Let a torus $T$ act on a compact symplectic manifold $M$ in a Hamiltonian way, with discrete fixed point set $M^T$ . Then the moment map can be used to construct a basis for the $T$-equivariant cohomology of $M$ (as a module over the equivariant cohomology of a point). In recent work, R. Goldin and S. Tolman used the symplectic structure of $M$ to compute the restriction of the elements of this basis to the fixed points of the action, which gives all the information about the equivariant cohomology ring of $M$, since in this case the restriction map to the equivariant cohomology of the fixed point set is injective. I will first explain recent results obtained jointly with S. Tolman on how it is possible to simplify their formula using special equivariant symplectic fibrations, and get in some nice cases positive integral formulas. Then I’ll focus on generic coadjoint orbits, where positive integral formulas are already known, and explain how to use “GKM fiber bundles” to get positive integral formulas, which are not equivalent to the ones found previously.