### 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.*