Oberseminar Geometrie, Topologie und Analysis

H. Geiges, A. Lytchak, G. Marinescu, S. Sabatini, G. Thorbergsson

Wintersemester 2014/15

Freitag, 10:30-11:30, Seminarraum 2 (Raum 204)

21.11.14 Mehdi Lejmi (Université Libre de Bruxelles)
The J-flow and stability

Abstract: The J-flow is a parabolic flow introduced by Donaldson. In this talk, we present a new algebro-geometric stability condition which is equivalent conjecturally to the existence of solutions of the critical equation of the J-flow. We present also examples due to Fang-Lai and explain how this is related to the stability condition. This is a joint work with Gabor Székelyhidi.
28.11.14 Christian Miebach (Calais)
On homogeneous Hamiltonian and Kähler manifolds

Abstract: Elementary examples show that holomorphic actions of complex reductive groups G on Kähler manifolds can be rather pathological. For instance, the topological closures of orbits need not be complex analytic. In my talk I will explain that the situation is much better if one assumes that there exists an equivariant moment map for a maximal compact subgroup of G: In this case the holomorphic action of G shares many properties of an algebraic one. This is joint work with Bruce Gilligan and Karl Oeljeklaus.
12.12.14 Marco Radeschi (Karlsruhe)
Metrics on spheres all of whose geodesics are closed

Abstract: Riemannian manifolds in which every geodesic is closed have been studied since the beginning of the last century, when Zoll showed the existence of a non-round metric on the 2-sphere all of whose geodesics are closed. Among the many open problems on the subject, a conjecture of Berger states that for any simply connected manifold all of whose geodesics are closed, the geodesics must have the same length. The result was proved in the case of the 2-sphere by Grove and Gromoll. In this talk, I will show recent work with B. Wilking, where we prove that the Berger conjecture also holds for spheres of dimension greater than three. If time permits, I will discuss current work on how to extend the result to the 3-sphere and to projective spaces.
16.1.15 Ursula Ludwig (Bonn)
The Witten deformation for singular spaces and radial Morse functions

23.1.15 Hartmut Weiß (Kiel)
The Higgs bundle moduli space - geometric and analytic aspects

Abstract: I will report on joint work with Rafe Mazzeo, Jan Swoboda and Frederik Witt in which we describe solutions of Hitchin's equation which lie close to the ideal boundary of the Higgs bundle moduli space. These solutions are constructed via gluing methods. This is the first step of a larger project aimed at studying the asymptotics of the complete hyperkähler metric on this moduli space.
30.1.15 Boris Okun (University of Wisconsin at Milwaukee/MPI Bonn)
Action dimension of right-angled Artin groups

Abstract: The action dimension of a group G, actdim(G) is the least dimension of a contractible manifold which admits a proper G-action. I will explain why actdim is interesting from the point of view of L2-cohomology, and give a computation of actdim for the right-angled Artin groups. This is based on a joint work with Grigori Avramidi, Mike Davis and Kevin Schreve.
6.2.15 Pablo Ramacher (Marburg)
Quantum ergodicity and symmetry reduction

Abstract: We study the ergodic properties of eigenfunctions of Schrödinger operators on a closed connected Riemannian manifold M in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, let M carry an isometric effective action of a compact connected Lie group G. We prove an equivariant quantum ergodicity theorem assuming that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of M is ergodic. We deduce the theorem by proving an equivariant version of the semiclassical Weyl law, relying on recent results on singular equivariant asymptotics. It implies an equivariant version of the Shnirelman-Zelditch-Colin-de-Verdière theorem, as well as a representation-theoretic equidistribution theorem. In case that G is trivial, one recovers the classical results.

H. Geiges, 5.4.02