Oberseminar Geometrie, Topologie und Analysis

S. Friedl, H. Geiges, G. Marinescu, G. Thorbergsson

Wintersemester 2010/11

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

22.10.10 Prof. Stefan Friedl
The Taubes conjecture

19.11.10 Dr. Joel Fine (Université Libre de Bruxelles)
Quantisation and canonical metrics in Kähler geometry

Abstract: I will explain the relation between the following two questions in complex geometry:
Q1. Given a Kähler manifold and fixed degree-two cohomology class, is there a "best" Kähler metric representing that class?
Q2. Given a complex submanifold X of CPn, is there a "best" representative of the projective equivalence class of X?
In Q1, "best" means Kähler metric of constant scalar curvature; in Q2 "best" means a submanifold with minimal centre of mass. I will describe a picture due to Donaldson of how Q2 can be seen as the "quantisation" of Q1. I will then outline some results which have been inspired by this picture.
26.11.10 Dr. Leonardo Biliotti (Parma)
Satake-Furstenberg compactifications, the moment map and the first eigenvalue

Abstract: In this talk, following the main ideas of Bourguignon, Li and Yau, we get sharp upper bounds for the first eigenvalue of the Laplacian on functions for an arbitrary Kähler metric on a Hermitian symmetric space. This is joint work with Dott. Alessandro Ghigi.
3.12.10 Prof. Juan Carlos Álvarez Paiva (Lille)
Can Humpty Dumpty increase his waistline without distending his skin?

Abstract: In this talk we shall explore the relation between the volume of a compact Riemannian n-dimensional manifold (M,g) and the length of its shortest closed geodesic. For this purpose we define the systolic volume S(M,g) as the volume of (M,g) divided by the n-th power of the length of its shortest closed geodesic. Unfortunately, almost nothing is known about this quantity - at least when M is simply connected - beyond the following result:

Theorem (Croke, Rotman). The systolic volume of any Riemannian metric on the 2-sphere is greater than 1/32.

It is thought that compact rank-one symmetric spaces are local minima for the systolic volume. In this talk I shall show how some simple contact geometry and bifurcation theory yields that they are at least critical points of the systolic volume considered as a functional on the space of Riemannian metrics.

Definition. The metric g is a critical point of the systolic volume if for any smooth deformation g(t) of g the limit as t tends to zero of the increment quotient (S(M,g(t)) - S(M,g(0)))/t is equal to zero whenever it exists.

Theorem (Alvarez-Balacheff). If all geodesics of (M,g) are closed and of the same length, then (M,g) is a critical point of the systolic volume. Moreover, if we relax the variational problem and allow deformations by smooth Finsler metrics, then these are the only critical points.
10.12.10 Dr. Raphael Zentner
Vanishing results for Casson-type instanton invariants on negative definite 4-manifolds

Abstract: After a short introduction to gauge theory and its prominent applications to the topology of smooth four-manifolds we will discuss Casson-type instanton invariants defined over negative definite manifolds. Similar situations have been studied by Furuta-Ohta and Ruberman-Saveliev where the invariant is known to be non-zero in general. In our situation we get the somewhat surprising result that the invariant is always zero. We sketch the proof which uses moduli spaces of non-abelian Seiberg-Witten monopoles with holonomy perturbations. Finally, we will discuss a few thoughts towards possible applications of this vanishing result.
21.1.11 Dr. William Kirwin
Adapted complex structures and the geodesic flow

Abstract: Let M be a compact, real-analytic Riemannian manifold. An adapted complex structure is a certain complex structure on a neighborhood of M in its tangent bundle which provides a way to understand the "complexification" of M (i.e. the Grauert tube). I will explain how adapted complex structures can be constructed from the "imaginary time" geodesic flow, and how the construction is related to a general construction known as Thiemann's complexifier method. I will also describe some applications, as well as other settings (for example, toric varieties and twisted cotangent bundles) where the complexifier method can be made to work.
28.1.11 Prof. Stephan Tillmann (University of Queensland)
Volume optimisation on triangulated 3-manifolds

Abstract: In 1978, Thurston introduced an affine algebraic set to study hyperbolic structures on triangulated 3-manifolds. Recently, Feng Luo discovered a finite-dimensional variational principle on triangulated 3-manifolds with the property that its critical points are related to both Thurston's algebraic set and to Haken's normal surface theory. The action functional is the volume. This is a generalisation of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary. Combining the work of Luo, Futer-Gueritaud, Segerman-Tillmann and Luo-Tillmann, this gives a new, finite-dimensional variational formulation of the Poincaré conjecture, and is expected to give insights that lead to a discrete interpretation of the 3-dimensional Ricci flow.

H. Geiges, 5.4.02