Sylvain Lavau

The notion of Singular Foliation most adapted for applications to dynamical systems and geometry is the following: a locally finitely generated sheaf of vector fields closed under the Lie bracket. Their singular behaviour, however, makes it very challenging to advance the theory using classical differential geometry. Graded geometry on the other hand, offers a new class of objects that, although stated in more involved terms, are easier to manipulate and allows us to go beyond the difficulty raised by the singularities of the foliation. Thanks to these higher ‘more regular’ structures, it becomes possible to proceed to some computations that were otherwise very difficult to handle, or to generalize to the singular setting some notions that were until now only defined in the regular one. In this talk, we will explain how one can `replace' a singular foliation by an adequate graded manifold, and how the Lie bracket on the former lifts to a Lie infinity-algebroid structure on the latter. We will then give several applications of this construction regarding deformations and characteristic classes of singular foliations.