Free Analysis: Exploring the Interactions between Operator Theory and Noncommutative Function Theory

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This project belongs to the branches of mathematical analysis known as Operator Theory and Functional Analysis. These subjects were developed initially in the early part of the 20th century as part of the development of the mathematical foundations of quantum mechanics. These ideas have subsequently evolved in many unexpected directions, far from their original source, with applications not only in physics (such as the currently very active areas of quantum computing and quantum information theory), but also in electrical and mechanical engineering (where these ideas are applied in the design of automatic control systems, in signal and image processing), and even artificial intelligence and machine learning. A particularly new and exciting branch of this field is known as “noncommutative function theory” which has its origins in the study of certain kinds of optimization questions in engineering, but has grown to take on a life of its own. The adjacent area of “multivariable operator theory” is multi-faceted but is closely connected with many questions arising in these applications (such as the study of quantum channels, and the theory of “linear matrix inequalities” in optimization). The project is aimed at expanding the array of mathematical tools available for the study of these questions, and at deepening our understanding of the interplay between these diverse mathematical ideas. The project further will integrate research and education, and professional development of junior researchers. A “Math Circle” program at a local school is to be carried out.The project will employ a blend of techniques from operator theory (especially in several variables), functional analysis, and complex analysis to study the interrelation between operator theory and the rapidly developing field of "free analysis" or "noncommutative" function theory. These recent, rapid developments have made new tools available to the study of some basic questions in mathematical analysis and have already found applications to diverse areas of mathematics, including optimization theory, convex analysis, and quantum information theory. The goal of joint work with graduate students is to apply these new methods to questions at the intersection of functional analysis and complex function theory, particularly the study of function-theoretic operator theory in "noncommutative" domains, including questions about factorizations, zero sets, and realization theory for noncommutative rational functions. The project will draw on techniques from several currently active areas of mathematical analysis, with the aim of broadening and deepening our understanding of the interplay between function theory, operator theory, and complex analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.