géométrie et théorie spectrale

几何谱理论是一个数学领域，它位于分析、偏微分方程式和微分几何的交汇处。在很大程度上，它是由研究现实生活现象中的问题所驱动的，如振动、热传播、流体振荡和量子力学效应。提案的第一部分涉及Steklov问题的谱几何。这个本征问题由V.Steklov在1902年提出，有许多重要的应用，但相对来说，纯数学家还没有探索过。建议的研究主题包括奇异Steklov型问题的研究，如晃动问题；等谱不变量和谱不变量；Steklov特征函数的节点几何。第二部分讨论了拉普拉斯算子和薛定谔算子的谱渐近性研究中出现的一些问题。

Geometric spectral theory is an area of mathematics that lies at the intersection of analysis, partial differential equations and differential geometry. To a large extent, it is motivated by questions originating in the study of real life phenomena, such as vibration, heat propagation, oscillations of fluids and quantum mechanical effects. The first part of the proposal is concerned with the spectral geometry of the Steklov problem. This eigenvalue problem, introduced by V. Steklov in 1902, has many important applications, but is still relatively unexplored by pure mathematicians. The proposed topics of research include the study of singular Steklov type problems, such as the sloshing problem; isospectrality and spectral invariants; nodal geometry of Steklov eigenfunctions. The second part of the proposal focuses on some questions arising in the study of spectral asymptotics of Laplace and Schrodinger operators.