Duistermaat and Heckman found a striking formula, saying that the push-forward of the Liouville measure of a moment map for a torus action is given exactly by the stationary phase approximation (which is in general just an asymptotic expansion rather than exact). Atiyah and Bott showed that this could be deduced from a more general formula in equivariant cohomology, which was a consequence of well-known localization theorems. Atiyah showed that the moment map was closely related to geometric invariant theory, and this idea was later developed much further by his student F. Kirwan. Witten shortly after applied the Duistermaat–Heckman formula to loop spaces and showed that this formally gave the Atiyah–Singer index theorem for the Dirac operator; this idea was lectured on by Atiyah.
duistermaat and heckman found a striking formula saying that the push-forward of the liouville measure of a moment map for a torus action is given exactly by the stationary phase approximation (which is in general just an asymptotic expansion rather than exact) atiyah and bott showed that this could be deduced from a more general formula in equivariant cohomology which was a consequence of well-known localization theorems atiyah showed that the moment map was closely related to geometric invariant theory and this idea was later developed much further by his student f kirwan witten shortly after applied the duistermaat–heckman formula to loop spaces and showed that this formally gave the atiyah–singer index theorem for the dirac operator; this idea was lectured on by atiyah