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Monday, February 13, 2006

**Abstract:** According to the splitting theorem of A. Weinstein, a Poisson structure induces a (singular) foliation on the underlying manifold, making the global classification of Poisson structures extremely difficult. Locally, in dimension 2, there is a hierarchy of Poisson structures related to Arnold's classification of singularities. We present a complete global classification of several types of Poisson structures corresponding to the first few elements in the local hierarchy. In analogy with algebras, P. Xu called two Poisson manifolds Morita equivalent if there is a symplectic manifold playing the role of an invertible bimodule. This also associates to every Poisson manifold an invariant, called its Picard group. We show that for a certain type of structures, this group is isomorphic to a particular mapping class group of the underlying manifold. Host: Ely Kerman