Geometry and Topology of Two-Dimensional Symplectic Manifolds with Generic Singularities and Hamiltonian Systems on Them

The topological and symplectic classification of closed 2-dimensional symplectic manifolds whose symplectic structure has generic singularities are obtained. The Liouville foliations of Hamiltonian systems on such manifolds are classified in topological category. The properties of index-one surgery along a pair of Liouville tori are studied together with the singularities of symplectic structure it gives rise to. The change of Liouville foliation topology after the surgery in dimension two is described.