Invitation to the Arbeitsgemeinschaft at Oberwolfach

Symplectic Geometry has seen a rapid development since the 80's with the discovery of rigidity phenomena like the non-squeezing theorem based on Gromov's seminal theory of pseudo-holomorphic curves. A directly related development was Floer's method of using Morse theory in a new way, known as Floer homology, in order to solve the Arnold conjecture. In the 60's Arnold formulated his conjecture based on Poincar e's so-called \Last Geometric Theorem" proven by G. D. Birkhoo, about the existence of at least two xed points of area preserving homeomorphisms of the annulus rotating the boundary circles in diierent directions. Arnold put his conjecture in the following terms, A2]: Any Hamiltonian diieomorphism of a compact symplectic manifold has at least as many xed points as a smooth function on this manifold has critical points. Under the additional assumption that all xed points are non-degenerate, a generic property, this conjecture was modiied to Moreover, there is a corresponding variant in terms of intersections of Lagrangian submanifolds giving a priori lower bounds on the minimal number of intersection points provided that one Lagrangian is a Hamil-tonian deformation of the other and meets some additional conditions. This variant is a direct generalization of the version for symplectic xed points when considering the graphs of the Hamiltonian diieomorphism and of the identity in the symplectic manifold (M M; ! (?!)). For the symplectic xed points, one can relate the non-degenerate Arnold conjecture to the Lefschetz xed point theorem in the same way as Morse theory with the Poincar e-Hopf theorem. The Arnold conjecture for non-degenerate symplectic xed points has now been proved in full generality. It was rst connrmed by Eliashberg E] for Riemann surfaces and then by Conley and Zehnder CZ1] for the 2n-torus. In G] Gromov proved the existence of at least one xed point under the assumption 2 (M) = 0. The breakthrough came when Floer established the Arnold conjecture for Lagrangian intersections, and hence symplec-tic xed points, again under the assumption 2 (M) = 0. In a series of papers F1, F2, F3, F5] Floer combined the variational approach of 1