Research Seminar Global Analysis

Prof. Dr. W. Müller

Sommersemester 2008/2009

21.04.09 Shantanu Dave (Vienna)
Equivariant non-commutative residue

Abstract: We shall consider a compact manifold with action of a finite or compact group. We shall consider the cross-product under this group action of the algebra of complete symbols of pseudodifferential operators and obtain traces on these algebras in terms of residues of a certain zeta functions. This provides us with asymptotics of representations occurring as subrepresentations of eigenspaces of invariant operators. We shall see a few examples as applications.
28.04.09 Wolfgang Lück (Münster)
L²-Torsion, projective class groups of complex group rings and applications to group automorphisms.

Abstract: We briefly recall basic facts about analytic and topological L²-torsion. For many applications of the L²-torsion the following conjecture is relevant:
Conjecture: The L²-torsion the universal covering of a finite CW-complex which is L²-acyclic and of determinant class, depends only on the homotopy type.
We explain that this conjecture follows from Moodys induction conjecture about the projective class groups of compley group rings which has been proved for a large class of groups including hyperbolic groups and elementary amenalle groups by Bartels and the speaker.
Finally we introduce a kind of L²-torsion for group automorphisms and explain its main properties. In the case of a pseudo-Ansosov self homeomorphism of a closed surface this invariant is up to a conctant the volume of the closed hyperbolic 3-manifold given by the mapping torus.
05.05.09 Nikolai Saveliev (University of Miami)
Seiberg-Witten invariants and end-periodic Dirac operators

Abstract: Let X be a smooth spin 4-manifold with homology of S1 x S3. In our joint project with Tom Mrowka and Daniel Ruberman, we study the Seiberg-Witten invariant of X. It depends on choices of Riemannian metric and perturbation. We remove this dependency by adding a correction term which is essentially the index of the end-periodic Dirac operator on a non-compact manifold with periodic end modeled on the infinite cyclic cover of X. I will discuss Fredholmness of such operators and a spectral flow formula for the change in their index.
12.05.09 no talk
19.05.09 Victor Nistor (Penn State University)
Analysis on Lie manifolds and Schrödinger operators

Abstract: What is common to locally symmetric spaces, real algebraic varieties, polyhedral domains, or Schrödinger operators? An answer is that (often) the analysis of these objects can be modelled by manifolds with nice compactifications that are manifolds with corners. In my talk, I will describe these manifolds (which I call 'Lie manifolds'). A typical results reduces the analysis of a geometric operator on a Lie manifold M to the analysis of a family of other operators P_j on manifolds of the form M_j = Z_j \times G_j, of the same dimension, where G_j is a Lie group. The operators P_j are invariant with respect to the natural action of G_j on M_j by right multiplication. The proof (but not the statement) of results of this kind requires a `Lie's third theorem' for Lie algebroids, which is not always true, but it is in the cases in which we are interested. At the end, I will discuss some applications of these results to the Hartree method for studying Schrödinger opeators. I will mention some applications to Classical Mechanics. The results of this talk are based on joint works with Bernd Ammann, Robert Lauter, Bertrand Monthubert, Anna Mazzucato, Eugenie Hunsicker, and Jorge Sofo.
26.05.09 Besma Ben-Ali (Orsay)
Maximal inequalities and Riesz transform estimates on L^p spaces for Schrödinger operators with magnetic field
02.06.09 no talk (Pfingsten)
09.06.09 Roberto Miatello (Cordoba, Argentina)
Distribution results for automorphic forms on Hilbert modular groups
16.06.09 Nader Yeganefar (Marseille)
L² harmonic forms on conformally compact manifolds

Abstract: Conformally compact manifolds may be seen as generalizations of the ball model of hyperbolic space. Rafe Mazzeo computed the space of L² harmonic forms on such manifolds. We will give a different (and hopefully simpler) proof of Mazzeo's result.
23.06.09 Matthias Lesch (Bonn)
Relative Connes-Chern character for manifolds with boundary

Abstract: The JLO version of the Connes-Chern character is a non-commutative analogue of the classical Chern character. On the one hand it provides a natural transformation from K-theory to cyclic (co)homology. On the other hand it comes naturally with a time parameter which generalizes the heat trace of an elliptic operator. Therefore there is a strong connection to local index theory and short and large time limits of the JLO Connes-Chern character provide interesting information. In a current ongoing project with Henri Moscovici and Markus Pflaum we are investigating these issues in a relative context. More precisely we study the Connes-Chern character for manifolds with boundary using the so called b-calculus of Melrose. Our main results identify the short and large time limits of the Connes-Chern character in this context.
30.06.09 no talk (Noncommutative Geometric Methods in Global Analysis)
07.07.09 Rafe Mazzeo (Stanford)
Toward an iterated edge calculus, with applications in geometry and topology

Abstract: Despite some existing purely microlocal approaches for the study of `iterated edge' operators, there is a need to develop methods which are adaptable to specific linear and nonlinear geometric problems. I will describe two results, one more general and directed toward a higher signature theorem on general stratified Witt spaces, and the other, which provides more detailed results for `low depth' singularities and an application to the Stoker conjecture for convex polyhedra.
14.07.09 Nicole Raulf (Lille)
Distribution of eigenvalues of Hecke operators


• Tuesday, 14:15 in room 008, Endenicher Allee 60
• Talks last for about an hour plus discussion

Past semester programm

Sommersemester 2008
Wintersemester 2007/2008
Sommersemester 2007
Wintersemester 2006/2007
Sommersemester 2006
Wintersemester 2005/2006