Speakers

Mini-course Speakers

Mini-course titles are tentative

  • Alexandre Girouard == Université Laval
    Isoperimetric-type problems in spectral geometry: beyond convexity
    This crash course will focus on spectral geometry, with particular emphasis on isoperimetric problems for the eigenvalues of the Laplace and Dirichlet-to-Neumann operators. I will briefly review the basic spectral theory of these operators on bounded domains in Euclidean space and compact Riemannian manifolds, highlighting variational characterizations of eigenvalues. Then, I will explore situations where the spectral gap is large, and explain how this relates to geometric concentration and clustering phenomena. This connection will be illustrated through quantitative isoperimetric inequalities linking eigenvalues to isoperimetric ratios—both for the Laplace operator on closed manifolds and the Dirichlet-to-Neumann operator on manifolds with boundary.

  • Antoine Henrot == Institut Élie Cartan de Lorraine
    Inequalities for Dirichlet and Neumann eigenvalues in convex domains
    In this course, we focus on the eigenvalues of the Laplace operator with Dirichlet boundary conditions (denote by \(\lambda_k(\Omega\)) and with Neumann boundary conditions (denoted by \(\mu_k(\Omega\)).
    Our aim is to explore bounds, if possible optimal, for the \(\lambda_k\) and the \(\mu_k\) especially when the domain \(\Omega\) is convex. These bounds should be expressed in terms of geometric quantities like the volume, the perimeter or the diameter. We will also give some interesting and challenging open problems.

  • Bo’az Klartag == Weizmann Institute
    Isoperimetric inequalities in high-dimensional convex sets

  • Galyna Livshyts == Georgia Tech
    Sharp inequalities for log-concave measures related to isoperimetry
    The most famous classical isoperimetric inequality roughly states that among plane figures with the boundary of a given length the largest area is achieved by a disk. It was vaguely known already in Ancient Greece, and since then, many generalizations and extensions were understood. In this mini-course, we discuss isoperimetric-type inequalities, such as Log-Sobolev inequality, Blaschke-Santalo inequality and inequalities related to Gaussian isoperimetry — classical versions as well as novel variants. We interpret these inequalities as concavity principles and employ the idea of linearization to understand better topics related to functional isoperimetry. Furthermore, we discuss new variants of the functional Blaschke-Santalo inequality in which the isoperimetric minimizers are not round. Some ramifications of this unusual phenomenon will also be explored.

  • Ramon van Handel == Princeton University
    Spectral methods in convex-geometric inequalities

Research Talk Speaker