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。 该提案概述了几何谱理论的几个新研究方向。 虽然本征函数的几何性质已经被积极研究了几十年，但对谱问题解的拓扑特征却知之甚少。我们建议使用各种方法研究拉普拉斯本征函数的节点和子级集的拓扑性质，包括最近开发的持久同调技术。 我们还旨在通过研究势论中出现的积分算子的特征值和特征函数的几何性质来扩大谱几何的范围，传统上处理微分和伪微分算子。 拟议的研究计划开辟了光谱几何和拓扑在计算机科学某些领域的新颖应用。此类应用近年来迅速兴起。特别是，光谱方法已被积极地应用于形状分析和几何处理。这些领域在现实生活中有许多应用，包括计算机动画和 3D 打印。大多数现有的谱算法利用拉普拉斯算子的数据来“编码”物体的内在几何形状。我们的目标是开发类似的技术，允许捕获欧几里德空间中表面边界区域的外在几何形状。看来，Steklov 问题的谱几何和称为单层势的密切相关的积分算子的谱几何提供了用于此目的的适当工具。 这是一个跨学科项目，涉及数学和计算机科学领域的合作者。