 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.With Hitchin he worked on magnetic monopoles, and studied their scattering using an idea of Nick Manton.