Oberseminar Globale Analysis/ M. Lesch, W. Müller

Research Seminar Global Analysis

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

Winter semester 2010 / 2011

01.02.11	Anna Maria Paolucci (Free University of Bozen/MPI) Title: A closer look at connections between wavelets, fractals, extended zeta functions and operator algebras.
	Abstract: I plan to give an overview of connections between wavelets and representations of $C^*$-algebras namely the Cuntz algebras and the Cuntz-Krieger algebras. Measures arising from certain families of Cuntz algebras turn out to be related to some extended zeta functions.
25.01.11	Joachim Hilgert (Paderborn) Title: Microlocal analysis on locally symmetric spaces
	Abstract: The geometric pseudo-differential calculus on Riemannian manifolds introduced by Sharafutdinov has the advantage that it has good invariance properties under isometries. Therefore it is well-suited to describe microlocal lifts (also called Wigner distributions) on locally symmetric spaces. In the context of compact hyperbolic surfaces Anantharaman and Zelditch introduced a related family of distributions, which they call Patterson-Sullivan distributions, using a pseudo-differential calculus based on the non-euclidian Fourier transform. We will show how to generalize this approach to rank one locally symmetric spaces, discuss the problems occurring in higher rank, and show how the geometric calculus helps in that case.
11.01.11	Rafe Mazzeo (Stanford/Nantes) The Yamabe problem on singular spaces
	Abstract: I will report on joint work with Kazuo Akutagawa and Gilles Carron concerning the Yamabe problem on compact singular spaces X. There is a general existence result under very mild hypotheses on X revolving around existence of a Sobolev inequality, and a more detailed result including sharp regularity when X is a stratified pseudomanifold. This generalizes the work of Akutagawa and Botvinnik in the conic case.
14.12.10	Wolfgang Lück (Münster/Bonn) Title: Approximating L^2-torsion and homological growth
	Abstract: We discuss the conjecture that the L^2-torsion of the universal covering of an aspherical closed manifold of dimension 2n+1 can be approximated by the orders of the torsion of the n-th homology of a tower of finite coverings converging to the universal covering. We present the very recent result that this conjecture holds in the case that the fundamental group contains an infinite elementary amenable normal subgroup.
07.12.10	Julie Rowlett (Bonn) Title: The dynamics of complete manifolds with variable negative curvature and interactions with their spectral theory
	Abstract: The main focus of this talk is joint work initiated here in Bonn(!) with P. Suarez-Serrato and S. Tapie concerning the dynamics of convex cocompact manifolds. This is an extremely rich class of manifolds containing many interesting examples: convex cocompact hyperbolic manifolds, asymptotically hyperbolic manifolds (with variable negative curvature), and the conformally compact manifolds (with variable negative curvature) introduced by Fefferman and Graham in the 1980s to study conformal invariants and more recently studied in mathematical physics for their role in AdS-CFT correspondence in string theory. After reviewing definitions and examples, I will discuss our main results which include: meromorphically extending generalized dynamical zeta functions, prime orbit theorems, and analytic variation of the topological entropy of the geodesic flow. This will be followed by a discussion of main ideas from the proofs. Finally, I will present some of my own work in the case of strongly- asymptotically-hyperbolic manifolds, interactions between the dynamics and spectral theory of these spaces, and the type of interactions we hope to obtain between the spectral theory and dynamics on convex cocompact manifolds with variable curvature.
30.11.10	Christian Blohmann (MPI) Title: Generalized symplectic reduction of Lagrangian field theories with external symmetries
	Abstract: If a Lagrangian field theory admits local symmetries, its initial value problem is not well-posed. Making the initial value problem well-posed generally involves two steps: imposing constraints and gauge fixing. It is well known that in the case of gauge theories this two step procedure can be understood as symplectic reduction. I will explain how to generalize this approach to field theories with external symmetries such as general relativity, where the symmetry group acts also on the base of the configuration bundle.
23.11.10	Matthias Lesch (Bonn) Title: Moving boundary conditions for the de Rham complex and the analytic torsion
	Abstract: Extending the celebrated Cheeger-Mueller Theorem on the equality of analytic and combinatorial torsion to singular spaces is one of the current challenges in the analysis on such spaces. In my talk I will show that there exists a cut-and paste framework, based on Vishik's moving boundary conditions, which allows to reduce the problem to a problem on the model of the singularity. The talk is based on unpublished work which dates back about a decade or so.
16.11.10	Mauricio Garay (MPI) Title: Classical and quantum integrability (joint work with Duco van Straten)
	Abstract: We consider the space R^{2n} equipped with its standard symplectic structure and quantised with the Moyal star product. Given a system of k-functions f_1,\dots,f_k on R^{2n} which Poisson commute, we ask for the existence of commuting operators whose principal symbols are the given functions. (This is of course a standard question going back to the early days of quantum mechanics). In case k=n and the functions define a lagrangian fibration. Twenty years ago, in an unpublished work, Duco van Straten constructed topological obstructions for performing the quantisation. Unexpectedly, it turns out that these obstructions vanish under mild conditions and that this leads to a quantisation theorem for integrable systems. There are nonetheless several open problems which I will discuss during the talk.
09.11.10	Christian Bär (Potsdam/MPI) Title: Parametrized integrals and the heat equation
	Abstract: TBA
02.11.10	Pablo Ramacher (Marburg) Title: Invariante Integraloperatoren auf der Oshima-Kompaktifizierung eines Riemannschen symmetrischen Raumes und regularisierte Spuren
	Abstract: We study invariant integral operators on the Oshima compactification of a Riemannian symmetric space, and characterize them within the framework of pseudodifferential operators. In particular, we describe the singular nature of their kernels, which enables us to define a regularized trace for them.
26.10.10	Frank Pfäffle Title: Gravity, Torsion and the Spectral Action Principle
	Abstract: In this talk I will consider Riemannian manifolds equipped with orthogonal connections with torsion. First, I will review the classical Einstein-Cartan theory. Then I will comment on Connes' Spectral Action Principle which is supposed to detect the Levi-Civita connection. The Chamseddine-Connes spectral action plays a prominent role in the noncommutative approach to Particle Physics. Finally, a formula for the torsion case will be given.
19.10.10	Boris Vertman (Bonn) Title: Analytic Torsion of Manifolds with Conical and Edge Singularities
	Abstract: We present the latest results on the analytic torsion of spaces with conical and edge singularities. The heat trace asymptotics guarantees existence of the analytic torsion in this singular setup. Rather unexpectedly, analytic torsion turns out to be independent of a particular choice of an edge Riemannian metric under additional dimensional assumptions. These assumptions exclude the case of (isolated) conical singularities, where we discuss the available formulas and their geometric interpretation. The talk is based on projects joint with Rafe Mazzeo and Werner Mueller.
12.10.10	Jens Kaad (Bonn) Joint torsion transition numbers of several commuting operators.
	Abstract: We introduce joint torsion transition numbers of several commuting operators satisfying a Fredholm condition. These new secondary invariants generalize the Carey-Pincus joint torsion of a pair of commuting Fredholm operators. The joint torsion transition numbers are non-zero elements of a field and they measure how certain Koszul complexes are contained in a bigger Koszul complex. Furthermore, these numbers are linked together by a cocycle condition and satisfy algebraic properties such as symmetry and multiplicativity. As an example, under more restrictive invertibility assumptions, we recover the Lefschetz numbers. We will also provide a link between the joint torsion transition numbers and the Cauchy integral formula for holomorphic functions on the poly-disc. Finally, we will discuss the case of Toeplitz operators over the circle. In this case, thanks to a result of Richard Carey and Joel Pincus, there is an interpretation of the joint torsion in terms of the Deligne tame symbol. It is our long term hope to find a generalization of this interpretation to the poly-disc setup.