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.