Recall that a symplectic manifold is a 2n-dimensional smooth manifold M together with a closed nondegenerate 2-form ω. A symplectomorphism is a diffeomorphism of M which preserves ω and an n-dimensional submanifold L ⊂ M is called Lagrangian if ω vanishes on TL. Such structures arise naturally from Hamiltonian dynamics and geometric optics and they have been studied for many decades. The past ten years have seen a number of important developments and major breakthroughs in symplectic geometry as well as the discovery of new links with other subjects such as dynamical systems, topology, Yang-Mills theory, theoretical physics, and singularity theory.

Many of these new developments have been motivated by Gromov's paper on pseudoholomorphic curves in symplectic geometry. The role pseudoholomorphic curves play in Gromov's work is reminiscent of the role of self-dual Yang-Mills instantons in Donaldson's theory on smooth 4-manifolds. Gromov used pseudoholomorphic curves to prove a number of surprising and hitherto inaccessible results in symplectic geometry. For example he proved that there is no symplectic isotopy moving the unit ball in R2n through a hole in a hypersurface whose radius is smaller than 1 (a symplectic camel cannot pass through the eye of a needle). The paper by McDuff and Traynor below gives a proof of this theorem which is based on Eliashberg's techniques of filling by pseudoholomorphic discs.

Moduli spaces of pseudoholomorphic curves also play an important role in McDuff's work on symplectic 4-manifolds.