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Monday, March 14, 2016

**Abstract:** I will give a brief overview on the recent joint work with R.L. Ferandes and D.M. Torres on Poisson manifolds of compact types. While Poisson manifolds generalize symplectic manifolds as well as (duals of) Lie algebras, the "compactness" we are referring to generalizes compactness of symplectic manifolds as well as compactness of Lie groups. While general Poisson structures are very flexible and there is quite little one can say about them in full generality, the compact ones come with some very rich and beautiful geometry. In particular, one discovers new connections with Lie theory, integral affine geometry, symplectic topology, orbifolds, integrable systems, gerbes, Duistermaat-Heckman measures and formulas etc, all interacting in a very interesting way. Returning to (duals of) Lie algebras, one may say that a large part of the very fundamental results on compact Lie groups are of a purely Poisson geometric nature: one just has to feed the dual of the Lie algebra into the theory of PMCTs (therefore making use only of the Poisosn structure). Since the list of interesting properties of PMCTs is very large, this talk will be rather superficial: just an overview of the most important features of PMCTs.