Let A be a commutative unital Banach algebra. Its maximal ideal space Max (A) is a compact Hausdorff space which plays a role somewhat similar to that of the space Spec (R) in algebraic geometry. In particular, A can be represented, not necessarily faithfully, as an algebra of continuous functions on Max (A).
The Taylor problem asks for a construction of the (Cech) cohomology of Max (A) "directly from A". For example, H^1 (Max (A); Z) is the quotient of the invertible group of A by the image of the exponential map. This, and related descriptions of H^0 (Max (A); Z) and H^2 (Max (A); Z), have been known since the 1970s, but the program seemed to stop there.
In this talk, we describe a solution for the _rational_ (Cech) cohomology H^s (Max (A); Q) for arbitrary s, in terms of the rational homotopy groups of the spaces of last columns of invertible n by n matrices over A for suitable n. This is joint work with Greg Lupton, Claude Schochet, and Samuel Smith.