Spectral geometry and topology and their applications

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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. **