Nekrasov has shown how the low-energy behaviour of certain super-symmetric quantum field theories can be derived from calculating equivariant volumes of moduli-spaces of instantons. From a mathematical point of view this work generates a number of remarkable conjectures in algebraic geometry. We review this and related recent work, and then explain an alternative method in symplectic geometry for calculating the equivariant volumes, using varying compactifications of the ADHM spaces and the Jeffrey-Kirwan-Witten method of non-abelian localization. This cohomological method parallels earlier work in K-theory of Nekrasov-Shadchin, in each case expressing the volumes as the iterated residue of a single rational function. This talk is based upon the preprint math.SG/0609841.