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.

光谱问题位于许多物理现象的数学模型的核心，例如波传播，热扩散和量子力学效应。研究建议涉及对光谱的几何和拓扑特性以及Laplace和Steklov型特征值问题的几何和拓扑特性的研究。我们打算在奇异领域探索这些问题的光谱渐进性，旨在开发新技术，并找到有关流体动力学和量子混乱中的一些长期开放问题的答案。特别是，我们的方法应导致Fox和Kuttler在1983年对二维sloshing特征值的两项渐近学提出的猜想的解决方案，这代表了运河中流体振荡的频率。