11:00 Robert Vandervorst (Amsterdam)
Braid Floer homology.
Abstract: Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on S¹xD², periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of the Floer homology originally used for the Arnol’d Conjecture yields a Morse-type forcing theory for periodic points of area-preserving diffeomorphisms of the 2-disc based on braiding.
Contributions of this paper include (1) a monotonicity lemma for the behavior of the nonlinear Cauchy-Riemann equations with respect to algebraic lengths of braids; (2) establishment of the topological invariance of the resulting braid Floer homology; (3) a shift theorem describing the effect of twisting braids in terms of shifting the braid Floer homology; (4) computation of examples; and (5) a forcing theorem for the dynamics of Hamiltonian disc maps based on braid Floer homology.14:15 Andrei Pajitnov (Nantes)
Arnold conjecture, Floer chain complexes and the augmentation ideals of finite groups.
Abstract: Let H be a time-dependent Hamiltonian H on a symplectic manifold M. Assume that the periodic orbits of the corresponding vector field are non-degenerate. The Arnold Conjecture says that the number P(H) of the periodic orbits is not less than the Morse number of M. The Floer chain complex, associated to the pair (M,H) allows in many cases to prove that P(H) is not less than the sum of Betti numbers of M.
We construct a refined version of the Floer chain complex associated to (M,H) and any regular covering of M. This chain complex gives rise to a sequence of numerical invariants, which provide new lower bounds for the number of periodic orbits of H.
For the case of finite fudamental group the first of these invariants is computable from the augmentation ideal of the group ring.Using these invariants we prove in particular that if the fundamental group of M is finite and solvable or simple, then P(H) is not less than the minimal number of generators of the fundamental group.
This is joint work with Kaoru Ono.
16:00 Andy Wand (Nantes)
Tightness and Legendrian surgery.
Abstract: A well known result of Giroux tells us that isotopy classes of contact structures on a closed three manifold are in one to one correspondence with stabilization classes of open book decompositions of the manifold. We will introduce a characterization of tightness of a contact structure in terms of corresponding open book decompositions, and show how this can be used to resolve the question of whether tightness is preserved under Legendrian surgery.
Prochaines séances: 5/12 (N. Vichery, M. Mazzuchelli, J. Kang) - 9/01 (J. Fine, D. Gayet, ?).