Desingularization in Poisson geometry

Andreas Schüßler will work on desingularization of various geometric structures which fit in the framework of Poisson geometry. A first step relates to blow-ups of Lie algebroids and of the corresponding Lie groupoids, as worked out for instance by Debord-Skandalis. In general, blowing up makes the structures less singular, and we wish to make the construction more explicit in the case of Lie algebroids. An aim is to describe the cohomology of a Lie algebroid in terms of the simpler cohomology of the blown-up Lie algebroid, and to compute this explicity in some interesting cases. With this tool at hand, Andreas will address the question: under what conditions does a geometric structure lift to the blow up? The geometric structures we have in mind include Poisson structures, Dirac structures, generalized complex structures, singular foliations,... Notice that these geometric structures have an associated Lie algebroid. Blow-ups can be iterated. An aim, in each of these cases, is to provide methods to make these structures less singular by suitable iterated blow-ups, and to establish 'how regular' the blown-up structure becomes.