Mathematics Colloquium

Anke Pohl

(Universität Göttingen)

"Geodesic flows, Laplace eigenfunctions, and resonances"

Date: Tue, April 21, 2015
Time: 17:15
Place: West Hall 4

Abstract: Eigenfunctions of Laplace operators on Riemannian manifolds and orbifolds are objects of common interest in various fields, e.g. number theory, spectral theory, harmonic analysis, and mathematical physics, in particular quantum chaos. It is long known that these eigenfunctions are intimately related to geometric properties of the orbifold. However, the full extent of this relation and its consequences is still an active field of research.

In this talk we will consider the family of Hecke triangle surfaces. This family starts with the modular surface \(\operatorname{PSL}_2(\mathbb{Z})\backslash\mathbb{H}\) (here, \(\mathbb{H}\) denotes the hyperbolic plane) which has one cusp and two elliptic points. By iteratively increasing the order of one of the elliptic points, this point is slowly pulled to infinity until it deforms into a second cusp. Then this cusp is opened to form a funnel of increasing width.

We will use this family to discuss parallel ''slow'' and ''fast'' discretizations for the geodesic flows on these surfaces. We will see that the fast discretizations serve for thermodynamic formalism approaches to dynamical zeta functions. The transfer operators arising from the slow discretizations however allow (classical dynamical) characterizations of Laplace eigenfunctions in the finite area case. Moreover, these results and the deformation properties lead to natural conjectures on resonances and vector-valued automorphic forms as well as two new formulations of the Phillips-Sarnak conjecture on nonexistence of even Maass cusp forms.

The colloquium is preceded by tea from 16:45 in the Resnikoff Mathematics Common Room, Research I, 127.