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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.
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TL;DR: In this article, Katzarkov introduced theories and constructions for cohomology of symplectomorphism groups, including the existence of invariant polynomials in the Lie algebra, the Chern-Simons-type secondary classes, and the symplectic Chern-Weil theory.
Abstract: From the cohomological point of view the symplectomorphism group $Sympl (M)$ of a symplectic manifold is `` tamer'' than the diffeomorphism group. The existence of invariant polynomials in the Lie algebra $\frak {sympl }(M)$, the symplectic Chern-Weil theory, and the existence of Chern-Simons-type secondary classes are first manifestations of this principles. On a deeper level live characteristic classes of symplectic actions in periodic cohomology and symplectic Hodge decompositions. The present paper is called to introduce theories and constructions listed above and to suggest numerous concrete applications. These includes: nonvanishing results for cohomology of symplectomorphism groups (as a topological space, as a topological group and as a discrete group), symplectic rigidity of Chern classes, lower bounds for volumes of Lagrangian isotopies, the subject started by Givental, Kleiner and Oh, new characters for Torelli group and generalizations for automorphism groups of one-relator groups, arithmetic properties of special values of Witten zeta-function and solution of a conjecture of Brylinski. The Appendix, written by L. Katzarkov, deals with fixed point sets of finite group actions in moduli spaces.
45 citations
45 citations
TL;DR: In this paper, the authors derived several fundamental inequalities about the eigenvalues of 2n × 2n real positive definite matrices, such as the relation between the symplectic eigenvalue of A and those of At, between the Riemannian mean of m matrices A1, A2, Am, and Am and their mean of their mean, a perturbation theorem, and some variational principles.
Abstract: If A is a 2n × 2n real positive definite matrix, then there exists a symplectic matrix M such that MTAM=DOOD where D = diag(d1(A), …, dn(A)) is a diagonal matrix with positive diagonal entries, which are called the symplectic eigenvalues of A. In this paper, we derive several fundamental inequalities about these numbers. Among them are relations between the symplectic eigenvalues of A and those of At, between the symplectic eigenvalues of m matrices A1, …, Am and of their Riemannian mean, a perturbation theorem, some variational principles, and some inequalities between the symplectic and ordinary eigenvalues.
45 citations
TL;DR: In this article, the authors combine tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results, including a dimension-independent bound for the symplectic capacity of a convex body.
Abstract: In this work we bring together tools and ideology from two different fields, symplectic geometry and asymptotic geometric analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body ,
44 citations
TL;DR: Based on the analogy between structural mechanics and optimal control theory, the eigensolutions of a symplectic matrix, the adjoint symplectic ortho-normalization relation and the eigenvector expansion method are introduced into the wave propagation theory for sub-structural chain-type structures, such as space structures, composite material and turbine blades.
Abstract: Based on the analogy between structural mechanics and optimal control theory, the eigensolutions of a symplectic matrix, the adjoint symplectic ortho-normalization relation and the eigenvector expansion method are introduced into the wave propagation theory for sub-structural chain-type structures, such as space structures, composite material and turbine blades. The positive and reverse algebraic Riccati equations are derived, for which the solution matrices are closely related to the power flow along the sub-structural chain. The power flow orthogonality relation for various eigenvectors is proved, and the energy conservation result is also proved for wave scattering problems.
44 citations