Spectral geometry and topology and their applications

光谱问题是许多物理现象的数学模型的核心，如波传播、热扩散和量子力学效应。本研究计划是研究在几何对象上定义的拉普拉斯和斯特克洛夫型特征值问题的光谱和解的几何和拓扑性质。我们打算在奇异域上探索这些问题的谱渐近性，旨在开发新技术，并找到一些长期存在的开放性问题的答案，这些问题起源于流体力学和量子混沌。特别是，我们的方法应该导致Fox和Kuttler在1983年提出的关于二维晃动特征值的两项渐近性的猜想的解，这些特征值表示管道中流体振荡的频率。

Spectral problems lie at the core of the mathematical models of many physical phenomena, such as wave propagation, heat diffusion and quantum-mechanical effects. The research proposal is concerned with the investigation of geometric and topological properties of spectra and solutions of Laplace and Steklov type eigenvalue problems defined on geometric objects. We intend to explore spectral asymptotics for those problems on singular domains, aiming to develop new techniques and find answers to some long standing open questions with origins in hydrodynamics and quantum chaos. In particular, our approach should lead to a solution of the conjectures put forward by Fox and Kuttler in 1983 on the two-term asymptotics of the two-dimensional sloshing eigenvalues, representing the frequencies of fluid oscillations in a canal.