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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711256
• 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. Several new directions of research in geometric spectral theory are outlined in the proposal. While the geometric properties of eigenfunctions have been actively studied for decades, rather little is known about the topological features of the solutions of spectral problems. We propose to study the topological properties of nodal and sublevel sets of Laplace eigenfunctions using a variety of methods, including the recently developed techniques of persistent homology. We also aim to broaden the scope of spectral geometry, which traditionally deals with differential and pseudodifferential operators, by investigating the geometric properties of eigenvalues and eigenfunctions of integral operators arising in potential theory. The proposed research program opens up novel applications of spectral geometry and topology to some areas of computer science. Such applications have been rapidly emerging in recent years. In particular, spectral methods have been actively used in shape analysis and geometry processing. These fields have many real life applications, including computer animation and 3D printing. Most existing spectral algorithms make use of the data for the Laplace operator that "encodes'' the intrinsic geometry of an object. We aim to develop similar techniques that would allow to capture the extrinsic geometry of surfaces bounding regions in the Euclidean space. It appears that the appropriate tools for this purpose are provided by the spectral geometry of the Steklov problem and of a closely related integral operator called the single layer potential. This is an interdisciplinary project involving collaborators both in mathematics and computer science.

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。