Mathematical Sciences: Spectral Geometry and Inverse Problems

本研究涉及谱几何学的三个方面：（1） 谱几何、迹公式和Kleinian行列式 群的有限性和紧性定理，（2）等谱群的有限性和紧性定理 黎曼流形;（3）自由边界的谱几何 问题 本研究涉及微量元素的开发 公式和拉普拉斯行列式的研究 任意几何有限离散双曲群 等距线 重点是共无限卷组， 目标是理解散射算子的极点 几何学和研究的行列式的行为 变形下的Laplacian和Selberg zeta函数 这项研究是在一般的几何领域，在 特别地，Kleinian群作用于双曲n维 空间 该研究对几何学和 数学物理

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.