The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking bundle-theoretic questions. However, in general, many non-equivalent bundles represent the same K-theory class. Bridging the gap between K-theory and actual bundles is challenging even for the simplest CW complexes.

For example, given random r and n, the number of rank r bundles on complex projective r-space that are trivial in K-theory is unknown. In this talk, we will compute the p-primary portion of the number of rank r bundles on $\mathbb CP^n$ in infinitely many cases. We will give lower bounds for this number in more cases.

Building on work of Hu, we use Weiss-theoretic techniques in tandem with a little chromatic homotopy theory to translate bundle enumeration to a computation of the higher real K-theory of particular simple spectra. The result will involve actual numbers!  This is joint work with Hood Chatham and Yang Hu.