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.

A New Integrable Symplectic Map of Neumann Type

Abstract: The nonlinearization approach is generalized to the case of the Neumann constraint associated with a discrete 3×3 matrix eigenvalue problem. A new symplectic map of the Neumann type is obtained by nonlinearization of the discrete eigenvalue problem and its adjoint one. A scheme for generating the involutive system of conserved integrals of the symplectic map is proposed, by which the symplectic map of the Neumann type is further proved to completely integrable. As an application, the calculation of solutions for the hierarchy of lattice soliton equations connected to the discrete eigenvalue problem is reduced to the solutions of a system of ordinary differential equations plus a simple iterative process of the symplectic map of the Neumann type.

A combinatorial construction of symplectic expansions

Abstract: The notion of a symplectic expansion directly relates the topology of a surface to formal symplectic geometry. We give a method to construct a symplectic expansion by solving a recurrence formula given in terms of the Baker-Campbell-Hausdorff series.

The quantum Lefschetz principle for vector bundles as a map between givental cones

Abstract: Givental has defined a Lagrangian cone in a symplectic vector space which encodes all genus-zero Gromov-Witten invariants of a smooth projective variety X. Let Y be the subvariety in X given by the zero locus of a regular section of a convex vector bundle. We review arguments of Iritani, Kim-Kresch-Pantev, and Graber, which give a very simple relationship between the Givental cone for Y and the Givental cone for Euler-twisted Gromov-Witten invariants of X. When the convex vector bundle is the direct sum of nef line bundles, this gives a sharper version of the Quantum Lefschetz Hyperplane Principle.

Hofer-Zehnder semicapacity of cotangent bundles and symplectic submanifolds

Abstract: We introduce the concept of Hofer-Zehnder $G$-semicapacity (or $G$-sensitive Hofer-Zehnder capacity) and prove that given a geometrically bounded symplectic manifold $(M,\omega)$ and an open subset $N \subset M$ endowed with a Hamiltonian free circle action $\phi$ then $N$ has bounded Hofer-Zehnder $G_\phi$-semicapacity, where $G_\phi \subset \pi_1(N)$ is the subgroup generated by the homotopy class of the orbits of $\phi$. In particular, $N$ has bounded Hofer-Zehnder capacity. We give two types of applications of the main result. Firstly, we prove that the cotangent bundle of a compact manifold endowed with a free circle action has bounded Hofer-Zehnder capacity. In particular, the cotangent bundle $T^*G$ of any compact Lie group $G$ has bounded Hofer-Zehnder capacity. Secondly, we consider Hamiltonian circle actions given by symplectic submanifolds. For instance, we prove the following generalization of a recent result of Ginzburg-G\"urel: almost all low levels of a function on a geometrically bounded symplectic manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold.