The PI intends to study complex geometry using the techniques of operator theory. This includes studying analytic subvarieties of bounded domains in several variables, where the theory of vector-valued Hardy spaces introduces extra techniques to several complex variables. The PI also intends to study Hilbert spaces of Dirichlet series, with the intention of using Hilbert space geometry to shed light on classical questions about Dirichlet series.
Feedback mechanisms have been used in engineering for a long time, largely to stabilize systems. The design of such systems is difficult, not least because the system that one wants to stabilize is, in practice, not known exactly, but only approximately. Many ad hoc approaches were used, but about 25 years ago it was realized that many problems can be approached systematically using techniques from an area of mathematics called Operator Theory. This has been so successful that the range of questions engineers now attempt to answer has increased dramatically, and there is a great need to find mathematical answers to the questions now being formulated. One area of importance is the multi-dimensional generalizations of questions that are well understood in one dimension. The PI proposes to work on several such problems.
