04/06/2013, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática
Daniele Sepe, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos
On complete isotropic realisations of Poisson manifolds
Poisson geometry can be viewed as the study of those manifolds which admit a (possibly singular) foliation by symplectic manifolds (e.g. \(\mathbb{R}^3\) foliated by spheres centred at the origin of increasing radius and the radius). A complete symplectic realisation of a Poisson manifold is a ‘desingularisation’, in the sense that it consists of a symplectic manifold together with a submersion onto the original Poisson manifold which reflects its Poisson structure. A beautiful theorem of Crainic and Fernandes proves that existence of such a desingularisation is equivalent to the Poisson manifold (viewed as an infinitesimal object, the analogue of a Lie algebra) admitting a global ‘integration’ (the analogue of a Lie group). In this talk, we consider complete isotropic realisations of Poisson manifolds, which are desingularisations as above of minimal dimension; these are related to non-commutative integrable Hamiltonian systems. The fundamental driving question behind this talk is the following: what can be said about Poisson manifolds that admit such desingularisations? Some results in this direction, both old and new, will be presented. This is ongoing joint work with Ioan Marcut.
