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

9707051 Perry这个项目解决了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的迹公式和动力学Zeta函数；散射算子及其作为Klein群形变空间上的函数的行列式。所使用的技术包括微扰理论、整体分析和泰希穆勒理论。这个项目的目的之一是证明对于接近常曲率的度量，紧致曲面的等谱集是有限的。谱几何是关于黎曼流形(具有度量的曲面空间)的微分几何性质与与之相关的自然微分算符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捉到原本难以处理的微分算子的各种分析性质。因此，阐明各种光谱数据的几何含量是光谱几何学的主要问题。

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.