Stable Polynomials, Rational Singularities, and Operator Theory

This project concerns a classical area of mathematics called mathematical analysis with specific attention on two of its subfields, complex analysis and operator theory. Complex analysis is a mature subject with wide applicability – from mapping the globe to understanding the runtime of algorithms. Operator theory was originally created to study quantum mechanics and has since grown into a similarly mature field with applicability to engineering and optimization. Part of the power of analysis is the ability to convert concrete tasks, such as designing a thermostat or understanding the distribution of prime numbers, into questions about mathematical objects called functions. This project concerns modern fundamental research in these subjects with a focus on questions with an inherent multivariable nature necessarily requiring a deeper understanding of multivariable functions (and more specifically multivariable rational functions). The research will be incorporated into educational roles at both the undergraduate and graduate levels and by mentoring of students on these modern and important areas of analysis. This project focuses on three main areas: (1) characterizing the boundedness of rational functions on domains in several variables, (2) understanding the integrability of rational functions, and (3) systematically determining the asymptotics of coefficients of multivariable rational functions. This research is also closely tied to the theory of stable polynomials, an area which has enjoyed numerous surprising applications in the last decade. Development will be continued of the local theory of stable polynomials in old and new settings to get detailed information about the behavior of a rational function near a singularity. In addition, certain polynomials and rational functions built out of natural operator theoretic constructions such as determinantal representations represent important special cases of interest both in applications and to operator theory. 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.