The talk will be on recent progress in a series of joint papers with Filip Živanović, about a large class of non-compact symplectic manifolds, which includes semiprojective toric varieties, quiver varieties, and conical symplectic resolutions of singularities. These manifolds admit a Hamiltonian circle action which is part of a pseudo-holomorphic action of a complex torus. The symplectic form on these spaces is highly non-exact, yet we can make sense of Hamiltonian Floer cohomology for functions of the moment map of the circle action. We showed that Floer theory induces a filtration by ideals on quantum cohomology. I will explain recent progress on equivariant Floer cohomology for these spaces, in which case we obtain a filtration on equivariant quantum cohomology. If time permits, I will also mention a presentation of symplectic cohomology and quantum cohomology for semiprojective toric varities.