Spectral theory of Dirac and Laplace operators

• Applicant: Professor Dr. Werner Müller
• Subject Area: Mathematics
• Term: from 2003 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5406725

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1154: Global Differential Geometry
• Participating Person: Professor Dr. Werner Ballmann