A symplectic structure on a manifold is a closed non-degenerate 2-form. Each of these conditions is easy to understand individually; it is the interaction between the two which makes the problem of existence and classification of symplectic manifolds and interesting one. In this talk we report on some recent developments - particularly in 4 dimensions - in this area. We describe work of the speaker on the existence of codimension-2 symplectic submanifolds and work of C. Taubes relating the new Seiberg-Wihen invariants of symplectic 4-manifolds to Gromov’s pseudo-holomophic curves.

Important features of both of these developments are the extension of ideas from Kahler geometry to the symplectic situation, and the role of curvature of complex line bundles. There are interesting problems to understand in the extension of these ideas from 4 to higher dimensions and we discuss these briefly.