géométrie et théorie spectrale

几何谱理论是数学的一个领域，它位于以下两个领域的交叉点： 分析，偏微分方程和微分几何。在很大程度上，它是由起源于研究真实的生活现象的问题所激发的，如振动，热传播， 流体振荡和量子力学效应。第一部分的建议是有关的谱几何的斯捷克洛夫问题。这个特征值问题， 由V. Steklov在1902年提出，有许多重要的应用，但仍然相对 未被纯数学家探索过建议的研究课题包括奇异Steklov型问题的研究，如晃动问题;等谱 和谱不变量;节点几何的斯捷克洛夫特征函数。的第二部分 本文主要讨论了在研究拉普拉斯和薛定谔算子谱渐近性时出现的一些问题。*

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.* *