In this paper we study overcomplete systems of coherent states associated to compact integral symplectic manifolds by geometric quantization. Our main goals are to give a systematic treatment of the construction of such systems and to collect some recent results. We begin by recalling the basic constructions of geometric quantization in both the Kähler and non-Kähler cases. We then study the re...

Studying the space of Lagrangian submanifolds is a fundamental problem in symplectic topology. Lagrangian spheres appear naturally in the Leftschetz pencil picture of symplectic manifolds. In this paper we demonstrate the uniqueness up to Hamiltonian isotopy of the Lagrangian spheres in some 4-dimensional Stein symplectic manifolds. The most important example is the cotangent bundle of the 2-sp...

In the symplectic category there is a ‘connect sum’ operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum Z = X#Y in terms of the relative GW invariants ofX and Y . Several applications to enumerative geometry are given. Gromov-Witten invariants are co...

In this paper, we study the integration of Hamiltonian wave equations whose solutions have oscillatory behaviors in time and/or space. We are mainly concerned with the research for multi-symplectic extended Runge-Kutta-Nyström (ERKN) discretizations and the corresponding discrete conservation laws. We first show that the discretizations to the Hamiltonian wave equations using two symplectic ERK...

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on “convenient” vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,ω), we construct a weak symplectic structure on each leaf Iw of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree fo...

There is a canonical exact symplectic structure on the unit tangent bundle of a Riemannian manifold M given by pulling-back the symplectic two form ω and Liouville one form λ from the cotangent bundle T ∗M using the Riemannian metric. The pull-back of λ gives a contact form on level-sets of the length function on TM . The geodesic flow of M is given by the Reeb vectorfield of this contact struc...

This is an expanded version of the talk given by first author at conference “Topology, Geometry, and Dynamics: Rokhlin – 100”. The purpose this was to explain our current results on classification rational symplectic 4-manifolds equipped with anti-symplectic involution. A detailed exposition will appear elsewhere.

The natural sum operation for symplectic manifolds is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n − 2)submanifold V . Given a similar pair (Y, V ) with a symplectic identification V = V and a complex anti-linear isomorphism between the normal bundles of V and V , we can form the symplectic sum Z = X# V=V Y . This no...