Functor calculus and vector bundle enumeration

Enumerating vector bundles of a fixed rank over a given manifold is a classical question in topology. While vector bundles are stably computable via K-theory, in the unstable range they become much harder to detect.

In this talk, we will demonstrate how the orthogonal/unitary calculus of Weiss—a version of functor calculi—can be applied to enumerate unstable topological vector bundles. We will present counting results for complex vector bundles over complex projective spaces in the metastable range and, time permitting, introduce an equivariant version of the calculus theory along with some potential applications in equivariant geometry. The talk includes joint work with Hood Chatham and Morgan Opie, and with Prasit Bhattacharya.