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。 该提案中概述了几何光谱理论研究的几个新方向。 尽管已经积极研究了数十年的本征函数的几何特性，但对光谱问题解决方案的拓扑特征知之甚少。我们建议使用多种方法（包括最近开发的持久性同源技术技术）来研究拉普拉斯本征函数的淋巴结和级别集合集的拓扑特性。 我们还旨在扩大光谱几何形状的范围，传统上可以通过研究潜在理论中产生的积分运算符的特征值和集成运算符的特征值的几何特性来处理差异和伪差算子。 拟议的研究计划向计算机科学的某些领域开辟了光谱几何学和拓扑的新应用。近年来，此类应用已迅速出现。特别是，光谱方法已在形状分析和几何处理中积极使用。这些字段具有许多现实生活中的应用程序，包括计算机动画和3D打印。 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.这是一个跨学科项目，涉及数学和计算机科学的合作者。