77 Avenue Denfert Rochereau, Paris, 75014, France
Department of Mathematics-UPC and BGSMath, Barcelona, Spain; CEREMADE (Universite de Paris Dauphine), IMCCE (Observatoire de Paris), and IMJ (Universite de Paris Diderot)
Miranda E., Planas A.
Equivariant Classification of $b^m$-symplectic Surfaces
2018, vol. 23, no. 4, pp. 355-371
Inspired by Arnold’s classification of local Poisson structures  in the plane using the hierarchy of singularities of smooth functions, we consider the problem of global classification of Poisson structures on surfaces. Among the wide class of Poisson structures, we consider the class of $b^m$-Poisson structures which can be also visualized using differential forms with singularities as bm-symplectic structures. In this paper we extend the classification scheme in  for $b^m$-symplectic surfaces to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for nonorientable surfaces. The paper also includes recipes to construct $b^m$-symplectic structures on surfaces. The feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in  is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.
Martínez-Torres D., Miranda E.
Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds
2018, vol. 23, no. 1, pp. 47-53
We prove that, for compact regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group, and we give some applications. In particular, we show that, for regular unimodular Poisson manifolds, top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what a perfect Poisson manifold is. We use these Poisson homology computations to provide families of perfect Poisson manifolds.
Miranda E., Kiesenhofer A.
Noncommutative Integrable Systems on $b$-symplectic Manifolds
2016, vol. 21, no. 6, pp. 643-659
In this paper we study noncommutative integrable systems on $b$-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a $b$-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the $b$-symplectic structure.