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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 (¾ÅÐãÖ±²¥)

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 (¾ÅÐãÖ±²¥)

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.

Math & Stats Number Theory Seminar
Thursday, October 3rd, 2024

Time: 4:30 p.m.  Place: Jeffery Hall, Room 202

Speaker: Sunil Naik (¾ÅÐãÖ±²¥)

Title: Coprimality of values of certain sequences

Abstract: One of the classical problems in number theory is to find consecutive coprime elements in arithmetic sequences. In this talk, we will discuss the coprimality of values of certain regular sequences. This is a joint work with Prof. Jean-Marc Deshouillers.

Math & Stats Department Colloquium
Friday, October 4, 2024

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

Speaker: Hitay Özbay (Bilkent University)

Title: Analytic Interpolation Methods for Strong Stabilization of Time Delay Systems

Abstract: Strong stabilization of a plant is defined as finding a stabilizing feedback controller that is itself stable. This problem has attracted interest since the 1970s, and various design techniques have been proposed. On the other hand, many variations of the problem with additional conditions, such as robust stability, H2, or H∞ optimality, are difficult and typically solved under certain sufficient conditions. This presentation will briefly review interpolation methods for constructing strongly stabilizing controllers for LTI finite-dimensional plants and illustrate their extensions to systems with time delays. We also present a new family of controllers for unstable time delay systems and discuss related open problems.

Math & Stats Number Theory Seminar
Thursday, September 26, 2024

Time: 4:30 p.m.  Place: Jeffery Hall, Room 202

Speaker: Ram Murty (¾ÅÐãÖ±²¥)

Title: TURAN'S THEOREM ON MAXIMAL CLIQUES IN GRAPHS

Abstract: In 1941, Turan gave an upper bound for the number of edges of a graph that does not contain a complete subgraph of prescribed size. This theorem has had many applications and proofs that it has even made it into "Proofs from THE Book''. I will discuss a few of these proofs in the talk. No knowledge of graph theory is assumed.