Oberseminar Geometrie, Topologie und Analysis

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

Sommersemester 2014

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

16.5.14 Karsten Grove (Notre Dame)
Reflection groups in nonnegative curvature

Abstract: We provide an equivariant description/classification of all complete (compact or not) non-negatively curved manifolds M together with a cocompact action by a reflection group W, and moreover, classify such W. In particular, we show that the building blocks consist of the classical constant curvature models and generalized open books with non-negatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover.
This is joint work with F. Fang.
23.5.14 Bernhard Hanke (Augsburg)
The space of metrics of positive scalar curvature

6.6.14 Dima Roytenberg (MPI Bonn)
What is a derived manifold?

Abstract: This talk will be a gentle introduction to derived differential geometry intended for mathematicians who may have heard of the notion and are curious to find out more. Generally speaking, derived geometry is an approach to singular spaces that replaces them with manifestly smooth objects using the tools of homotopy and higher category theory. After making a few motivating remarks, we will introduce some basic notions of the subject, such as differential graded manifolds, derived intersections, Koszul complexes and the virtual dimension. This will be followed by some examples and applications.
20.6.14 Steve Zelditch (Northwestern)
Geometric approximation theory in Kähler geometry

Abstract: Classical approximation theory is about approximating continuous functions by polynomials of large degree N in various norms. In recent years, the Yau-Tian-Donaldson program of relating existence of Kähler metrics with constant scalar curvature to GIT stability of holomorphic line bundles has given rise to a new field of geometry in which one approximates Kähler metrics by polynomial-type metrics called Bergman (or Fubini-Study) metrics of degree N. The infinite dimensional space K of Kähler metrics is formally a symmetric space, and the Bergman metric spaces BN are finite dimensional ones. As N goes to infinity, BN tends to K not just pointwise but in many local geometric ways. I will survey the different types of problems and results in this geometric approximation theory, concentrating on convergence of geodesics, which is a kind of optimal transport of metrics.
27.6.14 Anton Petrunin (Penn State)
Smoothing and faceting

Abstract: I will discuss bilateral approximations and related curvature conditions between Riemannian and polyhedral spaces
4.7.14 Vitali Kapovitch (Toronto)
Bounding lengths of closed geodesics on 3-manifolds with lower curvature bounds

Abstract: We prove that on a closed 3-manifold with sectional curvature bounded below and diameter bounded above the length of the shortest closed geodesic is uniformly bounded. This is joint work with R. Rotman.
11.7.14 Nena Röttgen (Freiburg)
Aus der Existenz einer vollständigen Geodätischen folgt nicht
die Existenz einer geschlossenen Geodätischen

18.7.14 Laurent Charles (Paris 6)
On the quantisation of compact symplectic manifolds

Abstract: For compact Kähler manifolds, there exist a well established quantisation scheme. The quantum space consists of the holomorphic sections of a prequantum bundle, and to any classical observable is associated a Berezin-Toeplitz operator. From the physical point of view, we expect that the complex structure plays an auxiliary role in this story. So the quantisation should extend to symplectic manifolds. Furthermore it should not depend on the complex structure. I will present some results supporting these ideas.

H. Geiges, 5.4.02