Oberseminar Geometrie, Topologie und Analysis

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

Sommersemester 2013

Freitag, 10:30-11:30, Seminarraum im Container bei der Physik

12.4.13 Viet Anh Nguyen (Paris-Sud, z.Zt. Köln)
Ergodic theorems for Riemann surface laminations

Abstract: We study the dynamics of possibly singular foliations by Riemann surfaces. The main examples are holomorphic foliations by Riemann surfaces in projective varieties. We introduce the heat equation relative to a positive harmonic current and apply it to the directed currents associated with Riemann surface laminations, possibly with singularities. We prove two kinds of Birkhoff ergodic theorems in this context. This is joint work with T.-C. Dinh (Paris 6) and N. Sibony (Paris 11).
19.4.13 Thomas Haettel (Bonn)
Visual limits of maximal flats in symmetric spaces and Euclidean buildings

Abstract: We study the space of maximal flats of symmetric spaces of non-compact type or of Euclidean buildings: we define a geometric compactification by looking at the visual limits of a diverging sequence of flats. We completely determine the possible degenerations of flats when X is of rank 1, associated to a classical group of rank 2 or to PGL(4). In particular, we exhibit unexpected behaviours of visual limits of maximal flats in various symmetric spaces of small rank and surprising algebraic restrictions that occur.
3.5.13 Neil Hoffman (MPI Bonn)
Big Dehn surgery space and the link of S3

Abstract: Thurston described a graph with a vertex for every closed oriented 3-manifold and an edge between two vertices vM and vN if there is a Dehn surgery along a curve in M that yields N. Famous results of Lickorish and Wallace show this graph is connected, and following those constructions we may ask "How few edges are needed to get from the three sphere S3 to a manifold M?" After providing the necessary background, I will provide an overview of known results and examples which shed light on the difficulty of this problem. This is joint work with Genevieve Walsh.
17.5.13 Thomas Vogel (MPI Bonn)
Uniqueness of the contact structure approximating a foliation

Abstract: According to a theorem of Eliashberg and Thurston a C2-foliation on a closed 3-manifold can be C0-approximated by contact structures unless all leaves of the foliation are spheres. Examples on the 3-torus show that every neighbourhood of a foliation can contain infinitely many non-diffeomorphic contact structures. In this talk we show that this is rather exceptional: In many interesting situations the contact structure in a sufficiently small neighbourhood of the foliation is uniquely determined up to isotopy. This fact can be applied to obtain results about the topology of the space of taut foliations.
7.6.13 Sorin Dumitrescu (Nice)
Quasihomogeneous analytic affine connections on surfaces

Abstract: In this joint work with Adolfo Guillot, we classify real-analytic torsion-free affine connections on compact oriented surfaces which are quasihomogeneous, in the sense that they are locally homogeneous on nontrivial open sets, without being locally homogeneous on all of the surface. The proof relies on a local result which classifies quasihomogeneous germs of real-analytic torsion-free connections on surfaces. I will also explain our motivations which come from Gromov's open-dense orbit theorem.
14.6.13 Hugues Auvray (MPI Bonn)
Analytic construction of dihedral ALF gravitational instantons

Abstract: In this talk we deal with an analytic construction of ALF gravitational instantons, or: complete hyperkähler manifolds of real dimension 4, with cubic growth of the ball volume. We give the construction of dihedral ALF instantons, which are not well understood by comparison with heir cyclic homologues. We first consider minimal resolution of Kleinian dihedral singularities. In particular, we shall see how the treatment of a complex Monge-Ampère equation in a general ALF context allows us on these examples to correct a simple prototype to obtain the sought hyperkähler metric. If time allows so, we shall also evoke how our construction can be generalized to deformations of dihedral Kleinian singularities, and in particular how this involves the asymptotics of Kronheimer's ALE hyperkähler metrics.
21.6.13 Christine Laurent-Thiébaut (Institut Fourier, Grenoble)
Embedding problems in complex analysis

Abstract: After recalling the notions of complex and CR structure we will try to partially answer the following natural question: is any compact abstract CR manifold globally CR-embeddable in a complex manifold ? Beside the classical result by Boutet de Monvel in the strictly pseudoconvex case, very little is known. We will mostly give results on the stability of the embeddability under small perturbations of the CR structure.

H. Geiges, 5.4.02