Oberseminar Globale Analysis/ M. Lesch, W. Müller

Research Seminar Global Analysis

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

Wintersemester 2012-2013

23.10.2012	Zhizhang Xie (IHES Paris) Title: A relative higher index theorem, diffeomorphisms and positive scalar curvature
	Abstract: I will talk about my recent joint work with Guoliang Yu on a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold, we define a secondary higher index class. Non-vanishing of this higher index class is an obstruction for the two metrics to be in the same connected component of the space of metrics of positive scalar curvature. In the special case where one metric is induced from the other by a diffeomorphism of the manifold, we obtain a formula for computing this higher index class. In particular, it follows that the higher index class lies in the image of the Baum-Connes assembly map.
06.11.2012	Martin Olbrich (Luxembourg) Title: Selberg zeta functions, transfer operators, and a conjecture 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.
27.11.2012	Clara Aldana (Potsdam) Title: Compactness of relatively isospectral sets of open surfaces.
	Abstract: We consider surfaces that have boundaries and ends that are asymptotic to cusps or asymptotic to funnels. We define the concept of being relatively isospectral. I will explain how we prove compactness of relatively isospectral sets using conformal surgeries. The results to be presented in the talk are joint work with Pierre Albin and Frederic Rochon.
04.12.2012	Michael Hoffmann (Bonn) Title: L^2-index theory of twisted Dirac operators and the Chern conjecture
	Abstract: We present a generalization of Atiyah's L^2-index theorem to twisted Dirac operators over good orbifolds. Chern asked if the Euler characteristic of closed Riemannian manifolds of dimension 2n with negative sectional curvature is negative for n odd and positive for n even. Gromov used a variant of Atiyah's L^2-index theorem that is covered by our result to prove the Chern conjecture for Kähler manifolds. We generalize Gromov's result to Kähler orbifolds and give partial results for quaternionic Kähler manifolds.
11.12.2012	Janna Lierl Bonn (IAM) Title: Estimates for the Dirichlet heat kernel in inner uniform domains
	Abstract: I will present sharp two-sided estimates for the heat kernel (with Dirichlet boundary condition) on domains that satisfy an inner uniformity condition. The results are applicable to any convex domain, to the complement of any convex domain, and to the interior of the Koch snowflake. The heat kernel estimates imply the intrinsic ultracontractivity of the associated semigroup. I also plan to explain the so-called boundary Harnack principle. This result is crucial in the proof of our heat kernel estimates, but it is also of interest independently of these. In this talk I will focus, for simplicity, on domains in Euclidean space. However, we have proved the heat kernel estimates in a very abstract setting of a metric measure space equipped with a bilinear form that resembles a Dirichlet form. In this setting, it is assumed that the form is associated with a heat kernel that satisfies two-sided Gaussian bounds globally on the underlying space. For instance, we can estimate the heat kernel (more precisely, diffusion kernel) associated with a uniformly elliptic divergence form operator with symmetric second order part and bounded measurable coefficients on inner uniform domains in Euclidean space.