Winter 2025 Projects and Mentors
Here are all the graduate student mentors that have signed up for Winter 2025. 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:
Categorification, quantum groups, geometric representation theory
1. Computation of Reshetikhin-Turaev invariants of links for non-simply laced types using folding of root systems. 2. Representation theory of hypertoric universal envelopping algebras. 3. Suggestions are welcome.
Antoine Labelle
Topics:
Category theory, enumerative geometry, representation theory of algebraic groups/Lie groups/Lie algebras, toric varieties, geometric Langlands program, mathematical physics
Some possibilities of books to follow include: Toric varieties by Cox-Little-Schenck, 3264 & All That - Intersection Theory in Algebraic Geometry by Eisenbud-Harris or Between electric-magnetic duality and the Langlands program by Ben-Zvi. I am also very open to other suggestions of topics.
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Jonah Saks
Topics:
profinite topology & residually-finite groups, or left-orderable groups and knots
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Maninder Dhanauta
Topics:
Galois Theory and Profinite groups
A Galois group is roughly a subgroup of the set of linear automorphisms of a vector space. The vector spaces come from field extensions, and when the dimension of the vector space is infinite, the Galois group has a non-trivial topology(Profinite). Although the original goal of Galois theory was proving the insolvability of the quintic by radicals, today's Galois theory is used in diverse areas of math. We will study some Galois theory, limits, profinite groups, modules. We will see an application of profinite groups to geometric group theory (tbd). The topics can be adjusted based on your background and preference. All levels welcomed!
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Rudy Ariaz
Topics:
Riemann surfaces and algebraic curves, or elliptic curves
Option 1: building up to the Riemann-Roch theorem using Miranda's "Algebraic Curves and Riemann Surfaces". Prerequisites: complex analysis, basic topology and algebra. Option 2: studying Silverman's "The Arithmetic of Elliptic Curves". Prerequisites: abstract algebra (especially ring theory). Note: our weekly meetings will typically be held on Zoom.
Analysis
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Kevin
Topics:
functional analysis, random matrices, spectral analysis
Miguel Ayala
Topics:
Scientific Computing, Numerical Analysis, Computer-Assisted Proofs.
William Holman-Bissegger
Topics:
numerical ODEs and PDEs; topological methods in PDEs; dynamical systems
Xiao Xiao
Topics:
spectral geometry (google "Chladni plates")
(Easy) Understand the basics of spectral geometry. (Hard) Compute eigenvalues of exotic planar shapes, possibly with singularities or junctions, with the help of computer numerics. (Might lead to something publishable.) You can of course propose other topics. I am interested in analysis, geometry, PDEs in general.
Geometric Group Theory
Carl Kristof-Tessier
Topics:
Word problem for groups, regular languages
I plan to go over topics in geometric and computational group theory. Namely, I would like to discuss the Word Problem for groups. The problem essentially asks for an algorithm allowing one to compare any two elements in the group, and test whether these elements are equal or not. In general, no such algorithm exists: the Word Problem for groups is undecidable. There is a wonderful theory of groups with decidable word problems, and more specifically automatic groups. My primary plan would be to explore the theory of automatic groups through David Epstein’s book, Word Processing in Groups. Beyond this, I would also be interested in reviewing other subjects in the field, such as those appearing in The Handbook of Computational Group Theory of Holt, Eick and O'Brien.
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Chris Karpinski
Topics:
Constructions of groups with exotic properties (e.g. Tarski and Gromov Monsters), small cancellation theory of groups (and rings)
Jessie Meanwell
Topics:
Geometric group theory, Cayley graph, group actions, free group, group presentation, the word problem, graph symmetries, visualizations
Have you been longing for a different way of understanding groups? Do you love thinking visually? In this DRP, we'll venture into the world of geometric group theory (GGT) - a field which uses geometry to better understand groups. We'll learn how Cayley graphs help us visualize a group's structure and encode its properties. We will explore various key topics in GGT (possibly including free groups acting on trees, hyperbolic groups, the word problem, etc.) by reading from the book 'Office Hours with a Geometric Group Theorist'. We may try to tackle some research problems or come up with our own visualizations.
Owen Rodgers
Topics:
Cohomology, Betti numbers, Classical descriptive set theory
I plan to explore some of the basic notions related to the L^2 spaces of groups. We would build from an introductory level and see notions related to Hilbert space theory, geometry, and operator algebra theory and hopefully culminating in describing some L^2 invariants of groups such as cohomology and Betti numbers. I am also happy to instead go through an introduction to descriptive set theory. We would read through an excellent set of notes and see beautiful result such as the Perfect Set Theorem and the determinacy of many kinds of games.
Tasmin Chu
Topics:
Random walks on Cayley graphs of groups, random walks, discrete probability, descriptive set theory, infinite groups.
Option 1: When does a random walk return to its origin infinitely often? What about in finite expected time? How can we define and compute the growth rate of infinite groups, trees, and graphs? These questions have been studied extensively in the field of discrete probability and geometric group theory. In general, probabilistic techniques have powerful applications to group theory, especially the theory of infinite groups. It also tells us a lot about random walks on graphs more generally. This is a fascinating field with lots of major results that have emerged in the last 30 years. We can read a very good book by Lyons and Peres together called Probability and Trees on Networks, or a set of notes by Gabor Pete (warning: considerably less polished). This is my field so I would probably be able to easily answer many questions here. A first course in probability (majors or Honours) should be enough. Analysis 2 would also be good. MATH 447 would be particularly good, but is not necessary. We may spend the first two weeks reviewing basic Markov Chain theory anyway. Option 2: Reading through my advisor Anush's notes on Descriptive Set Theory or Ergodic Theory. Analysis 3 would be good for this project.
Combinatorics, Probability and Random (Walks, Graphs, Fields, etc)
Antoine Poulin
Topics:
Combinatorics and Discrete Math, Descriptive Set Theory, Geometric Group Theory, Probability
Mackey's notion of virtual subgroups
Caelan Atamanchuk
Topics:
Combinatorics, Random graph theory, Graph Colouring, Temporal Networks
There are a wide variety of problems, papers, and textbooks in the field of probabalistic combinatorics that we could focus on. To see the kinds of topics this includes I'd recommend taking a look at "Random Graphs and Complex Networks" by Remco Van Der Hofstad, "Graph Colouring and the Probabilistic Method" by Reed and Molloy, or "Information, Physics and Computation" by Mezard and Montanari. If you have your own ideas too I am happy to hear them out. Background in probability theory and combinatorics would be helpful.
Gabriel Crudele
Topics:
combinatorics, intersecting families, graph theory, linear algebra, characteristic vectors
I plan to follow a set of notes by Natasha Morrison for a graduate course on algebraic methods in combinatorics. The common theme is turning combinatorics and graph theory problems into linear algebra problems with a clever encoding, then using more or less standard techniques from linear algebra to solve the problem in that setting.
Louis-Roy Langevin
Topics:
Random graphs
Find the starting vertex of a random graph with high probability. A special case would be finding the root of an attachment tree (in particular, a uniform attachment tree). We may also study behaviours of random graphs the start from a complete graph, but each edge is removed with some probability. Other ideas with random graphs may come up through the next weeks. We may also study combinatorics without probability is the student prefers it.
Noah
Topics:
Applied Mathematics, Probability
I want to read topics in the book: Random Fields and Geometry by Taylor and Adler. When applicable we'll look at applications to random matrix theory and or physics.
A background in some high-dimensional probability will be helpful and let us get to more interesting topics quicker. That being said any upper year undergrad will have enough to get something good out of it.Sasha
Topics:
one-dimensional random walks with decreasing step size
I want to read about random walks; specifically, unbiased random walks in one dimension with decreasing step size. Imagine a person going on a "walk" on the real line, where at each time step, they will step left or right with equal probability. Typically, their step size will be 1 each time, but we can consider the variation when their step size decreases over time. For instance, one could consider a "harmonic random walk" where their step size at time n is 1/n. There are a few papers studying distributions for such walks, conditions for the walker to stay in the same region with high probability, etc., that I would like to read and understand better. There are some interesting/surprising results involving Cantor sets and the golden ratio. Random walks also tie into a descriptive set theory problem I have been trying to solve for a while; I'd be happy to discuss this and work on it together if we have time. This project would require first courses in probability and analysis; a course in stochastic processes would be helpful but is not necessary. I also have some other ideas for projects in probability and descriptive set theory, and I am happy to plan a project with input from students.
Sophia Howard
Topics:
Probability, Brownian Motion, Random Graphs
Vincent Painchaud
Topics:
Random matrix theory
Statistics, Machine Learning and Data Analysis
Daniel Krasnov
Topics:
Machine Learning, Bayesian statistics, Fuzzy clustering, Dimensionality reduction, image analysis, NLP, Bayesian neural networks
I'm happy to research any area of machine learning the student is interested in. If they prefer, I select a project I have two choices in mind. An image segmentation project would involve applying a new Fuzzy clustering algorithm to aerial image data of Canada's forests. The goal would be for it to separate out different sections of the imaging to identify forest fire fuel types. The current system used by the government does hard clustering for this problem and there is lots of literature to suggest a fuzzy approach could be better. There is potential for this work to be published in an Environometrics journal. A dimensionality reduction project would be about learning dimensionality reduction techniques and then developing and publishing an R package for one of them that I'm currently working on. We could then apply our package to a dataset of interest to the student and do an analysis. Introductory machine learning knowledge would be helpful for both projects. The matrix factorization project would be best done by someone with some familiarity with Bayesian statistics though this isn't strictly necessary. The matrix factorization problem requires good R programming skills as the goal would be to publish a package. The image segmentation project requires fair R programming skills. If neither of these sound interesting I have experience in NLP, Bayesian neural networks, gaussian processes, recommendation systems, and various regression problems so we could study a model from one of these fields or we could simply find a dataset of interest and develop a model to analyze it.
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Julien Cheng
Topics:
Persistent Homology (a kind of Topological Data Analysis)
We will start by studying simplicial homology and persistent homology. From there the project could cake a very theoretical or a very applied to direction or something in between (depending on student interest). In recent years, people have developed a strong theoretical framework to apply persistent homology to analyze data. But often interpreting the analysis is difficult. We could trying running the analysis on different data sets and interpreting the results. (See https://donut.topology.rocks/ for inspiration.) On the theoretical side of things, there are also many things we could look at. A current area of research is multiparameter persistence. For this we could look at these course notes https://www.albany.edu/~ML644186/AMAT_840_Spring_2019/Math840_Notes.pdf This particular direction would be heavy on advanced concepts from algebra. The only prerequisite for this project is basic linear algebra (linear maps, their kernels and images, quotient spaces, and the dimension of a vector space). But I would be happy to review these things with students if desired.
Yanees Dobberstein
Topics:
Extreme Value Theory, Directed Algebraic Topology, Deep Learning, Higher Category Theory
Zayd Omar
Topics:
State-space modelling, correlation matrix structures, statistical inference, Bayesian analysis, Econometrics, Causal Inference
1) Bayesian analysis of random correlation matrices, dynamic correlation structures in state space models. There's a lot of interesting literature about in statistics and econometrics that looks dynamic correlation structures. But very few have proposed a mode based on a Bayesian framework. Further the models proposed have complicated algorithms with sometime very poor statistical properties. A Bayesian framework will allow us to propose new methods to estimate the parameters and pin down statistical properties of these dynamic correlation matrices. We can then apply the theory to practical examples. 2) Or we can do some other statistical topic of your choice.