报告题目：Eigenvalue estimates and a conjecture of Yau

报告时间：2024-09-23　16:30-17:30

报 告 人 ：Yannick Sire  美国约翰·霍普金斯大学教授

报告地点：理学院东北楼四楼报告厅（404）

ZOOM ID：896 7414 3298（ Password:301029）

报告摘要：

I will describe various upper and lower bounds on the spectrum of the Laplace-Beltrami (and associated Schrödinger) operator on Riemannian manifolds. The upper bounds lead to some important results in spectral geometry establishing a link between the so-called conformal spectrum (and extremal metrics) and branched minimal immersions into Euclidean spheres. I will as well describe some estimates on spectral projections, which amount to understand some concentration properties of eigenfunctions. Derivation of those various bounds are aimed at understanding the geometry and topology (and dynamical properties) of manifolds in terms of spectral data.

I will then describe a conjecture by Yau on the first eigenvalue on minimal submanifolds of the sphere, which is known only for some examples and still largely open. I will then present some recent results where we improve quantitatively the best known lower bound (in the general case) of Choi and Wang of the mid 80’s. I will address some open problems and possible directions to generalize the results.

专家简介：

Professor Yannick Sire is a Professor and the Director of Graduate Studies in the Department of Mathematics at Johns Hopkins University. He completed his Ph.D. at the Institut National des Sciences Appliquées de Toulouse, France, in 2005. Professor Sire's research spans a diverse range of mathematical disciplines, including partial differential equations, harmonic and geometric analysis, and dynamical systems. He has made significant advancements in areas such as non-local diffusion processes, the KAM theory, free boundary problems and many others, contributing deeply to both theoretical and applied mathematics. In recognition of his outstanding contributions to analysis and geometry, Professor Sire was awarded the Simons Fellowship for Mathematical Sciences in 2018 and was honored as a Fellow of the American Mathematical Society in 2021. Among his other accolades are the Simons Visiting Professorship and the distinction of JHU Professor of the Year in 2023.