Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

Symplectic groupoids of log symplectic manifolds

Abstract: A log symplectic manifold is a Poisson manifold which is generically nondegenerate. We develop two methods for constructing the symplectic groupoids of log symplectic manifolds. The first is a blow-up construction, corresponding to the notion of an elementary modification of a Lie algebroid along a subalgebroid. The second is a gluing construction, whereby groupoids defined on the open sets of an appropriate cover may be combined to obtain global integrations. This allows us to classify all Hausdorff symplectic groupoids of log symplectic manifolds in a combinatorial fashion, in terms of a certain graph of fundamental groups associated to the manifold. Using the same ideas, and as a first step, we also construct and classify the groupoids integrating the Lie algebroid of vector fields tangent to a smooth hypersurface.

Localization theorems by symplectic cuts

Abstract: Given a compact symplectic manifold M with the Hamiltonian action of a torus T, let zero be a regular value of the moment map, and M 0 the symplectic reduction at zero. Denote by κ 0 the Kirwan map H T * (M) → H*(M 0). For an equivariant cohomology class η ∊ H T * (M) we present new localization formulas which express \( \int_{M_0 } \kappa 0(\eta ) \) as sums of certain integrals over the connected components of the fixed point set M T. To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect to a certain cone, and then, if necessary, iterate this process using other cones. When all cones used to produce the formula are one-dimensional we recover, as a special case, the localization formula of Guillemin and Kalkman [GK]. Using similar ideas, for a special choice of the cone (whose dimension is equal to that of T) we give a new proof of the Jeffrey-Kirwan localization formula [JK1].

On the minimal number of periodic Reeb orbits on a contact manifold

Abstract: This thesis deals with the question of the minimal number of distinct periodic Reeb orbits on a contact manifold which is the boundary of a compact symplectic manifold.The positive S1-equivariant symplectic homology is one of the main tools considered in this thesis. It is built from periodic orbits of Hamiltonian vector fields in a symplectic manifold whose boundary is the given contact manifold.Our first result describes the relation between the symplectic homologies of an exact compact symplectic manifold with contact type boundary (also called Liouville domain), and the periodic Reeb orbits on the boundary. We then prove some properties of these homologies. For a Liouville domain embedded into another one, we construct a morphism between their homologies. We study the invariance of the homologies with respect to the choice of the contact form on the boundary.We use the positive S1-equivariant symplectic homology to give a new proof of a Theorem by Ekeland and Lasry about the minimal number of distinct periodic Reeb orbits on some hypersurfaces in R2n. We indicate how it extends to some hypersurfaces in some negative line bundles. We also give a characterisation and a new way to compute the generalized Conley-Zehnder index defined by Robbin and Salamon for any path of symplectic matrices. A tool for this is a new analysis of normal forms for symplectic matrices.