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，例如波传播、热扩散和量子力学效应。该研究方案涉及定义在几何对象上的Laplace和Steklov型特征值问题的谱和解的几何和拓扑性质的研究。我们打算探索奇异域上这些问题的谱渐近，旨在开发新的技术，并找到一些长期悬而未决的问题的答案，这些问题起源于流体动力学和量子混沌。特别是，我们的方法应该解决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，这些特征值代表了渠道中流体振荡的频率。虽然特征函数的几何性质已经被活跃地研究了几十年，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次水平集的拓扑性质，包括最近发展起来的持久同调技术。通过研究位势理论中积分算子的本征值和本征函数的几何性质，我们还旨在拓宽传统上处理微分和拟微分算子的谱几何的范围。*拟议的研究计划为计算机科学的某些领域开辟了谱几何和拓扑学的新应用。近年来，这类应用迅速涌现。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。大多数现有的谱算法利用拉普拉斯算符的数据，拉普拉斯算符对物体的内在几何进行编码。我们的目标是开发类似的技术，允许捕获欧几里德空间中曲面边界区域的非本征几何。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提供了适当的工具。这是一个涉及数学和计算机科学的合作者的跨学科项目。**

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. **