231205 Anke Pohl

Resonances of Riemannian manifolds play an important role in many areas of mathematics, e.g., analysis, dynamical systems, mathematical physics, and number theory. It has been long known that resonances of hyperbolic surfaces enjoy fundamentally different properties depending on whether the considered hyperbolic surface is compact or of finite area (but noncompact) or of infinite area. While the former two situations are fairly well understood, the latter still offers a lot of surprises despite intensive contemporary research efforts. A particularly successful approach to study the localization and distribution of resonances for hyperbolic surfaces of infinite area are transfer operator methods. Roughly speaking, this is a methodology that allows us to connect the spectral theory of hyperbolic surfaces with their dynamical properties. I will provide a gentle overview of this methodology, focusing on insights and heuristics, and discuss some recent results on the distribution of resonances.