Oberseminar Globale Analysis/ M. Lesch, W. Müller

Research Seminar Global Analysis

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

Wintersemester 2011/2012

24.01.12	Alexander Strohmaier (Univ. of Loughborough) Title:Eigenvalues and the Spectral Zeta function on Riemann Surfaces: Analytic and Computational Aspects
	Abstract:
17.01.12	Boris Vertman (Bonn) Title:The Yamabe flow on manifolds with edges
	Abstract: We present the setup and geometric conditions for short time existence of Yamabe flow on compact manifolds with incomplete edge singularities. We then continue with a discussion of long time existence of the Yamabe flow for negative initial scalar curvature.
10.01.12	Gilles Carron (Univ. de Nantes) Title:Sobolev and Yamabe constants
	Abstract: The Yamabe constant plays an important role in the prescription of the scalar curvature. And when this invariant is positive, it gives a Sobolev inequality. I will talk about several rigidity results that are consequences of this interpretation. One is a generalization of a result of Gursky about the second Betti number of 4-Manifolds with positive Yamabe invariants. This is a joint work with V. Bour (Grenoble).
13.12.12	Julie Rowlett (MPI Bonn) Title: The existence of bound states in quantum layers
	Abstract: (joint work with Z. Lu) A quantum layer is a layer of uniform width built over a complete surface embedded in R^3. The width is of mesoscopic scale: the length scale at which quantum effects are observed. This mathematical model describes the physical situation of a quantum particle trapped between hard walls. The existence of bound states in the physical model is mathematically equivalent to the existence of the ground state, the smallest eigenvalue in the discrete spectrum of the naturally associated Laplace operator. We shall discuss geometric conditions on the surface which imply the existence of the ground state. In the proofs of these results, we will see how the geometry of the surface influences the analysis of the Laplace operator.
06.12.11	Alan Carey (ANU Canberra und Münster) Title: The local index formula in NCG for noncompact noncommutative geometries
	Abstract: The local index formula in NCG gives an expression for the index pairing associated with a spectral triple. The unital case is due to Connes-Moscovici and is now a widely used tool in examples. I will explain a way to generalise the theorem to the nonunital case and also to problems that use the Breuer-Fredholm index.
29.11.11	Tilmann Wurzbacher (Metz/Bochum) Title: The geodesic flow on a Riemannian supermanifold
	Abstract: We consider a notion of geodesics on a Riemannian supermanifold that naturally extends the classical, ungraded case. We show the existence of a geodesic flow on the tangent bundle of the supermanifold. Integral curves of this flow are shown to be in bijection with geodesics. We also construct an expoential map and apply it to show that isometries faithfully linearize. We also briefly comment on the case of "odd metrics".
22.11.11	George Marinescu Title:Equidistribution of zeros of holomorphic sections of high tensor powers of line bundles
	Abstract: We present some equidistribution results for sequences of random sections of high tensor powers of positive line bundles over non-compact manifolds (e.g. Riemann surfaces with cusps, arithmetic quotients or, more generally, quasiprojective manifolds). We also examine the equidistribution of sections of big line bundles endowed with singular Hermitian metrics.
15.11.11	Clara Aldana Title: Heat kernels and their asymptotic expansions on surfaces with asymptotic cusps
	Abstract: In this talk I will briefly introduce the heat kernels associated to the Laplace operator on a Riemannian manifold. Then I restrict to the case of surfaces with asymptotically cusp ends. I consider two Laplacians on these surfaces and the relative heat operator associated to them. I will explain how to build an asymptotic expansion of the trace of the relative heat operator for small values of t. In the talk I will present the technical details of the construction. If I have time, I will mention other ways to construct these expansions.
25.10.11	Jonathan Pfaff (Bonn) Title: Analytic torsion of complete hyperbolic manifolds
	Abstract: This talk is based on joint work with Werner Müller. We define the analytic torsion for a complete oriented hyperbolic manifold of finite volume. It depends on a representation of the fundamental group. For manifolds of odd dimension, we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group.
18.10.11	Matthias Lesch (Bonn) Title: Semiclassical heat expansions, multiparameter resolvent expansions and regularized determinants
	Abstract: In this talk I will first recall the parameter dependent pseudodifferential calculus which is a fundamental tool in microlocal analysis. I will then discuss a new multiparameter resolvent expansion which follows easily from the parameter dependent calculus. As a first application a short proof of the semiclassical heat expansion for arbitrary elliptic differential operators will be given. Finally I will report on an ongoing joint project with Boris Vertman. First we prove a Fubini type Theorem for regularized integrals. As a new feature a correction term shows up when exchanging the order of integration. Combining this result with (an extension of) the above mentioned multiparameter expansion and the Euler McLaurin formula we are able to give formulas for the zeta-regularized determinants of Schrödinger operators on surfaces of revolution. So far, such formulas have been around only in the 1D case.