Homological "stability" of weakly exact Lagrangians

A symplectic manifold is a manifold endowed with a non-degenerate closed 2-form (the symplectic form). A Lagrangian is a submanifold of middle dimension on which the symplectic form vanishes. Lagrangians and Hamiltonian diffeomorphisms (diffeomorphisms induced by the flow of certain vector fields) have been extensively studied in different contexts. In this talk, I will show that a Hamiltonian diffeomorphism of a symplectic manifold which preserves a (weakly exact) Lagrangian acts trivially on its homology. The proof is algebraic and relies on standard tools of symplectic geometry (Floer homology and Seidel's morphism) whose construction will be sketched.