Mentor profiles
Summer 2026
Summer 2026
Mathematical Interests: In one phrase, I am interested in applied probability. This spans game theory, machine learning, financial mathematics, and much more! If you have a specific domain which you want to study mathematically, could be social choice theory, or epidimeology, then we can explore these concepts from a mathematical standpoint.
Some possible projects: Mathematics, especially randomness is ubiquitous in the world. There are three broad directions, one is pure theoretical development on a theory like stochastic control, or finite game theory. Second is a survey of applied methods in for example machine learning. The third is a specific application domain which we will drill into (social choice theory, political game theory, eg. gerrymandering, or epidemiology). If you are interested in any real world application of probability or machine learning, then there are a plethora of interesting projects to choose from!
Mathematical Interests: I do research in number theory and arithmetic geometry. My research involves L-functions (generalizations of the Riemann zeta function) and automorphic forms (generalizations of modular forms). More specifically, I am interested in special values of L-functions and the Kudla program.
Some possible projects: 1. Learn about L-functions associated with Hecke characters. These are generalizations of the Riemann zeta function and Dirichlet L-functions, yet simple enough—essentially the GL(1) case—that we understand them better than other L-functions. These L-functions are important tools in studying class field theory and more. We will learn about Tate's proof of the functional equation and the analytic class number formula. Familiarity with complex analysis and ring theory is desired but not strictly necessary.
Main reference: Algebraic Number Theory by S. Lang
2. Learn about modular forms, which are ubiquitous in number theory. Modular forms are indispensable tools for studying elliptic curves, and they also have broad applications, most famously in the proof of the sphere packing problem in dimension 8 and 24 and Fermat's last theorem. We will learn about basic properties of modular curves and modular forms. Then, we can choose to learn a specific application such as Gauss' circle problem or the congruent number problem. Background in complex analysis is recommended.
Main reference: A First Course in Modular Forms by F. Diamond and J. Shurman
Mathematical Interests: I am generally interested in algebraic combinatorics and combinatorial algebraic geometry. I study the classical/Orthogonal/Lagrangian Grassmannian, their totally nonnegative parts, and their stratifications by Positroid/Orthopositroid/Electroid varieties. I also work on a relatively new field called positive geometry, which lies on the intersection of combinatorics, complex algebraic geometry, and real algebraic geometry and has applications to Quantum Field Theory and Cosmology. I am also interested in more classical topics in algebraic combinatorics, such as symmetric functions, matroids, etc.
Some possible projects: For students with little background, we could learn about matroids (recommended text: Matroid Theory by James Oxley), which is a combinatorial model of linear independence.
For students with little background, we could learn about the beautiful area of projective geometry, linear incidence theorems (like Pappus's hexagon theorem), and potentially the recent ""master theorem"" by Fomin and Pylyavskyy (recommended text: Foundations of Projective Geometry by Robin Hartshorne and this paper https://arxiv.org/abs/2305.07728).
For students with little background but who are relatively comfortable with basic abstract algebra, we could learn about Enumerative Combinatorics or Young Tableaux using text by Fulton. In particular, topics may include symmetric functions, Grassmannians, Schubert Calculus, etc. These topics lie in the center of algebraic combinatorics and basically almost everything in algebraic combinatorics has something to do with it.
For students who are relatively familiar with basic algebraic geometry, we could learn about total positivity (of matrices, Grassmannians, etc.) and/or the positroid stratification. (recommended text: note by Thomas Lam https://arxiv.org/abs/1506.00603). This is an active area of research and personally my favorite topic of math.
For students relatively familiar with basic algebraic geometry, we could learn about positive geometry (recommended text: Lecture Note by Simon Telen https://sites.google.com/view/simontelen/teaching/positive-geometry-of-polytopes-and-polypols?authuser=0). This is an active area of research that generalizes the previous total positivity topic and personally my favorite topic of math. It has applications to Quantum Field Theory and Cosmology.
Mathematical Interests: I am interested in Algebraic Geometry, Algebraic Topology, and other related topics.
Some possible projects: One idea is to learn about the Chern-Gauss-Bonnet formula. This will require learning about curvature, for which Tu's "Connections, Curvature, and Characteristic Classes" is a nice book. This topic is very nice because it is already very interesting yet also understandable for 2-dimensional manifolds. Understanding the proof of the general CGB formula will provide a great roadmap for the summer reading. A topic which is related, but more algebraic-topology flavored would be to learn about characteristic classes from Milnor and Stasheff's book. Another direction would be to do a project about Riemann surfaces. I am also open to discuss other ideas related to my area of interest!
Mathematical Interests: My research is in applied mathematics and scientific computing, with a focus on developing high-performance algorithms for the efficient and accurate simulation of complex dynamical systems. In particular, I work on Lagrangian particle methods combined with GPU-accelerated treecodes to solve the fundamental equations governing plasma dynamics, both in kinetic regimes and under fluid approximations.
Some possible projects: Project 1: Particle Methods for Solving Differential Equations
Study Lagrangian particle-based numerical methods for solving partial differential equations, incorporating Green’s function formulations, kernel regularization techniques, and adaptive mesh refinement for improved accuracy and efficiency.
Project 2: Fast Summation Algorithms
Study fast summation algorithms (e.g., treecodes) to reduce computational complexity and enable large-scale simulations. This work can also explore novel parallel computing strategies and GPU acceleration.
Mathematical Interests: I'm interested in fluid dynamics, biolocomotion, and dynamical systems.
Some possible projects: One possible project involves reproducing the snake locomotion simulation results from "Slithering Locomotion" by Hu and Shelley (2012). Another avenue is studying boundary integral equations for highly viscous fluid flow from the book "Boundary Integral and Singularity Methods for Linearized Viscous Flow" by Pozrikidis.
Mathematical Interests: Stability condition and geometry of moduli spaces, especially for those parameterizing objects of derived category
Some possible projects: Intro of Algebraic Topology/Abstract Algebra/Algebraic Geometry, Homological Algebra, Derived Category.
Mathematical Interests: I am interested in combinatorial and computational aspects of representation theory.
Some possible projects: Project 1: Coxeter groups
We'll study some aspects of Coxeter groups, such as the exchange property, the Bruhat order, and the geometric representation.
Main reference: Combinatorics of Coxeter Groups by A. Bjorner and F. Brenti
Other references: Reflection Groups and Coxeter Groups by J. E. Humphreys, Lie Groups and Lie Algebras Chapters 4-6, N. Bourbaki
Project 2: Lie algebras
We'll study Lie algebras and their representations, including Weyl's theorem and the theory of highest weights.
References: Introduction to Lie Algebras and Representation Theory by J. E. Humphreys, Introduction to Lie Algebras by K. Erdmann and M. J. Wildon
Project 3: Quantum sl2 and its representations
We'll study the quantum group U_q(sl(2)) and its representation theory.
Main reference: Quantum Groups by C. Kassel
Mathematical Interests: I am interested in geometry in the broadest sense. I enjoy studying manifolds and algebraic varieties in low dimensions, exploring the various geometric structures and properties they possess. I am particularly drawn to negative curvature and the striking visual intuition behind it—much like the hyperbolic tilings in M.C. Escher's artwork. My PhD research primarily sits at the intersection of geometric group theory and complex surfaces.
Some possible projects: Track 1: Geometric Group Theory
Geometric group theory is the study of the geometry of infinite groups. In this project, we could explore the theory through a rich class of examples like free groups, Coxeter groups, Out(F_n), lamplighter groups, and Thompson’s groups. For a quick taste of the subject, check out this short article by Matt Clay: https://mattclay.hosted.uark.edu/Papers/ggt.pdf
Potential Summer Reading:
* Office Hours with a Geometric Group Theorist edited by Matt Clay and Dan Margalit
* Course notes on Out(F_n) by Alex Wright: https://websites.umich.edu/~alexmw/Math636Notes.pdf
Track 2: Complex Curves and Surfaces
Complex curves and surfaces are algebraic varieties of complex dimension 1 and 2, respectively. Their study sits beautifully at the intersection of algebra, complex analysis, and topology. We could spend the summer focusing on specific, fascinating classes of examples like elliptic curves, ruled surfaces, cubic surfaces, and K3 surfaces.
Potential Summer Reading:
* Algebraic Curves and Riemann Surfaces by Rick Miranda
* Complex Algebraic Surfaces by Arnaud Beauville
Mathematical Interests: I like topology and geometry, especialy the kind where I get to draw pictures!
Some possible projects: Office Hours with a Geometric Group Theorist.
Mathematical Interests: I am interested in Algebraic Combinatorics. It is a broad area of mathematics that is built on the interplay of Combinatorics with diverse areas like Algebra, Toplogy and Geometry. A lot of the objects that I study can be understood and studied with tools from basic mathematics but also lend themselves to analysis by sophisticated methods.
Some possible projects: 1) For someone who has a solid linear algebra background with an interest in combinatorics reading about Coxeter groups, for eg. Jim Humphreys' Reflection Groups and Coxeter Groups.
2) For someone who has background in Linear Algebra and Multivariable Calculus, we could read about differential forms, for eg. Morita's Geometry of Differential forms
3) For someone with background in Linear Algebra and Multivariable Calculus and some familiarity with Abstract Algebra a reading course on Lie groups following Brian Hall's Lie Groups, Lie Algebras and Representations.
Mathematical Interests: Differential equations, dynamical systems, introduction to number theory and combinatorics
Some possible projects: Nonlinear dynamics and chaos by Strogatz;
Networks by Newman;
A friendly introduction to number theory by Silverman;
Introduction to graph theory by West;
Combinatorics: Topics, Techniques, Algorithms by Cameron
Mathematical Interests: I am interested in learning about whatever excites you! In the past, I have thought broadly about theoretical/mathematical physics (e.g. topological quantum field theory (algebraic topology and representation theory), opinion dynamics (dynamical systems on graphs)). However, these are not the only things I enjoy thinking about—truly, name an area of math and/or physics, and I would be excited to learn more about it!
Some possible projects: Combinatorics and statistical mechanics, foundations of quantum mechanics (causal inference), squigonometry
Mathematical Interests: I study algebraic combinatorics, which is an area where we study combinatorial objects (like permutations, graphs, lattices, etc.) and their connections to algebraic structures (think representations, group actions) in a way that illuminates amazing properties of both! In my current research for my dissertation, I am focusing on webs, which are certain planar bipartite graphs that encode quantum tensor invariants. I like that you can learn things about that representation theory by playing with little pictures of graphs, their colorings, and related combinatorial objects.
Some possible projects: * The representation theory of the symmetric group (from The Symmetric Group by Bruce Sagan) - some algebra background would be helpful
* Combinatorial game theory (could use Combinatorial Game Theory by Aaron N. Siegel or Game Theory: A Playful Introduction by Matt DeVos and Deborah A. Kent)
* Introduction to varieties and Grobner bases (using Ideals, Varieties, and Algorithms by Cox, Little, and O’Shea) - some algebra background would be helpful
* Open to other projects in combinatorics or algebra, including at the introductory level
Mathematical Interests: Algebra, Combinatorics, and their computational aspect
Some possible projects: 1. Topics in algebra and/or combinatorics that people who've taken 296, 465/565, 412/493/494 might be interesting in. Just to name a few: group representation theory, Galois theory, Lie algebra, Coxeter group, symmetric functions, algebraic graph theory
2. This direction is more specific and research-oriented. A lot of extremal combinatorics problems in Euclidean spaces can be transformed into a finite search or optimization problems, and therefore be handled with both math and computational tools. We will pick some paper(s), learn the math behind and reproduce the computational result. Some example papers:
- Bachoc and Vallentin, New upper bounds for kissing numbers from semidefinite programming
- Greaves et al., Equiangular lines in low dimensional euclidean spaces
- Viazovska, The sphere packing problem in dimension 8
These researches use tools from a various of fields, including linear algebra, optimization, analysis and/or algebra. Willingness to pick up new knowledge is more important than a solid background, though certain level of math and programming maturity is required.
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!"