Invariants of functions on symplectic surfaces and ideal hydrodynamics

We describe a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic surface. We also describe invariants of simple Morse functions on symplectic surfaces with respect to actions of those groups. This gives an answer to V.Arnold's problem on describing all invariants of generic isovortex fields for the 2D ideal fluids. For this we introduce a notion of anti-derivatives on a measured Reeb graph and describe their properties. This is a joint work with Anton Izosimov and Mehdi Mousavi.