In this talk I will discuss some problems related to the classification of some equivariant topological invariants of compact symplectic manifolds with a Hamiltonian circle action. In particular I will present some recent results of my work currently in progress, as well as those obtained with L. Godinho in arXiv:1206.3195 [math.SG], involving the Chern numbers of the manifold, and show how these can be used to classify the equivariant cohomology ring and Chern classes of the manifold.