Professor Douglas plans to continue his research in two areas: multivariable spectral theory and the study of invariants for differential operators on manifolds with boundary and on foliated manifolds. His research in multivariable operator theory will involve the study of Hilbert modules from the viewpoint of analytic and algebraic geometry. The study of invariants for differential operators will involve the Chern character for operators on a manifold with boundary. This research involves the theory of Hilbert space operators. These are essentially infinite matrices of complex numbers. Such objects have applications in every area of applied science as well as in pure mathematics. This type of research is an attempt to classify certain important classes of Hilbert space operators.
