Topological vector bundles on complex projective spaces

Given two complex topological bundles over $\mathbb CP^n$, one can ask whether the bundles are topologically equivalent. The first test is to compare their Chern classes, since equivalent bundles must have the same Chern data. The converse fails in general, which leads to the following question: given $n$ and $k$ positive integers, what invariants beyond Chern classes are needed to distinguish complex rank $k$ topological bundles on $\mathbb CP^n$, up to topological equivalence? In this talk, I will discuss the subtleties of using methods from stable homotopy theory to answer this question. I'll start by explaining how Atiyah--Rees classified all complex rank 2 topological vector bundles on $\mathbb CP^3$ via an invariant valued in the generalized cohomology theory of real K theory. I will then discuss my work classifying complex rank 3 topological vector bundles on $\mathbb CP^5$ using a generalized cohomology theory called topological modular forms. As time allows, I will discuss work in progress extending this work to other ranks and dimensions.