Analytic projections, the geometry of holomorphic vector bundles and applications to the corona problem

This thesis deals with several aspects of the Corona Problem on the disk and polydisk. The Corona Problem on the disk was originally solved by Carleson in the sixties. His proof and methods led to the development of much of the machinery that is used today in complex and harmonic analysis. There have been several proofs of this theorem given, and the one that this thesis most closely resembles is that given by T. Wolff in the late seventies. Wolff's proof focused more on the analytic aspects of the problem while Carleson's dealt more with the geometric aspects. In some sense this thesis serves as a bridge between some of the analysis and geometry behind the Corona Problem.; In Chapter 2 solutions with estimates to the H p Matrix Corona problem on the disk and polydisk are given. In Chapter 3 necessary and sufficient conditions to guarantee the existence of bounded analytic projections are given. The existence of these projections is intimately connected with solutions to the Corona Problem. The general strategy in both problems considered is to first find a trivial solution to the problem and then correct the solution to be analytic, which is usually accomplished through duality.