I will make rigorous, in particular in the case where the symplectic manifold is noncompact, an idea of Witten's to calculate the cohomology of the reduced space of a symplectic manifold with Hamiltonian group action. Specifically, when the group is compact and the moment map is proper, I'll show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, and that certain integrals of equivariant cohomology classes localize as a sum of contributions from critical sets, and bound the contribution from each critical set. When these critical sets are sufficiently well-behaved we can identify the polynomial part of these integrals as evaluations of certain cohomology classes on the symplectic quotient. I will suggest how to use this result to understand the cohomology of moduli space of a Riemann surface (as suggested by Witten, and first done explicitly by slightly different means by Jeffrey and Kirwan). Unlike in this abstract, in the talk I will assume only a knowledge of basic differential geometry.