Winter 2027 Projects and Mentors
Here are all the graduate student mentors that have signed up for Winter 2027. 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
Zackary Horton
Topics:
Algebra, Number Theory
Introduction to modular forms or local class field theory (depending on interest and level).
Zhaoshen Zhai
Topics:
Free Groups, Decision Problems, Computability Theory, Turing Degrees
Which questions about infinite groups can computers answer? We will study free groups through words and labelled graphs, solving equality and subgroup membership problems and exploring finite quotients. Are some unsolvable problems harder than others? Following Andrew Marks' notes, we will study Turing degrees and construct two undecidable problems, neither solvable using a solver for the other.
Owen Rodgers
Topics:
Uniform Space, Group Action, Polish Group, Dynamics
Uniform spaces and their applications in topological dynamics. Most likely reading a few sections of this book: Topological dynamics of Polish groups.
Aaron Shalev
Topics:
Colour spaces, Riemannian geometry, Lie groups
From digital displays and image processing to printing and computer vision, working with colour quantitatively requires a mathematical description of it. The familiar RGB system provides one such description, though many others exist, each with its own strengths and limitations. After briefly surveying some historical models of colour perception, we will use tools from differential geometry to model the perceived colours as a geometric space (more specifically, a Riemannian manifold) in which the "distance" between two colours more faithfully reflects how different they appear to the human eye. This particular model, built from a set of axioms that capture basic observed properties of colour perception, surprisingly leads to just two possible geometries for colour space.
Ludovic Rivet
Topics:
Elliptic Curves, Modular Forms
The goal would be to get an overview of some important objects in Number Theory: Modular forms and Elliptic Curves. The project would start by reading the first chapter of Diamond and Shurman's "A first course on Modular forms", and continue in whichever direction we want to go.
Analysis
Tommy Giroux
Topics:
From Ergodic Theory to Quantum Ergodicity
The goal of this project is to build the background needed for ergodic theory through measure theory and functional analysis, then study classical results and examples. The long-term goal is to explore quantum ergodicity together, with ideas from semiclassical analysis and pseudodifferential operators. Analysis 2 is recommended; PDE knowledge is helpful, but motivation is the main requirement.
Miguel Ayala
Topics:
Loss-functions Landscapes
Learn more about training dynamics.
Henry Nguyen
Topics:
Analysis, PDE, Variational Methods, Elliptic Equations
Many PDEs can be studied by considering their solutions as minimizers of an energy functional. One idea could be to explore how Sobolev spaces and variational methods can be used to study weak solutions of elliptic PDEs together with other tools to investigate qualitative properties of solutions.
Eunpyo Bang
Topics:
The Maximum Principle, Parabolic/Elliptic PDEs
You are going to learn about the maximum principles for elliptic/parabolic PDEs. Using the maximum principles and their variant, our goal is to verify that the constants for some elliptic/parabolic estimates (e.x. Nash-Moser-DiGiorgi estimates, Schauder estimates) are optimal.
Combinatorics and Probability
Caelan Atamanchuk
Topics:
Random Graphs, Network Science
My specialty is the structure of random graphs, and I'll bring some fun topics in that field. However, my desire is to settle on a topic (and project) together, and I am happy to discuss any theoretical or applied probability topics.
Hussin Suleiman
Topics:
Flag Algebras, Extremal Graph Theory, Subgraph Densities
This project will explore flag algebras, a modern method in combinatorics used to study how small patterns appear inside large graphs. We will begin with classical extremal graph theory problems, such as determining how many edges a graph can have while avoiding certain configurations, and then see how flag algebras provide a systematic way to approach more complicated versions of these questions.
Marcel Goh
Topics:
Additive Combinatorics
In the past, I've found it nice to focus the semester on building up theory and techniques required to prove one significant theorem (e.g., freiman-ruzsa, roth's theorem), or several interrelated smaller ones (e.g., sum-product estimates, entropy methods in additive combinatorics).
Noah Marshall
Topics:
Understanding statistical mechanics
I want to read Taligrand's book on spin glass and the Sherington Kirkpatrick model. Other related topics might be chosen instead (other spin models).
Statistics, Machine Learning and Data Analysis
Adrien da Silva and Helena Heinonen
Topics:
Extreme Value Theory, Copula Modelling, Multivariate Statistics
The project would be either an introduction to copula modelling or extreme value theory. Extreme value theory is the study of the tails of distributions. Copulas are a tool for multivariate statistical analysis. This project would consist of learning the basics of either field, tools for exploratory data analysis, and statistical inference. The project would finish with a data application.
Mila Pourali
Topics:
Statistics, Semiparametric Inference, Causal Inference, Missing Data, Model Misspecification
We can either read about semiparametric theory and missing data and/or go over some introductory causal inference notes. We will try to simulate some of the things we see along the way.
Jamie McCormick
Topics:
Statistics, Social Network Analysis, Biostatistics, Dependence modelling via graphical models
Basic introductory project on network statistics, most likely working from this book Statistical Analysis of Network Data with R.
Geoffrey Kelly
Topics:
Neural Networks, Chaos, Dynamics, PDE
Idea is to use time delays to predict trajectories of non-linear or chaotic systems using Neural Networks. Essentially can we train a Neural Network to predict into the future of a complicated system given a limited amount of data?