# One Day Ergodic Theory Meeting: Noncommutative Geometry, Number Theory and Dynamics

Wednesday, 21-05-2014, MS.02.

This is part of a series of meetings between Bristol, Liverpool, Leicester, Loughborough, Manchester, QMUL, Surrey and Warwick, supported by a Scheme 3 grant from the LMS.

• 1.30-2.30 pm: Markus Fraczek, Warwick

"Zeros of the Selberg zeta function and involutions of Maass wave forms"

Abstract: We introduce a method for evaluating the Selberg zeta function for Hecke congruence subgroups $\Gamma_0(n)$ and unitary characters $\chi$. We study certain deformations by characters $\chi_\alpha$ with deformation parameters $\alpha$, where in general these characters are non-arithmetic. The main motivation for the investigation of such character deformations of the Selberg zeta function is that by studying the deformation of its zeros we gain access to the deformation of the discrete spectrum and the resonances of the hyperbolic Laplace-Beltrami operator, both under singular and non-singular perturbations. The Selberg zeta function can be expressed in terms of Fredholm determinants of certain operators from statistical mechanics, the so-called transfer operators. We found certain symmetries of the transfer operator that are related to involutions of the eigenfunctions, the so-called Maass wave forms, of the hyperbolic Laplacian. By using these symmetries we can determine whether a zero of the Selberg zeta function corresponds to an odd or even Maass wave form for each involution. Our numerical experiments for $\Gamma_0(4)$ indicate that even Maass wave forms exist only for discrete values of $\alpha$ and are destroyed under the smallest infinitesimal change of the value of $\alpha$, which is in accordance with the conjecture of Sarnak and Phillips on the non-existence of Maass wave forms for non-arithmetic $\chi_\alpha$.

• 2.45-3.45 pm: Gunther Cornelissen, Utrecht

"Graph reconstruction and noncommutative geometry"

Abstract: Given a finite graph, we consider the crossed product algebra of the action of its fundamental group on the boundary of its universal covering tree. As an operator algebra, this contains only topological information about the graph, but when enhanced by a suitable one-parameter group of automorphisms, one can reconstruct any multigraph with minimal degree three. Joint work with Matilde Marcolli.

• 4.15 pm: Bogdan Nica, Goettingen

"Boundaries of hyperbolic groups: dynamics and analysis from a noncommutative viewpoint"

Abstract: C*-algebras associated to hyperbolic groups have many exciting properties from the perspective of Noncommutative Geometry. The focus of this talk is on those properties which exploit the action of a hyperbolic group on its boundary.