Oberseminar Globale Analysis/ M. Lesch, W. Müller

Research Seminar Global Analysis

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

Wintersemester 2011-12

12.07.11

Pierre Albin (Jussieu, Paris)
Title: The signature operator on stratified pseudomanifolds

Abstract: The signature operator of a Riemannian metric is an important tool for studying topological questions with analytic machinery. Though well-understood for smooth metrics on compact manifolds, there are many open questions when the metric is allowed to have singularities. I will report on joint work with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza on the signature operator on stratified pseudomanifolds and some of its topological applications.

05.07.11	Jens Kaad (University of Bonn) Title: Spectral flow over complete manifolds and the unbounded index pairing.
	Abstract: In a paper from 1992 Robbin and Salomon investigate the spectral flow of a family of unbounded selfadjoint operators parametrized by the real line. Under appropriate conditions they are able to show that the spectral flow agrees with the Fredholm index of the unbounded Dirac-Schrödinger operator given by the sum of the family and differentiation with respect to time. During the attempt of generalizing Robbin and Salomon's result to the case of complete manifolds it soon becomes apparent that the problem has an underlying and far more general formulation in the framework of Kasparov's bivariant K-theory. The spectral flow over complete manifolds can be viewed as a special case of the index pairing (or the interior Kasparov product) and the search for appropriate unbounded Fredholm operators is really the task of defining and unbounded version of the index pairing. In the talk I'll explain the problem in a geometric context, present it's more general KK-theoretic formulation and outline how to solve it in full generality using KK-theoretic methods. The talk is based on joint work with Matthias Lesch.
28.06.11	Christoph Thiele (University of California, Los Angel\ es) Title: The triangle Hilbert transform and the twisted paraproduct
	Remark: This week we collaborate with the Sonderkolloquium. The talk will be held in the Lipschitz Saal at 2PM.
	Abstract: The triangle Hilbert transform is a remarkably elegant bilinear operator. It is so powerful that conjectured estimates for this operator imply many theorems proved in the area of time-frequency analysis over the past decades, most prominently Carleson's theorem on almost everywhere convergence of Fourier series. It also relates to a celebrated open problem in ergodic theory about convergence of ergodic averages for two commuting transformations. In the first part of the lecture we will survey these connections. The most recent progress towards understanding the triangle Hilbert transform came through surprisingly fundamental new insights and estimates for the twisted paraproduct. This was the last remaining barrier to be understood within a family of averages of the triangle Hilbert transform. In the second part of the lecture we will discuss the twisted paraproduct and outline a proof of some of the new estimates proven by my PhD student Vjekoslav Kovac.
21.06.11	Leonardo Cano Garcia Title: Mourre estimates on manifolds with corners of codimension 2.
	Abstract: Mourre theory was originally a tool in the spectral analysis of Schrödinger operators. It was mainly needed to prove: i) absence of singular spectrum, ii) that the pure point spectrum could accumulate only at tresholds and iii) asymptotic completeness. In the talk we provide a description of how to adapt this theory to Laplacians on manifolds with corners of codimension 2, we focus on applications i) and ii).
07.06.11	Anton Petrunin (Penn State University) Title: The ghost of Riemann in Alexandrov geometry.
	Abstract: TBA
31.05.11	Matthias Lesch (University of Bonn) Title: The local-global principle for regular operators on Hilbert C-modules.*
	Abstract: This is a report on aspects of an ongoing project with Jens Kaad. Hilbert C* modules are the analogues of Hilbert spaces where the scalar field is replaced by an arbitrary C* algebra. They were introduced by Kaplansky, Paschke and Rieffel. With the advent of Kasparov's celebrated KK-theory they became a standard tool in the theory of operator algebras. While the elementary properties of C*--modules can be derived basically in parallel to Hilbert space theory the lack of an analogue of the Projection Theorem soon leads to serious obstructions and difficulties. In particular the theory of unbounded operators, which is important since natural operators are often unbounded, is notoriously more complicated. E.g. to have a nice spectral theory an additional axiom of regularity has to be introduced. So far there has not been a good criterion to check whether an operator is regular. Jens and I have made an, in our view, significant progress by discovering a criterion for regularity in terms of the Hilbert space localizations of an unbounded operator. I am going to explain all this and I will state our main result which is of a Local-Global nature.
17.05.11	Jeremy Marzuola (University of North Carolina) Title: Nonlinear bound states on manifolds.
	Abstract: We will discuss the results of several joint projects (with subsets of collaborators Pierre Albin, Hans Christianson, Jason Metcalfe, Michael Taylor and Laurent Thomann), which explore the existence stability and dynamics of nonlinear bound states and quasimodes on manifolds of both positive and negative curvature with various symmetry properties.
03.05.11	Urs Lang (ETH Zürich) Title: Currents in metric spaces and applications to nonpositive curvature.
	Abstract: After a brief review of the theory of currents in metric spaces I will discuss some joint work with Bruce Kleiner on Hadamard manifolds of rank n > 1 in an asymptotic sense. We show that (quasi-)minimizing locally integral currents of dimension n and polynomial volume growth of order n behave in many respects like (quasi-)geodesics in hyperbolic spaces. Among other things, we solve an asymptotic Plateau problem and prove some stability/persistence results. This also leads to another proof of the Kleiner-Leeb rigidity theorem for symmetric spaces. The extension of the theory of currents to metric spaces is needed in that the arguments make use of singular constructions such as the Tits cone.
26.04.11	Sebastian Herr (Universität Bonn) Title: Sharp well-posedness results for energy-critical nonlinear Schroedinger equations.
	Abstract: Since the 1980's the Cauchy problem for critical nonlinear Schroedinger equations with small initial data in the energy space H^1(R^n) is well understood. However, in a geometric setting with trapped geodesics even the fundamental questions of global existence, uniqueness and continuous dependence for small initial data in H^1 remained open. In this talk, recent results on global well-posedness for small initial data in the energy space in specific geometries such as T^3 and S^3 and certain product spaces will be presented. The methods of proof involve critical function space theory and multilinear estimates of Strichartz type.
12.04.11	Semyon Malamud (Ecole Polytechnique Federale de Lausanne) Title: Boltzmann-type evolution equations arising in economics
	Abstract: We will introduce a new class of Boltzmann- type evolution equations that recently arose in economics in the description of propagation of information in large populations. We will discuss asymptotic behaviour of their solutions, its economic consequences, and indicate several open problems and interesting directions for future research.
05.04.11	Evgenij Troitsky (Moscow Lomonosov State University/MPI) Title: Finite holonomy and an index theorem for gauge-invariant operators (joint with Victor Nistor)
	Abstract: We consider the gauge-equivariant K-theory groups of a bundle X over B endowed with a continuous action of a bundle of compact Lie groups G over B. These groups are the natural range for the analytic index of a family of gauge-invariant elliptic operators (a family of elliptic operators invariant with respect to the action of a bundle of compact groups) and are a version of twisted K-theory. We establish the Thom isomorphism and prove an index theorem in this situation. We plan to explain some natural finiteness conditions in our situation (a sort of typical restrictions in twisted K-theory).