Spectral geometry and topology and their applications

光谱问题是许多物理现象的数学模型的核心，如波的传播，热扩散和量子力学效应。本研究计画系探讨定义于几何物体上之拉普拉斯与斯泰克洛夫型特征值问题之谱与解之几何与拓扑性质。我们打算探索奇异域上这些问题的谱渐近性，旨在开发新的技术，并找到一些长期存在的开放问题的答案，起源于流体力学和量子混沌。特别是，我们的方法应导致Fox和Kuttler在1983年提出的关于二维晃动本征值的两项渐近性的解，该本征值代表运河中流体振荡的频率。提出了几何谱理论的几个新的研究方向。 虽然本征函数的几何性质已经被积极研究了几十年，而很少有人知道的谱问题的解决方案的拓扑特征。我们建议使用各种方法，包括最近发展起来的持续同源技术，研究节点和子级集的拉普拉斯特征函数的拓扑性质。 我们还旨在扩大范围的谱几何，传统上处理微分和pseudodiomatic运营商，通过调查的几何性质的特征值和特征函数的积分算子产生的潜在的理论。** 拟议的研究计划开辟了新的应用程序的频谱几何和拓扑的某些领域的计算机科学。近年来，这种应用迅速出现。特别地，谱方法已被积极地用于形状分析和几何处理。这些领域有许多真实的生活应用，包括计算机动画和3D打印。大多数现有的光谱算法利用拉普拉斯算子的数据，“编码”的内在几何形状的对象。我们的目标是开发类似的技术，这将允许捕获的外在几何表面边界区域在欧几里得空间。看来，适当的工具，为这一目的提供了光谱几何的斯捷克洛夫问题和密切相关的积分算子称为单层潜力。这是一个跨学科的项目，涉及数学和计算机科学的合作者。**

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. **