contact structures
and foliations
Supported by BudAlgGeo

November 17 - 20, 2005.

Program & Abstracts

9:00 - 10:00
Ozsváth Kotschick Kotschick Schönenberger
10:15 - 11:15
Ozsváth Bourgeois
Braun Szabó
11:30 - 12:30

14:30 - 15:30


15:45 - 16:45

Javier Fernandez Bobadilla: TBA

Frederic Bourgeois: Fundamental group of the space of tight contact structures on torus bundles We show that the fundamental group of the space of tight contact structures on torus bundles coincides with the infinite cyclic subgroup described by Geiges and Gonzalo. The proof uses results on characteristic foliations, convex surfaces and bypasses by Giroux and Honda. This work was initiated jointly with Fabien Ngo.

Gábor Braun: Recovering a surface singularity from its resolution graph Let us consider an isolated surface singularity at the origin given by a polynomial in three variables. For Newton non-degenarate singularities whose link is a rationally homology sphere, we give an algorithm which computes the singularity from the resolution graph.

Vincent Colin: Reeb vector fields and open book decompositions: the periodic case We prove that any contact structure supported by an open book whose monodromy is (isotopic to) a periodic diffeomorphism satisfies the Weinstein conjecture. The approach is to study holomorphic curves for a particularly nice Reeb vector field. It also allows to deal with the topology of the manifold. This is a joint work with Ko Honda.

Hansjörg Geiges: On the classification of Legendrian knots This is a report on joint work with Fan Ding about certain knot and link types whose Legendrian realisations (e.g. in the 3-sphere with its standard contact structure) are classified by the two classical invariants (Thurston-Bennequin invariant and rotation number). I shall try to present the general scheme (due to Etnyre and Honda) behind such classification results.

Magnus Jacobsson: A review of Khovanov homology with applications relevant to this workshop.

Dieter Kotschick: Foliations and symplectic structures We shall discuss the interplay between symplectic geometry and the theory of foliations, concentrating on foliations with symplectic holonomy. Even the special case when there are two complementary symplectic foliations is very interesting. We shall consider connections to the group homology of symplectomorphism groups as discrete groups, to the theory of bi-Hamiltonian systems, to symplectic pairs and to holomorphic symplectic structures.

Paolo Lisca: 2-bridge knots and the ribbon conjecture The long-standing ribbon conjecture states that a smoothly slice knot in the 3-sphere is ribbon. I will describe a proof of the ribbon conjecture for the special class of 2-bridge knots.

András Némethi: The canonical contact structure of isolated singularities. On the local link of any complex analytic isolated singularity on has a canonical contact structure induced by the complex structure. We show that this structure is supported by all the Milnor open book decompositions associated with analytic germs defined on the singularity. Moreover, we prove that for surface singularities, the contact structure is determined up to a contactomorphism by the topology of the link (a fact, which is not true in higher dimensions).

Peter Ozsvath: Floer homology and knots and links Given a Heegaard diagram for a closed three-manifold, one can associate an invariant defined by counting pseudo-holomorphic curves in a symmetric product of the Heegaard surface. This construction can be adapted to the case of knots. For knots in the three-sphere, this gives an invariant whose Euler characteristic in a suitable sense is the Alexander polynomial, and the invariant detects the Seifert genus of the knot. I will describe this construction and some of its applications. This material is joint work with Zoltan Szabo. In the second lecture I will discuss Floer homology and links -- some extensions of the earlier construction to the case of links.

Yann Rollin: Contact invariants and Monopole Floer homology A element of the monopole Floer homology is associated to every contact structure on 3-dimensional manifold. We show that this contact invariant is functorial for the category of special symplectic cobordisms. We use this property to relate the contact invariant to the Floer homology of mapping tori.

Stephan Schönenberger: Determining symplectic fillings from planar open books

Saso Strle: Definite four-manifolds with boundary According to a celebrated theorem of Donaldson, if the intersection form of a smooth closed four-manifold is definite then it is diagonalizable. Later proofs of this result use Elkies' characterisation of the diagonal definite unimodular form. I will describe a generalization of Elkies' theorem to forms of arbitrary determinant. Combined with a theorem of Ozsváth and Szabó this gives a generalization of Donaldson's theorem to four-manifolds with boundary. This is joint work with Brendan Owens.

Zoltán Szabó: Link Floer homology and the Thurston norm. In this lecture we compute the link Floer homology HFL for various links in S^3. We also study a relationship between HFL and the Thurston norm of the link complement.