Abstract: I will review the paper "Superconnections, Thom classes, and equivariant differential forms" by Mathai and Quillen, and explain how their results can be used to simplify the cohomological formula for the index of an elliptic operator on a compact manifold.

Abstract: We introduce a new cohomology theory constructed using homotopy theoretic methods that bears some formal resemblance to K-theory -- it is equipped with Adams operations, a form of periodicity, and an analogue of the J-homomorphism. However, the information that it encodes is of a higher "chromatic level" than K-theory, and so may be suitable for studying higher chromatic phenomena in stable homotopy theory. Unfortunately, its geometric meaning is far from apparent, although there are some hints of a relationship with n-gerbes.