Below are this year's projects. Each project is a small reading group led by an upper-year undergraduate mentor. Descriptions are written by the mentors. Check the prerequisites, then sign up through the Google Form.
Mentor: Mustafa Omer (Year 4)
About the mentor
I'm a 4th year mathematics undergrad. I am interested in complex analysis and differential geometry.
Description
We will study harmonic functions, which are the solutions to Laplace's equation. We will investigate the connections between harmonic and holomorphic function. In particular, one of the most surprising facts about harmonic functions is that they are locally the real part of holomorphic functions. We will explore some fundamental results, including the mean value property, the maximum principle, and Harnack's principle. Towards the end, we will discuss the Dirichlet Problem.
The goal is to develop an understanding of these concepts through proofs and examples, exploring the connections between complex analysis and partial differential equations. We will roughly follow Axler, Bourdon, and Ramey's Harmonic Function Theory.
Prerequisites: A course in multivariable calculus. Having some complex analysis background would be helpful but not necessary.
Hours per week: flexible · Would consider continuing in the winter: Maybe
Mentor: Hana Connelly (Year 4)
About the mentor
My name is Hana, I'm in my fourth and final year specializing in Mathematics and its Applications (stats stream). I really enjoy proof-based math and creative problem solving! Through my experience in MAT475 (Problem Solving Seminar), I have developed a strong interest in exploring how mathematical ideas can be used to solve unfamiliar problems. I would love to share this enthusiasm with other students through a reading project in combinatorial game theory while continuing to deepen my own understanding of the subject. My goal is to pursue further studies in math and hopefully get a PhD in pure mathematics!
Description
This project explores the mathematics of two-player games in which players take turns making moves according to specific rules. Through games such as Nim and other impartial games, students will investigate winning and losing positions, develop strategic arguments, and learn to prove when a winning strategy exists. Topics will include mathematical induction, invariants, recursive reasoning, and the Sprague–Grundy theorem, which provides a framework for analyzing and combining impartial games. Emphasis will be placed on creative problem solving, rigorous proof construction, and discovering the mathematical structures underlying strategic games. Students will conclude by independently analyzing a game or exploring a generalization of a known result.
Prerequisites: MAT137Y1/MAT157Y1; MAT223H1 or equivalent; MAT344 would be very useful but not required.
Hours per week: 1 · Would consider continuing in the winter: Maybe
Mentor: Hao Jason Cao Shi (Year 3)
About the mentor
I am a third year math specialist and Physics Major, I attended a DRP last fall about Knot theory and I found the readings to be very interesting and found the group discussion engaging. I am interested in mathematical physics, more specifically quantum mechanics and operator theory.
Description
I plan on using Fourier Analysis: An Introduction by Stein as the main reference and hopefully cover chapter 1 - 5, covering motivations and topics such as construction of Fourier series, convergence, Hilbert spaces, Fourier transform and application to the heat equation.
Prerequisites: Calculus and linear algebra
Hours per week: flexible · Would consider continuing in the winter: Maybe
Mentor: Slimane Alaoui Soulimani Valenti (Year 3)
About the mentor
I'm Slimane Alaoui Soulimani Valenti, a third year Math major at UofT.
Description
Look at the night sky and you'll see planets tracing paths astronomers can predict centuries ahead. But the solar system has eight planets and a sun, at least nine bodies all pulling on each other. Meanwhile, one of the most famous results in mathematics says even the three-body problem has no general closed-form solution, and its motion can be chaotic. So how do we predict anything at all?
We begin with the two-body problem, which reduces completely using conserved quantities and gives Kepler's conics. For n ≥ 3, we ask what goes wrong. Using the Hamiltonian formulation and Noether's theorem, we identify the classical first integrals (energy, momentum, angular momentum) and use them to reduce the system's dimension. We then see why this isn't enough: Bruns and Poincaré showed there are no further integrals of the right kind, and Poincaré's work on the restricted three-body problem led directly to the discovery of chaos.
Prerequisites: Currently taking a course with proofs. If possible, I would like either all of the "students" to either have the same experience with ODEs (not at all wouldn't be a problem as I could just add a session).
Hours per week: 2 · Would consider continuing in the winter: Yes
Mentor: Zander (Year 4)
About the mentor
Hi I'm Zander. I'm a third year math specialist. I am interested in all kinds of math, and especially like to get my hands on objects in order to understand them better.
Description
The prime number theorem is a central result in algebraic number theory. It approximates the asymptotic growth of the count of prime numbers. One classical approach to its proof uses Tauberian theorems to connect the behavior of Dirichlet series near their singularities to asymptotic information about their coefficients. This provides a natural starting point for exploring a broader principle: analytic properties of L-functions encode their arithmetic information.
The goal of this reading course will to first understand this proof of the prime number theorem. And then we will move to Rankin-Selberg theory. By studying L-functions associated with automorphic forms, we will investigate how ideas from the classical theory extend to more general arithmetic settings.
Prerequisites: MAT137/157, MAT224/247. Please be enrolled in MAT257 or MAT237 and M327.
Hours per week: 2 · Would consider continuing in the winter: Yes
Mentor: Isaac Clark (Year 4)
About the mentor
My name is Isaac Clark. I am a fourth-year math specialist at the St. George campus. My mathematical interests primarily lie in functional analysis, specifically operator algebras. Outside of math, I enjoy theatre and working with dogs.
Description
The goal of this DRP will be to survey various results which may be called “spectral theorems” and their corollaries. We will briefly recount the spectral theorem for finite-dimensional vector spaces before taking the plunge into Pedersen’s Analysis Now to discuss Banach spaces, compact and bounded operators, projection-valued measures, and the Gelfand transform. Extra topics may be covered as time permits and according to the interests of the participants.
Prerequisites: MAT257, or MAT247 and some other background in real analysis. Familiarity with complex analysis may be helpful, but is not necessary.
Hours per week: flexible · Would consider continuing in the winter: Yes
Mentor: Ruchir Varma (Year 4)
About the mentor
My name is Ruchir Varma and I am a 4th year persuing a math specialist, Physics major and philosophy major. Ive been getting more into optimal transport while taking MAT1502 and would like to share it with others since i don't think it's a very well discussed topic in undergrad. Ive been part of 2 DRPs in the past and seen how much that has motivated me to learn about a topic that usually seems unapproachable and would like to give that chance to others as well.
Description
The project will introduce optimal transport from an accessible Euclidean viewpoint. Some of the main thing that i think should be touched on are Monge and Kantorovich formulations, duality, the 1D monotone rearrangement, Wasserstein distances and geodesics, plus a glimpse of transport on manifolds and the formal Riemannian geometry of Wasserstein space. It'll be in a seminar style but ill make problems each week for the content being talked about as well so that there is some accountability for more than just your own talk.
Prerequisites: I would like to make it flexible so, preferably 357 (or some familiarity with measure theory) but i could cover the necessary content and have it just be MAT257 to have some degree of mathematical maturity and basic notions of the motivation behind the ideas discussed (also MAT327 since a lot of topology is used)
Hours per week: flexible · Would consider continuing in the winter: Maybe
Mentor: Kristopher Zhao (Year 4)
About the mentor
I am a 4th year math specialist student interested in number theory.
Description
It will be an introduction to analytic number theory by following the first 5 chapters of Apostol’s introduction to analytic number theory. We can change the topic depends on students interests.
Prerequisites: MAT157 and interests in NT
Hours per week: 1 · Would consider continuing in the winter: Maybe
Mentor: Zef (Year 4)
About the mentor
4th year, math specialist student interested in infinite dimensional linear algebra and analysis.
Description
Why do the eigenvalues of a large symmetric random matrix form a semicircle? We will cover the basics of operator algebras and quickly review central notions from classical probability. The focus of the project is free independence and free products. Our goal is to prove Wigner's semicircle law. We will follow Nica and Speicher, Lectures on the Combinatorics of Free Probability.
Prerequisites: MAT157 and MAT247. Some knowledge of probability is desirable.
Hours per week: 1 · Would consider continuing in the winter: Yes