Operator theory methods in pure and applied mathematics

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

The proposal unifies within a common operator theory framework several investigations of current interest to the community of mathematical analysts. Among the topics under consideration are the following: a study of the multivariate Beurling transform via Plemelj symmetrization and the Poincare variational principle in potential theory; the derivation of sharp estimates for Bergman orthogonal polynomials on disconnected planar domains, which represents a continuation of classical work of Carleman and of Widom; the elaboration of constructive decision methods (based on precise algebraic principles) in a free star-algebra; a detailed study of the qualitative features of moving boundaries under Hele-Shaw-type evolution laws. The completion of this project will involve a combination of operator theory techniques and real algebra, along with potential theoretic and complex analytical methods. The novelty of the project lies partly in the need to refine and upgrade the tools that currently exist for studying such problems. The project addresses several questions that are of current interest because of their relevance to a wide range of nonmathematical disciplines. Such disciplines include, but are not limited to, the following: classical field theory (electrostatic or gravitational); optimization theory as it arises in statistics, engineering, and economics problems; classical and quantum interpretations of fluid mechanics. The Principal Investigator will continue to play an active role in disseminating mathematical ideas among physicists, computer scientists, and economists. This will be accomplished through specialized publications and several scientific meetings that he plans to organize in the near future. The pedagogical component of the project will focus on the supervision of graduate students, the mentoring of postdocs (not confined to individuals specializing in mathematics), and the identification of young mathematical talent, especially among the undergraduates at the Principal Investigator's home institution.