The problem of classifying topological vector bundles is an old and difficult one. For example, a classification of complex, topological vector bundles of a given rank on CP^n is not known. Classical invariants, like Chern classes, are not complete: in general, there may be non-isomorphic bundles which have the same Chern data. To see these differences, one must go beyond invariants valued in ordinary cohomology. In this talk, I will discuss a classification of complex rank 3 topological vector bundles on CP^5, using a generalized cohomology theory called topological modular forms (localized at the prime 3). I will also explain how this is analogous to prior work of Atiyah–Rees for rank 2 bundles on CP^3, where real K-theory plays the role of topological modular forms.
The rank 2 on CP^3 and rank 3 on CP^5 classification problems suggest that Chromatic theories at different primes and of different heights may also yield interesting bundle invariants. With this in mind, I will give brief introduction to Hopkins–Miller higher real K-theories, and explain why they are good candidates to distinguish other classes of vector bundles which have the same Chern data.