RUI: Representations of Analytic Functions in Several Variables and Applications

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This project sits at the intersection of operator theory and complex analysis, two areas of mathematics. Complex analysis is concerned with functions involving complex numbers, which have broad use in mathematics, physics, and engineering applications. Complex analysis is often used when systems (such as a circuit) are encoded into a function which is studied as a means of understanding the system. A common approach in the analysis of functions is to transform them into more readily accessible objects. In classical analysis, this transformation is often an integral transform, such as the Fourier transform, where members of the same family of functions are transformed into a form with a common structure (called a kernel) and a geometric object that encodes individual behavior (called a measure). Another approach to studying families of functions representing systems common in engineering is called realization, which turns a system into a simple form involving matrices that represent properties of the system being studied. For functions with more than one input, both approaches are enveloped by operator theory, which arose from the mathematical foundation of quantum physics. This project will use operators to study representations of more general families of functions and apply them to questions in several complex variables. There are many questions in this project that will be explored by groups of student researchers.Specifically, this project is concerned with representations of functions that are bounded on certain domains as expressions of operators acting on associated Hilbert spaces and how the behavior of these functions at the boundary of their domains is encoded in the representations. Examples include the Pick functions, which play an important role in interpolation, probability, and engineering, and which possess well-known and useful representations in one variable (such as the Nevanlinna representation). One goal of the project is to extend methods of operator realizations into more general settings, including understanding concrete formulas for the operators in realizations and detailing the interplay between operator components and function regularity (in parallel with boundary measures in the classical theory) for functions that take matrix inputs. A second topic of the project is boundary problems in two or more variables, such as analogues of the Denjoy-Wolff problem on iteration of bounded functions on the complex disk. The connected goals of this project are motivated by the important historical and continuing interplay between the representation of functions bounded on special domains and the study of boundary properties. This research has potential applications in several complex variables and free 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.