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.

这项研究涉及谱几何的三个领域：(1) 谱几何、迹公式和克莱因群的行列式，(2) 等谱黎曼流形的有限性和紧性定理，以及 (3) 自由边界问题的谱几何。 该研究涉及迹公式的发展以及任意几何有限离散双曲等距群的拉普拉斯行列式的研究。 重点是余无穷体积群，目标是从几何角度理解散射算子的极点，并研究变形下拉普拉斯和塞尔伯格 zeta 函数行列式的行为。 这项研究属于几何学的一般领域，特别是作用于双曲 n 维空间的克莱因群。 这项研究对于几何学和数学物理都具有重要意义。