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New Advances through the boundaries of Poisson Geometry (New-Poetry)
Date du début: 1 sept. 2011, Date de fin: 31 août 2016 PROJET  TERMINÉ 

This project proposes new research directions that originate in the field of Poisson Geometry and which reach out towards other fields of Differential Geometry and Topology. It can also be seen as the development of a new field- Poisson Topology, the birth of which is clearly predicted by the recent results of the PI on stability of symplectic leaves.Aims:1. solving some of the most fundamental open problems in Poisson Geometry.2. breaking the current boundaries of Poisson Geometry and bringing it at the forefront of the interplay between other fields in geometry (Foliation Theory, Symplectic Geometry etc).Methods/tools:1. build on the breakthrough results of the PI (and his collaborators) such as the one on the integrability of Lie algebroids or the geometric approach to Conn-Weinstein theorem.2. new tools in Poisson Geometry such as Nonlinear Functional Analysis or the use of the fundamental ideas of Cartan that were not yet exploited in Poisson Geometry (G-structures, Exterior Differential Systems, etc ).New directions: I propose several interacting directions/subprojects, each one of independent interest. For example:- study of local invariants in Poisson Geometry (a wide-open problem) based on PI's work and the use of ideas from G-structures.- a new unified approach to stability theories, such as Mather's theory or Nijenhuis-Richardson's(apparently unrelated!). Poisson Geometry plays an unifying role. We expect new fundamental results in Poisson Topology and related fields (including moduli spaces of flat connections).- the study of global aspects of Poisson Geometry. E.g. the existence of codimension one Poisson structures on spheres (the 5-dimensional sphere was settled only last year!). Recall that the similar problem in Foliation Theory served as a driving force for the field (and will be used here). Global aspects will also take us to the world of Symplectic Topology- a high risk/high return journey that has never been taken before