Title of Course: The Structure of Hilbert Modules

Course Description: One approach to the study of multivariate operator theory on Hilbert space is by studying Hilbert spaces which are modules over certain natural algebras.  Although one could consider noncommutative algebras we focus on the commutative case for such algebras as the $\mathbb{C}[z_1 ,...,z_m ]$ or function algebras over bounded domains in $\mathbb{C}^m$.  The course will start with a review of single operator theory from this point of view setting the stage for studying the multivariate case.  Techniques which will be developed include those from several complex variables, commutative algebra, and analytic geometry.  There's an intimate relation between these topics and complex differential geometry.  Two open problems that will be considered are the so-called Arveson conjecture and the corona problem. For the former we will review the results obtained so far and describe further possibilities.  For the corona problem we will show its interpretation in the language of Hilbert modules and review existing results in this context.  Finally, we will develop the parallel in this context of the results of Quillen-Suslin in algebraic $K$-theory in modules over the polynomials.