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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)


Cardona R., Miranda E.
On the Volume Elements of a Manifold with Transverse Zeroes
2019, vol. 24, no. 2, pp.  187-197
Moser proved in 1965 in his seminal paper [15] that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold coincide. In particular, this yields a classification of compact symplectic surfaces in terms of De Rham cohomology. In this paper we generalize these results for volume forms admitting transversal zeroes. In this case there is also a cohomology capturing the classification: the relative cohomology with respect to the critical hypersurface. We compare this classification scheme with the classification of Poisson structures on surfaces which are symplectic away from
a hypersurface where they fulfill a transversality assumption ($b$-Poisson structures). We do this using the desingularization technique introduced in [10] and extend it to $b^m$-Nambu structures.
Keywords: Moser path method, volume forms, singularities, $b$-symplectic manifolds
Citation: Cardona R., Miranda E.,  On the Volume Elements of a Manifold with Transverse Zeroes, Regular and Chaotic Dynamics, 2019, vol. 24, no. 2, pp. 187-197
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 [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
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.
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
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.
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

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