Winter 2026 Projects and Mentors
Here are all the graduate student mentors that have signed up for Winter 2026. They are loosely arranged in terms of what field(s) their projects could follow. The categorization below is not perfect; there are overlaps and certainly mentors that could fit into multiple categories, but it might help to give you a sense of what field is about.Algebra and Geometry
Alexis Leroux-Lapierre
Topics:
Combinatoric of the product monomial crystal and representation theory in the BGG category O
Recently, a concrete combinatorial tool --- the product monomial crystal --- was introduced to better understand representation theory of a family of shifted quantum groups. As predicted by technical results, special cases of the product monomial crystal exhibit combinatorics similar to that of a Weyl group. The objective of this project is to better understand this connection.
Arihant Jain
Topics:
Quadratic Forms, Representation of primes as weighted sums of squares, Class Field Theory
This project explores a classical yet simple question: which primes can be written as $x^2+y^2$ or more generally as $x^2+ny^2$? Through the first five chapters of Cox’s Primes of the Form $x^2+ny^2$, we’ll uncover the answer while learning about two fundamental objects in number theory : quadratic forms and number fields.
Boris Zupancic
Topics:
(Pseudo)-Riemannian geometry and General Relativity, Poisson Geometry and Mathematical Physics
Two ideas: (1) A project on pseudo/semi-Riemannian geometry, with the goal of learning about the mathematical/geometric techniques underlying General Relativity, Einstein’s theory of gravitation, OR (2) A project on Poisson geometry (generalization of symplectic geometry, a framework for doing classical mechanics), with the goal of learning about some non-trivial examples (in math or physics).
Fabricio Dos Santos
Topics:
Geometry and Combinatorics of Coxeter groups
Coxeter groups are a type of group generated by reflections. Some examples include symmetry groups of regular polygons and polyhedra, or of some regular tilings of the Euclidean or hyperbolic plane. I plan to go over the basics of Coxeter groups, and then explore some of their nice combinatorial/geometric properties by reading (parts of) the book “Combinatorics of Coxeter Groups”.
Shereen Elaidi
Topics:
Lie group, symmetry methods, differential equations, nother’s theorem
How can we use symmetries to understand differential equations? Symmetries are another tool that allow us to understand differential equations, solve them, and develop better numerical approximations for their solutions.
Zhaoshen Zhai
Topics:
Model theory, (classical/effective) descriptive set theory, recursion theory, and set theory
We can isolate certain classes of mathematical structures (numbers, groups, graphs, spaces, etc) and understand their 'first-order properties' from multiple different angles. Depending on your interest, we can approach it via model theory, recursion theory, complexity theory, set theory, etc. Prerequisite: (familiarity with content from) MATH 318.
Analysis
Dominic Blanco
Topics:
Numerical continuation, Pattern Formation, Fourier series
Performing numerical continuation in various reaction diffusion systems. Investigation of pattern formation in these systems with different symmetries. Will use Fourier series with various symmetries.
Eunpyo Bang
Topics:
Brownian motion, Malliavin calculus
Brownian motion is very interesting topic in probability theory. We will read about Brownian motion and its properties focusing on the aspect that it is a measure on the space of continuous paths. Further we will study about a branch of stochastic calculus(Malliavin calculus), which is a version of calculus of variations, when we are given a functional on certain class of random variables.
Gabriel Remond-Tiedrez
Topics:
Topological dynamics, complex dynamics, Julia sets, chaos
We'll be looking at "An Introduction to Chaotic Dynamical Systems" by Devaney, to talk about how to define chaos in dynamical systems. Depending on how comfortable you are with topology and so on, we'll do a review/intro to that to go into topological dynamics and defining chaos. Another approach we could take is more computational and plot nice pictures.
Hugh Johnston
Topics:
Analysis, PDE, Geometry, Calculus of Variations, Riemannian Geometry
There are a few directions we can pursue depending on interest and background. We can study surface minimizing PDE (calculus of variations). We could study straight lines (geodesics) or curvatures on surfaces (Riemannian geometry). Finally, we could do a project looking at different methods of solving PDE in a non geometric (euclidean) context, outside of those typically covered in courses.
Owen Rodgers
Topics:
To be determined. Something dynamical likely.
To be determined
William Holman-Bissegger
Topics:
PDEs, fluid dynamics, numerics for differential equations
One idea could be to look at a hierarchy of PDEs used in the study of fluids, with their different qualitative and structural properties; e.g. effects of viscosity, compressibility, magnetization... And/or how some continuum PDEs arise from kinetic equations. Another could be to study and implement some numerical methods for fluids or dynamical systems.
Combinatorics and Probability
Catherine Fontaine
Topics:
Random Graph Theory, Discrete Probability
I propose an introduction to random graphs by reading the manual Random Graphs and Complex Networks by Remco van der Hofstad. We will explore how large networks behave and how to apply these theorical concepts to study real-world netwroks, such as the scale-free structure of the World Wide Web. If time permits, we could also run simulations to explore these properties on smaller-scale networks.
Gabriel Crudele
Topics:
Planar maps, bijective encodings, random geometry, trees
A planar map is a graph drawn on the sphere with no crossing edges. I hope to read about the geometry of random planar maps. The main tools that make analyzing random planar maps possible are very nice ways of encoding them into simpler looking objects (like trees) which still retain all the geometric information. A significant portion of the reading would be spent understanding these encodings.
Kevin Xiao
Topics:
Random matrix theory, high-dimensional probability
Exploring concepts in Random matrix theory, typically involving some form of approximation of a random matrix by a deterministic equivalent.
Mat Chaubet
Topics:
Probability theory
I'd like to review the theory of spin glasses. Not sure what it is? Me neither, but we've got some reading to do. Another possible project is to look at mixing times of particle systems, and read a very nice set of notes: Mixing time and cutoff for one dimensional particle systems by Hubert Lacoin.
Sasha Bell
Topics:
Game theory, Game of life, Conway's soldiers, cellular automaton, mathematical puzzles
I would like to study problems involving cellular automata, such as Conway's soldiers and Conway's game of life. How do these systems change over time, and how can we analyze them rigorously? I have a difficult but easy-to-explain open problem in this field that I would be happy to work on together.
Vincent Painchaud
Topics:
Random matrix theory, spectral theory
A possible project would be to understand one of the proofs of the semicircle law, which is one of the most important results in random matrix theory. There is a moment-based proof which is mostly combinatorics, one that uses Stieltjes transforms, and one that uses free probability (which is in a sense an alternative theory of probability). But it could be something else, I have other ideas!
Statistics, Machine Learning and Data Analysis
Adrien da Silva
Topics:
Spatial Statistics, Dependence Modelling, Extreme Value Theory, Bayesian Modelling
Spatial statistics is the field of statistics that studies geographical data. I would like to explore spatial statistics in frequentist or Bayesian frameworks by reading books, reading papers and completing a data analysis.
Daniel Krasnov
Topics:
distance metric, regression, classification
A distance metric tells us how far apart two things are. We can define a distance metric for vectors of mixed-type data. This means dimensions of the data vector are continuous and categorical. Recently a new distance metric has been invented that is very flexible. Our project will be to incorporate it into various machine learning models and see how we do.
Helena Heinonen
Topics:
statistics, extreme value theory
I don't have an exact project in mind yet but I was thinking of investigating serial dependence of extreme events. It will probably involve some data analysis and simulations.
Mila Pourali
Topics:
statistics, survey sampling, semiparametric theory, missing data, differential privacy
I have a couple of projects in mind: 1) Going over the basics of survey sampling. 2) Reading the book "Semiparametric Theory and Missing Data". 3) Reading "The Algorithmic Foundations of Differential Privacy". We might try to reproduce some interesting simulations. Some background in statistics, probability, linear algebra would be useful.
Peiyuan Huang
Topics:
Applied statistics, Spatial statistics, Bayesian inference
We can read a book about spatial statistics in Bayesian paradigm. Spatial statistics are mainly used for disease mapping, modelling marketing data, as well as environmental science. We will learn some softwares from R for such modelling. By the end we can do a toy project with real world dataset as an exercise.