0. Introduction. A symplectic groupoid is a manifold T with a partially defined multiplication (satisfying certain axioms) and a compatible symplectic structure. The identity elements in T turn out to form a Poisson manifold To? and the correspondence between symplectic groupoids and Poisson manifolds is a natural extension of the one between Lie groups and Lie algebras. As with Lie groups, und...

We give a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of closed symplectic surfaces. We also classify simple Morse functions on symplectic surfaces with respect to actions of those groups. This gives an answer to V.Arnold’s problem on invariants of generic isovorticed fields for 2D ideal fluids.

we study the notion of harmonicity in the sense of symplectic geometry, and investigate the geometric properties of harmonic thom forms and distributional thom currents, dual to different types of submanifolds. we show that the harmonic thom form associated to a symplectic submanifold is nowhere vanishing. we also construct symplectic smoothing operators which preserve the harmonicity of distri...