The PI will develop operator related function theory and its interaction with algebraic varieties. The main theme is that multi-variable polynomials whose zero sets have a natural relationship to the torus or the polydisk in complex euclidean space offer an interesting setting for the study of complex analysis and operator theory. The goal is to better understand "stable" polynomials (those without zeros on a specified domain), because of their frequent appearance in function theory (as in rational inner functions and interpolation problems), mathematical physics, and engineering, as well as to view algebraic varieties as domains on which to study function theory and operator theory. Function theory on varieties can enrich one variable function theory and while shining light on difficult problems in function theory in several variables.
Much of the work has its intellectual roots in the works of Norbert Wiener, Andrey Kolmogorov, and Arne Beurling (to name just a few) on areas of probability theory and mathematical analysis that formed the mathematical underpinnings of signals analysis (or communications), control theory (as in automatic pilots), and time series analysis (the study of sequential data like stock prices). This project will continue in this long and fruitful tradition by developing and generalizing the underlying mathematics further (most notably by emphasizing relations to algebraic topics). The project should have connections to areas of scientific endeavor with multidimensional data (e.g. an image) as opposed to the previous examples that feature primarily one dimensional data (the one dimension being time).
