
- Stochastic processes (following Lawler)
- Quantum chemistry (following Szabo and Ostlund, I promise its all just linear algebra in disguise)
- etc.
Mathematical Interests: My interests are broad but mostly related with algebraic geometry. More precisely, I have been interested in derived categories of coherent sheaves, moduli theory, (geometric) representation theory and their interactions. In the past, I was also interested in different flavors of algebraic topology.
Some possible projects: I am willing to mentor reading projects in any (reasonable) topic within algebraic geometry, homological algebra, and representation theory. Here is a list of projects related to my interests at different levels:
- Representation theory of complex semisimple Lie algebras (undergraduate). The goal is to study their highest weight theory, Weil's character formula and, perhaps, relationships with the representation theory of compact Lie groups. Background : basic linear algebra/abstract algebra. References : "Representation Theory : A First Course" by Fulton and Harris, "Introduction to Lie Algebras : Finite and Infinite Dimension" by Hall, and "Introduction to Lie Algebras and Representation Theory" by Humphreys.
- Geometric Invariant Theory (undergraduate). We will study what is the quotient of an algebraic variety by an algebraic group, an incredibly useful technique in algebraic geometry. Background : abstract algebra and point-set topology, some commutative algebra will be helpful, but I won't expect knowledge on classical algebraic geometry. References : "An Introduction to Invariants and Moduli" by Mukai, "Lectures on Invariant Theory" by Dolgachev, and the lecture notes "Moduli Problems and Geometric Invariant Theory" of Victoria Hoskins.
- Local Class Field Theory and Galois Cohomology (undergraduate /graduate). We will learn about local fields and their Galois theory. Later we will introduce the machinery of group cohomology and conclude with some results of Galois cohomology. Background : abstract algebra at the level of Galois theory and point-set topology, no knowledge of homological algebra is needed (the purpose of the project is to introduce homological methods for number theorists). References : "A Gentle Course in Local Class Field Theory" by Guillot, and "Galois cohomology and class field theory" by Harari.
- Algebraic Curves and their Moduli (graduate). The goal is to familiarize with the theory of smooth projective curves over an algebraically closed field (usually the complex numbers), and certain moduli spaces such their Jacobians or their moduli of vector bundles. Background : commutative algebra, point-set topology, some elementary algebraic topology, and some knowledge on classical algebraic geometry will be helpful but not strictly necessary. References : "An Introduction to Invariants and Moduli" by Mukai, "Algebraic Geometry" by Hartshorne, "The Practice of Algebraic Curves" by Eisenbud and Harris, and "Lectures on Vector Bundles" by Le Portier. This is a more ambitious version of the project on GIT.
- Cohomology of Flag Varieties (graduate). The goal is to study the relationships between the representation theory of GL(n) and the geometry of flag varieties. Ultimately, we would attempt to understand the statement of the Borel-Weil-Bott theorem, a cornerstone of geometric representation theory. Background : abstract algebra, commutative algebra, point-set topology and preferably some basic algebraic topology, some knowledge of representation theory is also helpful (for instance, of finite groups), and we will catch up with the necessary algebraic geometry along the way. References : "Flag Varieties" by V. Lakshmibai and J. Brown, and for an introduction to the topic I suggest skimming through the first three sections of "Lectures on Geometric Constructions of the Irreducible Representations of GL(n)" by J. Kamnitzer.
- Non-commutative Resolutions of Singularities (graduate). The goal is to go through the lecture notes "Lectures on Non-Commutative Resolutions" by Michael Wemyss. We will introduce ourselves into homological methods for studying singularities. This will take us to a whole bunch of different mathematics : representation theory of quivers, representation theory of Cohen-Macaulay modules, derived categories and categories of singularities... Background : commutative algebra, some exposure to homological algebra can speed up things but we can catch up with the necessary background along the way.
Mathematical Interests: I am interested in algebraic geometry and commutative algebra.
Some possible projects: (1) Fulton's Introduction to Toric Varieties, (2) Cavalieri and Miles' Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory, (3) Coutinho's A Primer of Algebraic D-Modules.
Mathematical Interests: I am mostly interested in number theory (analytic and algebraic) and algebraic/arithmetic geometry.
Some possible projects: Tom Apostol - Introduction to Analytic Number Theory
Tom Apostol - Dirichlet Series and L-functions in Number Theory
Henry McKean and Victor Moll - Elliptic Curves: Function Theory, Geometry, Arithmetic
Silverman - Elliptic Curves
Gortz and Wedhorn - Algebraic Geometry I & II
Mathematical Interests: I am currently especially interested in algebra (mainly representation theory) and combinatorics. I also like some topics in analysis, point-set topology and logic!
Some possible projects: - group theory (https://www.jmilne.org/math/CourseNotes/gt.html)
- logic via sudoku (https://arxiv.org/pdf/2212.01053)
- oligomorphic permutation groups (book by Cameron)
- permutation puzzles (https://www.sfu.ca/~jtmulhol/permutationpuzzles/)
I'm also open to other projects!
Mathematical Interests: My research interests lie in algebraic combinatorics. Roughly speaking, the kind of math I do involves taking a problem in algebraic geometry and translating it to a problem involving combinatorial objects.
Some possible projects: Hyperplane arrangements, cluster algebras, symmetric functions, graph colorings, and extremal graph theory.
Mathematical Interests: My research interests are centered around derived categories of coherent sheaves (of algebraic varieties). More broadly, I am interested in algebraic geometry, algebra, and number theory.
Some possible projects: Galois Groups and Fundamental Groups, Szamuely
Number Fields, Marcus
Algebraic Curves and Riemann Surfaces, Miranda
An Introduction to Invariants and Moduli, Mukai
Topology from the Differentiable Viewpoint, Milnor
An Introduction to Commutative Algebra, Atiyah and Macdonald
Mathematical Interests: Geometric analysis on graphs: do you want to know how convolution and Fourier Transform work on a graph? And differential equations defined in a graph?
Some possible projects: Introduction on analysis on graph
Mathematical Interests: My mathematical interests are quite diverse! My research involves probability and complex analysis in the context of random tilings. However, I enjoy learning about many areas about math, including: algebra, point-set topology, algebraic topology, ordinary differential equations, partial differential equations, probability, real analysis, complex analysis, functional analysis, and possibly more!
Some possible projects: I would be happy to mentor a project that serves as an introduction to any of the subject areas listed above! I would also be interested in mentoring an intermediate project; possible examples include:
1) For those with background in basic probability and analysis, we could use Vadim Gorin's "Lectures on random lozenge tilings" to study ideas related to random tilings.
2) When I was a DRP mentee, I did a project on the basics of distributions (also known as "generalized functions"). For those with background in basic analysis and differential equations, we could study distributions and their applications to functional analysis and differential equations. We would likely use a variety of sources for this.
3) For those who are familiar with algebraic topology (basics of fundamental groups and covering spaces), we could use Allen Hatcher's "Algebraic Topology" to study homology calculations, and possibly even higher homotopy groups (which we would be learning together).
Mathematical Interests: I'm a graduate student currently working in positive characteristic geometry; I'm a fan of all things algebra, e.g. commutative algebra, algebraic geometry, or algebraic number theory, but I'm happy to mentor in more general areas as well.
Some possible projects: Introduction to Algebraic Geometry: Vakil's "The Rising Sea," Reid's "Undergraduate Algebraic Geometry," or others. Mild commutative algebra + point-set topology background recommended. Happy to tailor towards future interests!
Introduction to Algebraic Number Theory: Lang's "Algebraic Number Theory" with personal notes as a supplement. Galois theory background recommended.
Introduction to Commutative Algebra: Atiyah-MacDonald "Introduction to Commutative Algebra" with a focus on solving exercises.
Introduction to Abstract Algebra: Artin's "Algebra." Several possible focuses here: finite groups, rings & modules, or Galois theory are all possible topics!
Mathematical Interests: I am interested in knot theory and low dimensional topology (i.e. shapes in the third and fourth dimension). I also love group theory, combinatorics, and anything where I get to draw a picture.
Some possible projects: Knot theory: knots, links, and braids. Prerequisite of linear algebra.
Possible texts: The Knot Book (Colin Adams). For students with abstract algebra or topology background, we could use Knot Theory (Charles Livingston) or Braid Groups (Kassel and Turaev)
Group theory, but with pictures. Prerequisite of linear algebra.
My favorite part of group theory is the pictures you can draw! While most algebra courses define groups, and then look at the symmetries or group actions that result, we can also go in the opposite direction. This could serve as a supplement to a more traditional algebra course (412, 493).
Possible texts: Groups and Symmetries (M.A. Armstrong), Visual Group Theory (Nathan Carter), Algebra (Michael Artin)
Mathematics and social justice.
Possible texts: Weapons of Math Destruction (Cathy O’Neil), Political Geometry (Moon Duchin, Olivia Walch)