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2013
Impact Factor

# Eva Miranda

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)

## Publications:

 Miranda E., Planas A. Equivariant Classification of $b^m$-symplectic Surfaces 2018, vol. 23, no. 4, pp.  355-371 Abstract Inspired by Arnold’s classification of local Poisson structures [1] 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 [24] 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 [10] is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail. Keywords: Moser path method, singularities, $b$-symplectic manifolds, group actions Citation: Miranda E., Planas A.,  Equivariant Classification of $b^m$-symplectic Surfaces, Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 355-371 DOI:10.1134/S1560354718040019
 Martínez-Torres D., Miranda E. Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds 2018, vol. 23, no. 1, pp.  47-53 Abstract 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. Keywords: Poisson homology, foliated cohomology Citation: Martínez-Torres D., Miranda E.,  Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 47-53 DOI:10.1134/S1560354718010045
 Miranda E., Kiesenhofer A. Noncommutative Integrable Systems on $b$-symplectic Manifolds 2016, vol. 21, no. 6, pp.  643-659 Abstract 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. Keywords: Poisson manifolds, $b$-symplectic manifolds, noncommutative integrable systems, action-angle coordinates Citation: Miranda E., Kiesenhofer A.,  Noncommutative Integrable Systems on $b$-symplectic Manifolds, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 643-659 DOI:10.1134/S1560354716060058