Spectral theory of Dirac and Laplace operators

在这个项目中，我们将研究各种黎曼流形上狄拉克算子和拉普拉斯算子的谱理论问题。在紧致黎曼流形上，我们将研究紧致黎曼流形的谱刚性与等谱可变形性的问题，与长度谱的关系，以及不由谱数据决定的度量或势的性质。我们还将研究更高的谱不变量，如δ不变量和解析扭转。本课题的另一个基本问题是理解一类非紧黎曼流形的连续谱与无穷远处几何的关系。应用于这个问题的基本工具之一是几何散射理论的方法。特别强调的是有限体积的局部对称流形。这类流形的谱理论在自同构形式理论和数论中有着重要的应用。

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.