Mathematical Sciences: Spectral Geometry and Inverse Problems

数学科学：谱几何和反问题

• 批准号: 9203529
• 负责人: Peter Perry
• 金额: $ 12.3万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-06-01 至 1996-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203529&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Spectral; Geometry; Inverse

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.

这项研究涉及光谱几何形状的三个领域：（1）kleinian群体的光谱几何形状，痕量公式和决定因素，（2）等riemannian歧管的有限和紧凑定理，以及（3）自由边界问题的光谱几何学。 这项研究涉及痕量公式的发展以及对任意几何有限的双曲线异构体离散群的laplacians的决定因素的研究。 重点是共同体积组，目的是通过几何形式理解散射操作员的极点，并研究在变形下Laplacian和Selberg Zeta功能的行为。 这项研究是在几何形状的一般领域，尤其是作用于双曲线N维空间的克莱琳群体。 该研究对几何和数学物理学具有重要意义。