This project is mathematical research in certain areas of function theory. Functions in the plane satisfying rather simple partial differential equations, for instance analytic and harmonic functions, turn out to have properties which make them both useful and fascinating. A fundamental property exhibited by such functions is that their values on a simple closed curve completely determine their values at points inside. One can study harmonic and analytic functions so to speak concretely, or one can take a more abstract approach by forming Banach spaces or algebras of appropriate kinds of functions and applying the methods of functional analysis. The investigators specialize in attacking concrete problems with abstract methods. More specifically, the topics to be investigated include uniform algebras, algebras of bounded analytic functions, the corona problem in the plane, estimates for harmonic measure, and the spectral theory of certain types of Schroedinger operators.
