Research Seminar Global Analysis

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

Summer semester 2012

Talks

10.07.2012 Alexander Gorokhovsky (Boulder)
Title: Higher analytic indices and symbolic index pairing
Abstract: .Higher index theory was started in the work of A. Connes and H. Moscovici on the Novikov conjecture. The goal of my talk is to reinterpret their theorem, extend the definition of higher indices to new situations, and to describe a theorem computing them in topological terms. This is joint work with H. Moscovici.
26.06.2012 Henri Moscovici (Ohio State University)
Title: Modular curvature for noncommutative tori
Abstract: This talk will discuss the main results of my joint work with A. Connes [http://arxiv.org/abs/1110.3500] on the conformal geometry of noncommutative 2-tori in the framework of modular spectral triples. We obtained explicit expressions for the curvature functionals determined by the value at zero of the zeta functions of Laplacians, in terms of the generating function for the Bernoulli numbers applied to the modular operator. Computer aided calculations based on Connes' adaptation of the pseudodifferential calculus lead to explicit but intricate curvature formulas, which besides the above generating function involve functions of two variables in the modular operator. The dependence of the latter on the Bernoulli generating function is elucidated by computing in two different ways the gradient of the holomorphic torsion functional on the space of conformal factors. Moreover, we proved (the analogue of the classical result for Riemann surfaces) that the maximum value of the determinant of the Laplacian for metrics of fixed area is attained only at the constant curvature metric.
12.06.12 Double Feature: Gerardo Menoza and Frank Lapp
14:15-15:15 Gerardo Mendoza (Temple University)
Title: Complex b-manifolds and their boundary
Abstract: A complex b-structure on a manifold with boundary is an involutive subbundle b^T^{0,1}M of the complexification of b^TM with the property that \C b^TM = b^T^{0,1}M + \overline{b^T^{0,1}M} as a direct sum; the interior of M is a complex manifold. The complex b-structure induces a rich structure on the boundary of M similar to that of the circle bundle of a Hermitian holomorphic line bundle over a complex manifold, however generically the circle action is replaced by an \R action. I will briefly show how the structure is obtained and then discuss a classification theorem for such boundary structures similar to that of line bundles by their first Chern class, and embedding (i.e., ampleness) and vanishing theorems similar to Kodaira's well known theorems for positive line bundles.
15:30-16:30 Frank Lapp (Humboldt University Berlin)
Title:The Fredholm index of Dirac operators on manifolds with perturbed metric horns
Abstract: A manifold with a metric horn has a singularity of the form [(0,1)xN, dr2+r^{2b}g_N], b>1. Unlike on a closed manifold, the Dirac operator on such a manifold is not essentially self-adjoint, but it has self-adjoint extensions. Splitting the operator and choosing a closed extension yields a Fredholm operator whose index is made up of an integral over the difference of the heat-kernel traces on the compact part of the singular manifold and the eta invariant and the kernel of the Dirac operator over the section manifold N. The Fredholm properties also hold for certain perturbations of such operators without changing the index formula.
05.06.12 Martin Olbrich (Luxemburg)
Title: Selberg zeta functions, transfer operators, and a conjectuer of Patterson
Abstract: Selberg zeta functions count closed geodesics of negatively curved locally symmetric spaces in a similar way as the Riemann zeta function counts primes. They are initially defined on some half plane in C. There are essentially two ways to establish their meromorphic continuation to the whole complex plane and to understand their singularities (zeroes and poles): either via Selberg trace formulas or via transfer operators associated with a Markov partition (Ruelle, Fried). The first method (if it succeeds) gives a description of the singularities in terms of spectral and topological data of the locally symmetric space. Due to the choices involved, the second method does not give immediately a canonical description of the singularities. However, it inspired Patterson to conjecture that they can be uniformly described in terms of the cohomology of the fundamental group with coefficients in principal series representations. In the meanwhile, the conjecture has been verified in many cases by U. Bunke and myself, but still using the trace formula approach. In the talk, I will explain the conjecture and indicate a more direct way towards its proof via transfer operators.
22.05.12 Michael Gruber (Hannover)
Title:Absolutely continuous spectrum for periodic magnetic fields
Abstract: The spectral theory of Schrödinger operators with periodic magnetic fields (non-zero flux) still poses surprisingly many unsettled questions. We review the main methods and results pertaining to the measure theoretic nature of the spectrum (absolutely continuous vs. pure point) and present new results for AC spectrum.
15.05.12 Hans Boden (McMaster Univ., MPI)
Title: Spliced sums and the SU(3) Casson invariant
Abstract: Given two knots in the 3-sphere, one can construct the spliced sum (a generalization of connected sum), and the resulting 3-manifold can be easily seen to be a homology 3-sphere. For instance, the SU(2) Casson invariant is additive under both connected sum and spliced sum. On the other hand, the SU(3) Casson invariant is not additive under connected sum, but subtracting a suitable multiple of the square of the SU(2) Casson invariant gives something that is additive under connected sum but not under spliced sum. In this talk we present methods for computing the SU(3) Casson invariant for certain spliced sums. The examples include splicing along complements of (p,q) torus knots, and the techniques involve developing a good working description of the SU(3) flat and perturbed flat moduli spaces and providing methods to compute the relevant spectral flows. This is a report on joint work with Chris Herald and Ben Himpel.
17.04.12 Jean Raimbault (Jussieu)
Title:Analytic torsion for hyperbolic 3-manifolds defined by congruence subgroups of the Bianchi groups
Abstract: For finite-volume hyperbolic (three)-manifolds one can define "regularized analytic torsion" simply by using the constant term of the Maass-Selberg expansion for the trace of the heat operator on forms. We are interested in the behaviour of this torsion (with coefficients in a fixed local system) for sequences of manifolds as the systole tends to infinity. More precisely we would like to generalize a recent result of Bergeron and Venkatesh from the compact to the noncompact setting, and we get the following kind of result: Let V be the tautological representation of G=SL_2(C), then if \Gamma_n are an infinite sequence of principal congruence subgroups of a fixed Bianchi group, M_n=\Gamma_n\backslash H^3, the normalized analytic torsion log T(M_n;V)/vol(M_n) tends to 11\pi/12. I will show how to define the regularized analytic torsion and try to explain some of the difficulties arising when studying its asymptotics."
03.04.12 Jens Kaad (Paris VII)
Title: An index theorem for commuting Toeplitz operators
Abstract: For a large class of Hilbert spaces of holomorphic functions on a domain in n-dimensional complex space there is an associated index problem for tuples of Toeplitz operators with holomorphic symbols. However, outside the strictly pseudoconvex world, concrete computations of the index seems quite scarce. This leaves out classical function spaces such as Bergman and Hardy spaces over the polydisc. The purpose of this talk is to prove an index theorem for a general class of Hilbert spaces satisfying appropriate boundary and density conditions. In this context the index only depends on the behaviour of the symbols near their common roots and can be computed as a sum of intersection numbers (or equivalently of local degrees). The main tool applied in the proof is an algebraic reduction theorem due to Douglas, Paulsen, Sah and Yan. The talk is based on an ongoing project with Ryszard Nest.
28.03.12 S. Wakatsuki i (Kanazawa Univ., Japan)
Title: On coefficients of unipotent orbital integrals for the symplectic group of rank 2

Informations


• Tuesday, 14:15 in room 0.008, Endenicher Allee 60
• Talks last for about an hour plus discussion

Past semester program


Winter semester 2011/2012
Summer semester 2011
Winter semester 2010/2011
Summer semester 2010
Winter semester 2009/2010
Summer semester 2009
Winter semester 2008/2009
Summer semester 2008
Winter semester 2007/2008
Summer semester 2007
Winter semester 2006/2007
Summer semester 2006
Winter semester 2005/2006