Spectral geometry and topology and their applications

谱几何和拓扑及其应用

• 批准号: RGPIN-2017-05565
• 负责人: Polterovich, Iosif
• 金额: $ 3.13万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635314
• 关键词: Spectral; geometry; topology; their; applications

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.

光谱问题是许多物理现象数学模型的核心，例如波传播、热扩散和量子力学效应。该研究计划涉及光谱的几何和拓扑性质的研究以及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解决方案。我们打算探索奇异域上这些问题的谱渐近性，旨在开发新技术并找到一些源于流体动力学和量子混沌的长期悬而未决的问题的答案。特别是，我们的方法应该能够解决 Fox 和 Kuttler 在 1983 年提出的关于二维晃动特征值（代表运河中流体振荡频率）的两项渐近性的猜想。