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。******提出了几何光谱理论研究的几个新方向。虽然特征函数的几何性质已经被积极地研究了几十年，但对于谱问题解的拓扑特征却知之甚少。我们建议使用各种方法来研究拉普拉斯特征函数的节点集和子水平集的拓扑性质，包括最近发展的持久同调技术。我们还旨在通过研究位势理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光谱几何的范围，光谱几何传统上处理微分和伪微分算子。******提出的研究计划为谱几何和拓扑学在计算机科学的某些领域开辟了新的应用。近年来，此类应用迅速涌现。特别是光谱方法在形状分析和几何处理中得到了积极的应用。这些领域在现实生活中有很多应用，包括计算机动画和3D打印。大多数现有的光谱算法都利用拉普拉斯算子的数据来“编码”一个物体的内在几何形状。我们的目标是开发类似的技术，以捕获欧几里得空间中表面边界区域的外在几何形状。Steklov问题的谱几何和与之密切相关的称为单层势的积分算子的谱几何似乎为实现这一目的提供了合适的工具。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. **