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Math & Stats Number Theory Seminar
Wednesday, October 23rd, 2024

Time: 1:30 p.m.  Place: Jeffery Hall, Room 319

Speaker: Tony Haddad (Université de Montréal)

Title: A coupling for prime factors

Abstract: We present a coupling between a random integer $N_x$, chosen uniformly from the interval $[1, x]$, and a Poisson-Dirichlet process $(V_i)_{i \ge 1}$ satisfying $$\mathbb E\, \sum_{i \ge 1} |\log P_i- V_i\log x| \asymp 1,$$ where $N_x = P_1P_2 \cdots$ is the unique factorization of $N_x$ into primes (or ones), and the $P_i$'s are non-increasing. This resolves a conjecture posed by Arratia in 1998. We also explain how to apply the coupling to extract statistical information about divisors. This is joint work with Dimitris Koukoulopoulos.

Math & Stats Algebra & Geometry Seminar
Tuesday, October 22nd, 2024

Time: 3:30 p.m.  Place: Jeffery Hall, Room 422

Speaker: David Wehlau (RMC & ºÚÁϳԹÏ×ÊÔ´)

Title: Rings of Invariants for Finite Groups of Lie Type

Abstract: This is joint work with H.E.A. Campbell (Queen's) and R.J. Shank (University of Kent at Canterbury). There has been significant interest for a number of years in the rings of invariants of the defining representations of the finite classical groups. These rings have been computed for several of the groups and have remained mysterious for several of the other cases. Recently, we succeeded in finding the first explicit description of the rings of invariants for the case of orthogonal groups in odd characteristics. This case has provided a single framework that allows us to handle all the finite classical groups. I will outline this framework and discuss some of the consequences of these computations.

Math & Stats Department Colloquium
Friday, October 18, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Hester Graves (Center for Computing Sciences)

Title: The Minimal Euclidean Function on the Gaussian Integers

Abstract: Motzkin proved that every Euclidean domain $R$ has a minimal Euclidean function, $\phi_R$ . He showed that when $R = \mathbb{Z}$, the minimal function is $\phi_{\mathbb{Z}} (x) =\ log 2_ |x|$. For over seventy years, $\phi_{\mathbb{Z}}$ was the only example of an explicitly-computed minimal function in a number field. We give the first explicitly-computed minimal function in a non-trivial number field, $\phi_{\mathbb{Z}[i]}$ . The proof introduces a new way to visualize quotients $\mathbb{Z}[i]$. This talk is accessible to undergraduates.

Math & Stats Department Colloquium - The L. Lorne Campbell Lecture
Friday, October 25, 2024

Time: 3:00 p.m.  Place: Jeffery Hall, Room 127

Speaker: Amaury Hayat (Ecole des Ponts ParisTech, CERMICS)

Title: How can Machine Learning Help Mathematicians?

Abstract: Artificial Intelligence (AI) has demonstrated remarkable achievements across diverse fields, from natural language processing to mastering complex games like chess. This naturally raises the question: can AI assist mathematicians in solving open problems in mathematics? This talk aims to address this question. We will explore how AI models can be trained to provide valuable insights into several mathematical questions from different areas of mathematics and applied mathematics. We will then present examples of AI models specifically trained to prove mathematical theorems on their own.

 

PDEs & Applications Seminar

Friday, October 11th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: Alexandre Girouard (Université Laval)

Title: Singular perturbations, homogenization and conformal approximation in spectral geometry: tools for eigenvalue optimization and flexibility

Abstract: In this talk, I will discuss recent advances in spectral geometry that use techniques from homogenization theory—originally developed in industrial mathematics—and conformal perturbations to address spectral optimization problems. These methods allow an elliptic eigenvalue problem (P) to be approximated as the limit of a family of related problems (P_t), often of a different character. The solutions of (P_t) are carefully controlled to approximate those of (P) in the limit. This approach yields sharp isoperimetric bounds for eigenvalues of the Dirichlet-to-Neumann map on planar domains and shows the instability of maximizers in well-known spectral optimization problems. I will also discuss a simpler singular perturbation: excision of thin tubular neighborhoods around submanifolds, which in some cases offers more direct proofs of similar results.

Math & Stats Department Colloquium
Friday, October 11, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Alexandre Girouard (Université Laval)

Title: Symmetry Breaking and Hidden Symmetries in Eigenvalue Optimization

Abstract: In classical shape optimization, maximizers are often highly symmetric. This goes all the way back to the classical isoperimetric inequality, where balls uniquely maximize the volume under perimeter constraint. In this talk, I will explore the symmetries of shapes that optimize eigenvalues of Laplace and Dirichlet-to-Neumann operators under various geometric constraints. While these optimizers tend to exhibit symmetry, there are interesting exceptions where symmetries appear to be broken. But are they truly broken—or might they be subtly hidden, waiting to be uncovered a different perspective?

Math & Stats Algebra & Geometry Seminar
Tuesday, October 8th, 2024

Time: 3:30 p.m.  Place: Jeffery Hall, Room 422

Speaker: Khoa Nguyen (Queen's University)

Title: On U(h)-free modules over sl(m|n)

Abstract: The study of modules over Lie algebras and superalgebras is usually divided into different categories. The first important category is the category of weight modules. In particular, such a category consists of modules that decompose into direct sums of their weight space with respect to a fixed Cartan subalgebra h. Free U(h)-modules are important non-weight modules. Such modules are free of finite rank when restricted to the Cartan subalgebra h. In this talk, I will provide the classification of U(h)-free modules of rank 2 over sl(m|1). I will also give an explicit family of weight modules of multiplicity 2 over sl(m|1) obtained from those U(h)-free modules. This is based on a joint work with Ivan Dimitrov.

PDEs & Applications Seminar

Friday, October 4th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: Liangbing Luo (Queen's University)

Title: Logarithmic Sobolev inequalities on sub-Riemannian manifolds

Abstract: The logarithmic Sobolev inequality has been first introduced and studied by L. Gross on a Euclidean space, and since then it found many applications. In particular, many existing results concern the question on how the constant in the logarithmic Sobolev inequality depends on the geometry of the underlying space. In this talk, I will review recent results on different perspectives to study the constant (and its dimension-independence) in the logarithmic Sobolev inequality on sub-Riemannian manifolds. As for many of such setting curvature bounds (or classical Bakry-Emery estimates) are not available, we use different techniques. Based on joint work with M. Gordina, R. Neel, A. Butaev and N. Shanmugalingam.