Research Seminar Global Analysis

Prof. Dr. M. Lesch, Prof. Dr. W. Müller

Summer semester 2014


20.05.2014 Boris Vertman (Bonn)
Title: Combinatorial QFT and gluing of determinants
Abstract: We define the combinatorial Dirichlet-to-Neumann operator and establish a gluing formula for determinants of discrete Laplacians using a combinatorial Gaussian quantum field theory. In case of a diagonal inner product on cochains we provide an explicit local expression for the discrete Dirichlet-to-Neumann operator We relate the gluing formula to the corresponding Mayer-Vietoris formula by Burghelea, Friedlander and Kappeler for zeta-determinants of analytic Laplacians, using the approximation theory of Dodziuk. Our argument motivates existence of gluing formulas as a consequence of a gluing principle on the discrete level.
03.06.2014 Ursula Ludwig (Bonn)
Title: The Witten deformation for singular spaces and radial Morse functions
Abstract: The Witten deformation is an analytical method proposed by Witten in the 80's which, given a Morse function on a smooth compact Riemannian manifold M, leads to a proof of the famous Morse inequalities. The aim of this talk is to present a generalisation of the Witten deformation to a singular space X with cone-like singularities and radial Morse functions. As a result one gets Morse inequalities for the L^2-cohomology, or dually for the intersection homology of the singular space X. Moreover, as in the smooth theory, one can relate the Witten complex, i.e. the complex generated by the eigenforms to small eigenvalues of the Witten Laplacian, to an appropriate geometric complex (a singular analogue of the smooth Morse-Thom-Smale complex). Radial Morse functions are inspired from the notion of a radial vectorfield on a singular space. Radial vectorfields have first been used by Marie-Helene Schwartz to define characteristic classes on singular varieties. They also appear in the work of David Trotman, who has studied the Poincare-Hopf theorem for radial vectorfields.
24.06.2014 Erez Lapid (Weizman Institute)
Title: Analytic and group-theoretic aspects of limit multiplicities
Abstract: The study of limit multiplicities started in the 1970's with a seminal paper of DeGeorge-Wallach. Since then many people extended their result, mostly for uniform lattices. I will describe recent work with Tobias Finis which establishes limit multiplicities for arbitrary sequences of congruence subgroups of SL_n(Z).
01.07.2014 Erez Lapid (Weizman Institute)
Title: An analogue of the Ichino-Ikeda conjecture for Whittaker coefficients
Abstract: Period integrals of automorphic forms occur in various arithmetic, topological and analytic contexts. A conjecture of Ichino-Ikeda, based on earlier results of Waldspurger, suggests a local-to-global principle for certain periods. I will describe recent work with Zhengyu Mao on an analogue of the Ichino-Ikeda conjecture for Whittaker coefficients of representations of classical groups.
08.07.2014 Ksenia Fedosova (Bonn)
Title: L^2 torsion for compact hyperbolic orbifolds
Abstract: We prove Selberg trace formula for compact hyperbolic orbifolds and calculate the orbital integrals, corresponding to elliptic elements. Then we discuss the growth of the L^2-torsion on a compact hyperbolic 3-manifold associated with the m-th symmetric power of the standart representation in SL(2,C) and its possible extension to the case of orbifolds.
15.07.2014 Xenia Spilioti (Bonn)
Title: Selberg and Ruelle zeta functions of compact hyperbolic manifolds
Abstract: In this talk, we will present some recent work on the zeta functions of Ruelle and Selberg on compact hyperbolic manifolds X of odd dimension d. These are dynamical zeta functions associated with the geodesic flow on the unite sphere bundle S(X). The zeta functions depend on a finite dimensional representation of the fundamental group of X. We prove that these zeta functions converge absolutely and uniformly on compact subsets of some half-plane Re(s)>c. Furthermore, they admit a meromorphic continuation to the whole complex plane. We describe the singularities of the Selberg zeta function in terms of the discrete spectrum of certain differential operators on X and we establish functional equations, relating their values at s with those at (-s). The main tool that we use is the Selberg trace formula. We generalize results of Bunke and Olbrich to the case of non-unitary representations of the fundamental group.