Oberseminar Geometrie, Topologie und Analysis

H. Geiges, M. Lesch, G. Thorbergsson

Wintersemester 2004/05

Freitag, 10:30-12:00, Seminarraum 1

15.10.04 Benjamin Himpel (MPIM Bonn)
SU(2) spectral flow of the twisted odd signature operator on torus bundles over S1.

Abstract: The spectral flow of a path of self-adjoint Dirac operators is roughly defined to be the algebraic intersection number of the track of the spectrum with 0.

In 1989 Edward Witten defined a topological quantum field theory using certain 3-manifold invariants involving the Feynman path integral and the Chern-Simons function. Although their definition is not mathematically rigorous, they have two main interpretations which Lisa Jeffrey compared in 1992. Only a detail about the SU(2) spectral flow of the twisted odd signature operator between flat connections on torus bundles over the circle is left to complete her analysis.

The talk is about a splitting formula for spectral flow, which is used to compute the spectral flow on torus bundles bundles over S1. Part of this is also an explicit spectral flow computation for the solid torus with certain Atiyah-Patodi-Singer boundary conditions.

22.10.04 Antonio J. Di Scala (Turin)
The normal holonomy group of Kähler submanifolds
29.10.04 Dan Burghelea (Columbus, z.Zt. MPIM Bonn)
The geometric complex associated to a Morse Bott function (analysis and topology).

Abstract:Elementary Morse theory associates to a Riemannian manifold and a Morse function a cochain complex of finite dimensional vector spaces whose underlying vector space is generated by the critical points of the function. This complex is fundamental both in topology and geometric analysis and, to a large extend, does the same job as the de Rham complex which is infinite dimensional.

If f is a Morse Bott function, the critical points set is a collection of smooth submanifolds, the critical manifolds. The geometric complex can be defined but is not finite dimensional; its underlying vector space is essentially the same as the space of differential forms of the critical manifolds.

Fortunately this geometric complex comes equipped with a natural filtration and therefore with a spectral sequence. The E2 term of this spectral sequence is however a finite dimensional cochain complex and a good substitute of the geometric complex. If the Morse Bott function f is actually a Morse function then this E2 term coincides with the geometric complex.

The observation is particularly important since in many situations there are no Morse functions around but only Morse Bott functions (for example in the case of G-manifolds, G a compact Lie group).

The observation has many interesting applications in geometric analysis. Some of them might be discussed or at least stated.
19.11.04 Felix Schlenk (Leipzig)
Symplectic covering numbers.

Abstract: We study the problem how many Darboux charts are needed to cover a closed symplectic manifold. It turns out that the answer depends only on the Lusternik-Schnirelman category of the underlying smooth manifold and the Gromov width. The proof is elementary but involved.
26.11.04 Klaus Mohnke (Humboldt Univ., Berlin)
Lagrange-Einbettungen und Symplektische Feldtheorie.

Abstract: Es wird erläutert, wie man mithilfe neuer Techniken bei der Untersuchung pseudoholomorpher Kurven Starrheitssätze für Lagrange-Einbettungen gewinnt: die Nichtexistenz einer Lagrange-Einbettung der Kleinschen Flasche in den CP2 und eine Abschätzung für den kleinsten symplektischen Flächeninhalt einer Scheibe mit Rand auf der Einbettung eines Torus in CPn. Die Hauptschwierigkeit bei der Umsetzung in den Beweisen wird erläutert und ein Lösungsvorschlag vorgestellt, der noch wesentlicher auf Ideen der Symplektischen Feldtheorie zurückgreift.
03.12.04 Dmitri Anosov (Moskau)
Infinite curves on closed surfaces.

Abstract: This work arose from studying flows (continuous 1-parameter transformation groups, usually defined by a vector field) on surfaces (closed 2-dim manifolds). Surfaces considered are those of nonpositive Euler characteristic χ, since for those with χ > 0 the well-known Poincare'-Bendixson theory provides a complete description of possible qualitative types of behavior of trajectories. A new feature in case χ ≤ 0 is that the universal covering surface is noncompact and can be endowed by a structure of Euclidean or hyperbolic (Lobachevskiy) plane; thus one can lift the trajectory to this plane and ask about behavior of the lifted trajectory, so to say, at infinity, using geometric notions appropriate to the above mentioned structure. Such approach in rather particular cases was suggested by A.Weil in 30-s, then forgotten and revived by N.Markley (USA) and myself in 60-s.

In the course of research its subject (originally belonging to the theory of ordinary differential equations (ODE) and dynamical systems) was somewhat extended in a natural way and now my work concerns geometrical questions which can be asked not only about the (semi)trajectory, but about any (semi)infinite continuous curve having no selfintersections (again lifted to the covering plane). Special attention is paid to the "intermediate" case of leaves of foliations on surfaces with only finite number of singularities. So, main content of this work is considering several classes of curves and compareing these classes with respect to several properties. The latter are such that they do not change if the lifted curve is replaced by another curve lying at a bounded Frechet distance from the first curve. E.g.,such is the property that the lifted curve is (or is not) bounded, that it tends (or does not tend) to infinity and has an asymptotic direction there. Such properties are, in a sense, more "elementary", "primitive" than properties of the limit behaviour of the trajectories usually considered in the qualitative theory of ODE.
10.12.04 Hessel Posthuma (Frankfurt)
Quantization and conformal field theory.

Abstract: In the talk, I will give an introduction to the geometry of the moduli space of flat connections over closed surfaces, and explain how the quantization of this space is related to ideas from conformal field theory. After that I will describe an approach to quantization of the infinite dimensional moduli space associated to surfaces with boundaries. The axioms for Topological Quantum Field Theory are verified by proving that "quantization commutes with reduction" in this context. Finally, the meaning of these results at the level of K-theory is explained.
14.01.05 Peter Albers (Univ. Leipzig)
Functoriality for Floer homology.

Abstract: Floer theory associates to a symplectic manifold (M,ω) and a Hamilton function H : S1 x M → R the symplectic invariant HF*(H), namely Floer homology groups. We address the question how to assign to a symplectic map f: (M,ω) → (N,σ) an induced homomorphism on Floer homology, and propose a construction for a large class of such maps. Inspection of special embeddings leads to new results on the topology and geometry of Lagrangian submanifolds.
21.01.05 Fan Ding (Peking University, Beijing)
E8-plumbings and exotic contact structures on spheres.

Abstract: We prove the existence of exotic but homotopically trivial contact structures on spheres of dimension 8k-1. Together with previous results of Eliashberg and Geiges, this proves the existence of such structures on all odd-dimensional spheres. This is a joint work with Geiges.
In this talk, I will give an introduction to some concepts such as almost contact structures, E8-plumbings and will explain briefly the ideas for proving the above-mentioned result.
28.01.05 Arkadi Onishchik (Jaroslavl Univ., z.ZT. Bochum)
Classification problems related to complex manifolds.

Abstract: After an introduction to the theory of complex analytic supermanifolds, the following two problems will be discussed:
1) To classify (up to isomorphy) all complex supermanifolds with a given retract.
2) To describe all homogeneous supermanifolds with a given homogeneous retract.
Approaches to these problems basing on non-abelian cohomology and representation theory will be proposed and some classification results will be presented.
04.02.05 Tillmann Jentsch (Köln)
On the geometry of II-parallel submanifolds.

Abstract: In Riemannian geometry, for any point p of a smooth submanifold there exists a tensor called the second fundamental form at p, which describes the extrinsic geometry of the submanifold at p up to second order. A submanifold is called II-parallel, if its second fundamental form is a parallel section in a suitable tensor bundle. Of course every totally geodesic submanifold is II-parallel, since its second fundamental form vanishes identically. Non totally geodesic examples are the arcs in the euclidian plane, the cylinders and spheres in the three dimensional euclidian space.

The aim of the talk is to give a generalisation of a theorem of Cartan concerning totally geodesic submanifolds; shortly Cartans theorem states necessary and sufficent tensorial conditions for the total geodesy of a geodesic umbrella. It can be shown that the geodesic lines of a II-parallel submanifold are helical arcs in the ambient space, which is obvious in the above examples. Thus one has to construct a more general umbrella, the rays of which are helical arcs. Necessary and sufficent tensorial conditions can be derived for the II-parallelity of that helical umbrella; and locally every II-parallel submanifold can be obtained in this way.

M. Bohn, 01/2005