Uniqueness of symplectic structures (Q356539)

The fundamental existence problem in symplectic topology is under which conditions a given cohomology class can be represented by a symplectic form. The present survey paper focuses mainly on the uniqueness questions for symplectic forms on compact manifolds without boundary, a problem related to some equivalence relations on the space of symplectic forms. Some precise formulations of relevant questions are given. Many results about symplectic structures in dimension four rely on Taubes-Seiberg-Witten theory. Several examples are provided and some conclusions and conjectures are discussed, as well as the Donaldson geometric flow approach to the uniqueness problem for hyper-Kähler surfaces.