Mathematical Sciences: Spectral Geometry and Inverse Problems

本研究涉及谱几何的三个领域：(1)Klein群的谱几何、迹公式和行列式；(2)等谱黎曼流形的有限和紧性定理；(3)自由边界问题的谱几何。这项研究涉及发展迹公式和研究任意几何有限的双曲等距离散群的拉普拉斯行列式。重点是余无限体积群，目标是从几何上理解散射算符的极点，并研究拉普拉斯函数和Selberg Zeta函数的行列式在变形下的行为。这项研究是在一般几何领域，特别是作用在双曲n维空间上的Klein群。这项研究对几何学和数学物理学都有重要意义。

This research concerns three areas of spectral geometry: (1) spectral geometry, trace formulas, and determinants for Kleinian groups, (2) finiteness and compactness theorems for isopectral Riemannian manifolds, and (3) spectral geometry of free boundary problems. The research concerns the development of trace formulas and the study of determinants of Laplacians for arbitrary geometrically finite discrete groups of hyperbolic isometries. The emphasis is on co-infinite volume groups, and the goal is to understand poles of the scattering operator geometrically and to study the behavior of the determinant of the Laplacian and Selberg zeta function under deformations. This research is in the general area of geometry and, in particular, Kleinian groups acting on hyperbolic n-dimensional space. The research has significance both to geometry and to mathematical physics.