Classification problems in equivariant symplectic geometry

In this talk I will discuss some problems related to the classification of some equivariant topological invariants of compact symplectic manifolds with a symplectic circle action. In particular I will present some recent results involving the Chern numbers of the manifold, and show how these can be used to:
(a) classify the equivariant cohomology ring and Chern classes when the action is Hamiltonian and has minimal number of fixed points (joint work with L. Godinho, arXiv:1206.3195 [math.SG]) and
(b) give a lower bound on the number of fixed points when the action is not Hamiltonian and the first Chern class of the manifold vanishes (joint work with A. Pelayo, arXiv:1307.6766 [math.SG]).