Lectures on groups of symplectomorphisms

These are notes for a lecture course given in Jussieu, Paris, July 2001. They discuss groups of symplectomorphisms of closed symplectic manifolds (M, ω) from various points of view. The first two lectures provide an overview of their algebraic, geometric and homotopy theoretic properties. Much of the recent progress in understanding the geometry and topology of these groups has come from studying the properties of fibrations with the manifold M as fiber and structural group equal either to the symplectic group or to its Hamiltonian subgroup Ham(M). The case when the base is S has proved particularly important. Lecture 3 describes the geometry of Hamiltonian fibrations over S, while Lecture 4 discusses their Gromov-Witten invariants via the Seidel representation. Finally in Lecture 5 we apply the ideas developed in the previous two lectures to demonstrate the existence of (short) paths in Ham(M, ω) that minimize the Hofer norm over all paths with the given endpoints.