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Documents 53D05 3 résultats

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Symplectic singularities of varieties - Domitrz, Wojciech (Auteur de la Conférence) | CIRM H

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We study germs of singular varieties in a symplectic space. We introduce the algebraic restrictions of differential forms to singular varieties and prove the generalization of Darboux-Givental' theorem from smooth submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geometric meaning. We show that a quasi-homogeneous variety $N$ is contained in a non-singular Lagrangian submanifold if and only if the algebraic restriction of the symplectic form to $N$ vanishes. The method of algebraic restriction is a powerful tool for various classification problems in a symplectic space. We illustrate this by the construction of a complete system of invariants in the problem of classifying singularities of immersed $k$-dimensional submanifolds of a symplectic 2n-manifold at a generic double point.

Keywords: symplectic manifolds - symplectic multiplicity and other invariants - Darboux-Givental's theorem - quasi-homogeneous singularities - singularities of planar curves[-]
We study germs of singular varieties in a symplectic space. We introduce the algebraic restrictions of differential forms to singular varieties and prove the generalization of Darboux-Givental' theorem from smooth submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geometric meaning. We show that a quasi-homogeneous variety $N$ is ...[+]

58K55 ; 32S25 ; 53D05

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Existence of Liouville structures on cobordisms - Murphy, Emmy (Auteur de la Conférence) | CIRM H

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Given a smooth cobordism with an almost complex structure, one can ask whether it is realized as a Liouville cobordism, that is, an exact symplectic manifold whose primitive induces a contact structure on the boundary. We show that this is always the case, as long as the positive and negative boundaries are both nonempty. The contact structure on the negative boundary will always be overtwisted in this construction, but for dimensions larger than 4 we show that the positive boundary can be chosen to have any given contact structure. In dimension 4 we show that this cannot always be the case, due to obstructions from gauge theory.[-]
Given a smooth cobordism with an almost complex structure, one can ask whether it is realized as a Liouville cobordism, that is, an exact symplectic manifold whose primitive induces a contact structure on the boundary. We show that this is always the case, as long as the positive and negative boundaries are both nonempty. The contact structure on the negative boundary will always be overtwisted in this construction, but for dimensions larger ...[+]

53D05 ; 53D10 ; 53D35

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The aims of this talk are two-fold: a) to give a biased overview of some structural properties of (smooth) symplectic mapping class groups, together with some key proof ingredients, and b) to present some open questions, both about those groups and about their $C^{0}$ counterparts, whose study was recently initiated by Alexandre Jannaud.

53D05 ; 53D40 ; 57R17

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