Winter 2024 Projects and Mentors

Algebra and Geometry

• Aaron Shalev

Non-Archimedean Analysis, (possibly Functional Analysis and/or Operator Theory)

• Alexis Leroux-Lapierre

Representation theory, BGG category O, Soergel Bimodules, Quantum groups

• Asa Kohn

Commutative algebra, Category theory

• Jessie Meanwell

Making fractal animations in desmos! Have you wanted to make art using math? In this DRP, we'll create fractal animations entirely in Desmos graphing calculator. We'll learn about dynamics and chaos, shapes which have dimension 1.5, and use structures related to the Mandelbrot set (complex function iteration) to create our own visualizations.

Combinatorics and Discrete Math

• Agnès Totschnig

Graph Theory and Algebraic Combinatorics

• Caelan Atamanchuk

Random Combinatorics, Graph Theory, Additive Combinatorics: We can work through any book or paper that you would like (I have many to recommend that are in the area of combinatorics for those without ideas). There are also a few small problems in the area of discrete probability (random graphs and random walks in particular) that could be fun projects to tackle for the semester.

• Jérémie Turcotte

Computer formalisation of mathematics: Lean is a computer langage to formalise mathematics. I learnt some of it last year, and would like to get back into it. We would mostly be following various introductory tutorials to the subject, and we can probably modulate the topics depending on students' interests.

• Marcel goh

Additive Combinatorics: sumsets and difference sets and how they relate to e.g. Roth's theorem, the Furstenberg--Sarkozy theorem

• Max Kaye

Graphs and Optimization

Number Theory

• Hazem Hassan

Number Theory, p-adic numbers, p-adic analysis: We will start by understanding the p-adic numbers and then looking briefly at the Riemann Zeta function and thinking how to translate it to a p-adic function.

• Hugues Bellemare

Gröbner Bases and Elimination Theory.

• Maninder Dhanauta

Diophantine Equations (Pythagorean triples, Four Square Theorem, Hilbert's 10th Problem and Computability) OR Riemann Mapping Theorem and Applications

• Marti Roset

Kummer's congruence via modular forms: Modular forms are holomorphic functions on the upper half plane that satisfy certain transformation properties with respect to Mobius transformations. In particular, they are periodic functions and admit a Fourier expansion. Their Fourier coefficients encode interesting arithmetic sequences. For example, the so-called Bernoulli numbers appear as coefficients of modular forms.  We will study the theory of modular forms and explore how their transformation properties with respect to Mobius transformations can be used to deduce congruences between Fourier coefficients of modular forms. In particular, we will deduce the so-called Kummer's congruence. This idea, due to Serre, exemplifies the fact that it is possible to prove algebraic statements (congruences between numbers) using analytic methods (holomorphic and periodic functions).

• Rudy Ariaz

Possible areas: basic algebraic geometry (classical or scheme-theoretic), basic topics in number theory: "This would be a reading project. Possible sources for algebraic geometry: Gathmann's notes or Vakil's ""The Rising Sea"" notes (available online), etc. Possible sources for number theory: Gouvea's ""p-adic Numbers: An Introduction"" or Ireland and Rosen's ""A Classical Introduction to Modern Number Theory"". The topic would be chosen according to student preference. (Note: weekly meetings would typically be conducted through Zoom.) "

• Simon Lapointe

Algebraic Number Theory, Local-Global Principle, Rational Points, Elliptic Curves

Numerical Analysis

• Miguel Ayala

Scientific computing, Dynamical systems, Numerical Analysis, Computer-assisted proofs.

• Seth Taylor

Shape Analysis for Computer Vision and Medical Imaging: Shape analysis for computer vision and medical imaging

• William Holman-Bissegger

Numerical methods for the Euler equations (for incompressible fluids): particle methods (e.g. point-vortex) and/or PDE methods (e.g. finite difference, Fourier-spectral methods)

Partial Differential Equations

• Arturo Arellano Arias

Monte Carlo simulation of stochastic processes.

• Jacob Reznikov

Riemannian Geometry, Symplectic Geometry, Mean Curvature Flow, Einstein Equations

• Samuel Zeitler

Calculus of Variations, Yamabe Problem, Sobolev Spaces, Einstein Constraints, Operators on Manifolds, Riemannian Geometry, Differential Topology, Blow up Analysis

Probability

• Sophia Howard

I don't have have an exact project in mind, but areas of probability: Brownian Motion, Random Voronoi Diagrams, other topics tbd

• Vincent Painchaud

Random Matrix Theory

Statistics and Machine Learning

• Hugo Latourelle-Vigeant

Theory of machine learning, random matrix theory, high dimensional probability, large-scale optimization

• Noah Marshall

I want to review some classical and modern results in stochastic optimization. We'll look at analysis of stochastic gradient descent. We'll work our way to Robbins-Monro from around the 50's to modern results. We'll try to implement some results.

• Peiyuan Huang

Missing Data, Regression Models, Bayesian Statistics, Biostatistics: We will be spending time studying an applied methodology that involve a specific regression model, by together reading a book or research papers. Simulation studies and a complete real data analysis will be done by the end as the DRP deliverable.

• Yanees Dobberstein

Unsafe Statistics: dependent data, rare events (Theory) [OR] Toying with ML (Programming)