Global Problems in Poisson Geometry and Related Structures

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This project aims to study properties of Poisson structures on manifolds as well as of related geometric structures such as Dirac structures, Lie algebroids or generalized complex structures. The project focus primarily on global aspects, drawing on ideas and techniques from Foliation Theory, Equivariant Geometry, Integral Affine Geometry and Symplectic Geometry. These ideas, together with recent results and techniques in Lie groupoid theory developed in the last 10 years, will lead to new methods to attack some long standing fundamental problems in Poisson Geometry, such as the existence of regular Poisson structures, the classification of Poisson manifolds of "compact type", or the existence of normal forms around leaves that go beyond linearization. The project also aims at going beyond the current boundaries of Poisson Geometry by proposing new interactions with the theory of Exterior Differential Systems and with the theory of Integrable Systems.Poisson Geometry lies on the intersection of Mathematical Physics and Geometry. It originates in the mathematical formulation of classical mechanics as the semiclassical limit of quantum mechanics. The field developed rapidly in the last 20 years, stimulated by the connections with a large number of areas in mathematics and mathematical physics, including differential geometry and Lie theory, quantization, noncommutative geometry, representation theory and quantum groups, geometric mechanics and integrable systems. This project shares this flavor of Poisson geometry, aiming not only at a deeper understanding of geometric properties of Poisson brackets, but also at developing new applications in other fields of Mathematics.