Oberseminar Geometrie, Topologie und Analysis

H. Geiges, G. Marinescu, U. Semmelmann, G. Thorbergsson

Wintersemester 2006/07

Freitag, 10:30-11:30, Seminarraum 1

10.11.06 Prof. Juan Carlos Álvarez (Lille)
What's the area of the unit sphere in a three-dimensional normed space?

Zusammenfassung: In this talk I'll present Serge Ivanov's proof of the sharp isosystolic inequality for Finsler metrics on the projective plane and use it to give a sharp lower bound for the areas of the unit spheres in three-dimensional normed spaces. This is joint work with Serge Ivanov and Tony Thompson.
17.11.06 Dr. Alexander Strohmaier (Bonn)
High energy limits of Dirac eigenfunctions and frame flows

Zusammenfassung: A classical theorem by Shnirelman, Zelditch and Colin de Verdière states that ergodicity of the geodesic flow on a compact Riemannian manifold implies quantum ergodicity for eigenfunctions. Roughly this means that most eigenfunctions become equidistributed on the manifold as the eigenvalue increases. We show that a similar statement is true for the Dirac operator on a Riemannian spin manifold, the role of the geodesic flow being replaced by the frame flow. We also discuss ergodicity questions for other canonical operators on vector bundles. The results have nice interpretations in physics.
24.11.06 Dr. Urs Frauenfelder (LMU München)
Rabinowitz action functional and obstructions for exact contact embeddings

Zusammenfassung: This is joint work with Kai Cieliebak. We define Floer homology for a Lagrange multiplier action functional which also appears in the work of Rabinowitz. As an application of the computation of this new Floer homology we get restrictions for exact contact embeddings.
1.12.06 Prof. Matthias Lesch (Bonn)
Relative pairing in cyclic cohomology and divisor flows
Mathematisches Kolloquium
Prof. Paolo Lisca (Pisa)
A knot theoretical application of Donaldson's theorem on the intersection forms of smooth four-manifolds.

Zusammenfassung: One of the basic open problems in classical knot theory is to settle the so-called ribbon conjecture, which states that a smoothly slice knot is ribbon. A natural test case for the conjecture, already considered over thirty years ago by Casson and Gordon, is given by the class of two-bridge knots. After reviewing the relevant definitions and the previously known results, I will explain how to apply Donaldson's celebrated theorem on the intersection forms of smooth four-manifolds to prove that the ribbon conjecture holds for two-bridge knots. The level of the exposition will be quite elementary -- I expect non-topologists as well as graduate students to have no problem following the talk.
8.12.06 Prof. Claudio Gorodski (São Paulo)
Singular Riemannian foliations with sections and transnormal maps

Zusammenfassung: A ``singular Riemannian foliation with sections'' (SRFS) on a complete Riemannian manifold is a singular Riemannian foliation in the sense of Molino which admits a transversal complete immersed manifold that meets all the leaves and meets them always orthogonally. They were introduced by Boualem and then by Alexandrino as a simultaneous generalization of orbital foliations of polar actions of Lie groups, isoparametric foliations in simply-connected space forms, and foliations by parallel submanifolds of an equifocal submanifold with flat sections in a simply-connected compact symmetric space.

A smooth map from a complete Riemannian manifold to an Euclidean space is called ``transnormal'' if it is an integrable Riemannian submersion in a neighborhood of any regular level set. We prove that the leaves of a given SRFS coincide with the level sets of some transnormal map in the case in which the ambient manifold is simply-connected, the sections are flat, and the leaves are compact. This extends previous results due to Carter-West, Terng and Heintze-Liu-Olmos. (Joint work with M. Alexandrino.)
12.1.07 Prof. Krzysztof Galicki (New Mexico)
Obstructions in Kähler-Einstein and Sasakian-Einstein Geometry

Zusammenfassung: I will discuss some new existence and obstruction results concerning Sasaki-Einstein metrics on 5-manifolds. The new obstructions are also obstructions to the existence of Kähler-Einstein metrics of positive scalar curvature on certain compact log del Pezzo orbifolds.
19.1.07 Prof. Iskander Taimanov (Sobolev Institute of Mathematics)
Frobenius manifolds via the finite-gap integration

Zusammenfassung: We shall expose how to apply the finite-gap integration theory to constructing new examples of Frobenius manifolds.
26.1.07 Prof. Nessim Sibony (Paris-Sud)
Unique ergodicity for holomorphic foliations in CP2

Zusammenfassung: The question is to study global properties of solutions of differential equations in the complex domain. Let F be a holomorphic foliation in CP2 without algebraic leaves and such that all singular points are hyperbolic, which is the generic situation for CP2 foliations. We show that appropriate averages of every leaf converge to a unique harmonic current directed by the foliation, independently of the leaf. The main tool is a theory of intersection of ddc closed currents. This is joint work with J. E. Fornaess.
2.2.07 Prof. Markus Pflaum (Frankfurt)
Die algebraische Indexformel für Orbifolds

Zusammenfassung: In dem Vortrag wird zuerst erläutert, wie man mit Hilfe von Gruppoiden Orbifolds repräsentieren kann, und anschließend das Konzept der Deformationsquantisierung über Orbifolds erklärt. Anschließend wird gezeigt, wie man mit Hilfe nichtkommutativer Methoden und der Konstruktion einer (universellen) Spur auf der deformierten Konvolutionsalgebra des Orbifolds die algebraische Indexformel ableiten kann. (gemeinsame Arbeiten mit N. Neumaier, H. Posthuma und X. Tang)
9.2.07 Dr. Oksana Yakimova (z.Zt. Köln)
Weakly symmetric spaces: main properties and classification ideas

H. Geiges, 5.4.02