The basic problem Wermer will study is the description of all commuting algebras of operators on a finite dimensional Hilbert space in which the k-fold von Neumann inequality holds for all k. A second part of the project will be to generalize the results of Alexander, Slodkowski, Forstneric and others on the polynomial hull of a set in C^2 lying over the unit circle S, to the corresponding problem when S is replaced by the boundary of the unit ball in C^2. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.