Recall that a square matrix P is called a projection matrix if P^2 = P.  It makes sense to talk about projection matrices with coefficients in any commutative ring; the image of a projection matrix is called a projective module.  This seemingly innocuous notion intercedes in geometric questions in the same spirit as the famous Hodge conjecture because of Serre's dictionary: projective modules are ``vector bundles''. If X is a smooth complex affine variety, we can consider the rings of algebraic or holomorphic functions on X.  Which of the holomorphic vector bundles on X admit an algebraic structure?  I will discuss recent progress on these questions, using motivic homotopy theory, and based on joint work with Tom Bachmann and Mike Hopkins.