Mathematical Sciences: Spectral Geometry and Inverse Problems

本研究涉及光谱几何的三个领域：(1)Kleinian群的光谱几何、迹公式和行列式；(2)等电黎曼流形的有限性和紧性定理；(3)自由边界问题的光谱几何。研究了任意几何有限双曲等距离散群的迹公式的发展和拉普拉斯行列式的研究。重点是协无穷体积群，目标是从几何上理解散射算子的极点，并研究拉普拉斯和塞尔伯格zeta函数的行列式在变形下的行为。本研究是在几何的一般领域，特别是在双曲n维空间上的Kleinian群。该研究对几何和数学物理都具有重要意义。

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.