 They solved the simplest unknown case, by showing that all rank 2 vector bundles over projective 3-space have a holomorphic structure.Horrocks had previously found some non-trivial examples of such vector bundles, which were later used by Atiyah in his study of instantons on the 4-sphere.Atiyah, Bott and Vijay K. Patodi gave a new proof of the index theorem using the heat equation.