Spectral theory of Dirac and Laplace operators

在这个项目中，我们将研究各种类型的黎曼流形上的狄拉克和拉普拉斯型算子的谱理论问题。在紧致黎曼流形上，我们将研究紧黎曼流形的谱刚性与等谱变形性，与长度谱的关系，以及谱数据不确定的度量或势的性质。我们还将研究更高的谱不变量，如η-不变量和解析挠率。本项目的另一个基本问题是理解给定的一类非紧黎曼流形的连续谱和无穷远处几何之间的关系。几何散射理论方法是解决这一问题的基本工具之一。特别强调将放在类的局部对称流形的有限体积。在这类流形上的谱理论在自守形式理论和数论中有重要的应用。

In this project we shall study problems of spectral theory of Dirac and Laplace type operators on various classes of Riemannian manifolds. On compact Riemannian manifolds we will study questions of spectral rigidity versus isospectral deformability of compact Riemannian manifolds, relations to the length spectrum, and properties of metrics or potentials which are not determined by spectral data. We shall also study higher spectral invariants such as eta-invariants and analytic torsion. Another basic problem in this project is to understand the relations between the continuous spectrum and the geometry at infinity of a given class of non-compact Riemannian manifolds. One of the basic tools that will be applied to this problem are methods of geometric scattering theory. Special emphasis will be put on the class of locally symmetric manifolds of finite volume. Spectral theory on such manifolds has important applications in the theory of automorphic forms and in number theory.