On hamiltonian fiber bundles and topology of symplectomorphism group

In the talk, I explain a method of finding non-trivial elements in homotopy groups of groups of symplectomorphisms via constructing suitable hamiltonian fiber bundles. This circle of ideas appared in works of Reznikov and was developed by McDuff. I apply this method to symplectic homogeneous spaces, and relate cohomology of the classifying space of the group of hamiltonian symplectomorphisms, to cohomology of lattices in semisimple Lie groups of non-compact type.