Symplectic manifolds play a central role in modern topology. Most notably, the revolution initiated by Seiberg-Witten invariants and the results of Taubes, Kronheimer, Mrowka, Fintushel and Stern have made the study of four-dimensional symplectic manifolds a central topic of low dimensional smooth topology. Fundamental results on the topology of symplectic manifolds have also been obtained by M. Gromov, Y. Eliashberg, R. Gompf, D. McDuff, D. Salamon and G. Tian, among many others. Their work has allowed the subject to mature into one of the most exciting parts of modern mathematics.
More recently, the existence result obtained by Donaldson for symplectic Lefschetz pencil structures on symplectic manifolds has opened a completely new direction in low dimensional symplectic topology, followed by work of Auroux and Katzarkov on the topology of 4-dimensional symplectic manifolds viewed as finite ramified coverings of CP2. These results allow one, in principle, to classify symplectic 4-manifolds using combinatorial data in certain groups (mapping class groups or braid groups respectively). They also make the study of four-dimensional symplectic manifolds accessible to methods from algebraic geometry.