Spectral theory with non-unitary twists

• Applicant: Professorin Dr. Anke Pohl
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 441868048

The spectral theory of Riemannian manifolds is an important area of mathematics with strong influence on many other fields of mathematics (e.g., representation theory, number theory, harmonic analysis, and mathematical physics). During the last few years, a huge interest was risen in spectral theory of noncompact spaces (in particular of so-called open systems) as well as in a spectral theory for Riemannian locally symmetric spaces with non-unitary twists.With this project we will study the spectral properties of noncompact Riemannian locally symmetric spaces with twists of non-expanding cusp monodromy. This class of twists reaches far beyond the set of representations that are unitary at cusps, and it constitutes a frontier until which we might currently expect a twisted spectral theory. Our investigations will focus on hyperbolic spaces (thus, Riemannian locally symmetric spaces of noncompact type and of rank 1) but we will also carry out first steps towards an extension of the expected results to locally symmetric spaces of higher rank as well as to non locally symmetric spaces with hyperbolic ends. We will investigate how the geometry and dynamics at the various types of ends govern the properties of the resonances, proper eigenvalues, eigenfunctions, etc. of these spaces. Further we will study the asymptotic properties of these (sets of) spectral objects. In order words, we will understand here the leitmotiv "geometry at infinity" in two ways.This project is closely related to other projects in the SPP 2026 with regard to research interests, considered spaces and techniques applied.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity