Mathematical Sciences: Spectral Geometry of Compact Riemannian Manifolds and Kleinian Groups

佩里9707051 该项目涉及谱几何中的几个问题：紧致曲面的逆谱问题;双曲流形和相关向量丛上拉普拉斯算子的迹公式和动态zeta函数;散射算子及其行列式作为克莱因群变形空间上的函数。所采用的技术包括微扰理论、全局分析和Teichmuller理论。这个项目的目的之一是表明，一个紧凑的表面的等谱集是有限的度量接近常曲率。 谱几何是研究黎曼流形（一个具有度量的弯曲空间）的微分几何性质与自然微分算子的谱之间的相互作用，算子的谱通常以离散和可计算的方式捕获一个难以处理的微分算子的各种分析性质。阐明各种光谱数据的几何内容，然后是光谱几何的主要关注。

9707051 Perry This project deals with several problems in spectral geometry: the inverse spectral problem for compact surfaces; trace formulas and dynamical zeta functions for the Laplacian on hyperbolic manifolds and associated vector bundles; the scattering operator and its determinant as a function on the deformation space of a Kleinian group. Techniques to be employed include perturbation theory, global analysis, and Teichmuller theory. One of the aims of this project is to show that the isopectral set of a compact surface is finite for metrics close to constant curvature. Spectral geometry is concerned with the interaction of differential-geometric properties of a Riemannian manifold (a curved space with a metric) with the spectra of natural differential operators associated with it. The spectrum of an operator often captures various analytic properties of an otherwise intractable differential operator in a discrete and computable manner. Elucidating the geometric content of various spectral data then is the main concern of spectral geometry.