RUI: Operator Theory and Its Connections

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This proposal revolves around two novel threads, both of which have already borne copious fruit for the PI, his students, and his collaborators in the last few years. The first concerns the dynamic interaction between operator theory (originally developed to provide a rigorous foundation for quantum mechanics), combinatorics (a field that often treats discrete enumeration problems), and probability theory (the study of chance and randomness). The second thread concerns the interplay between discrete geometry (the study of certain highly structured geometric objects) and operator theory. Many of the sub-projects are suitable for undergraduate involvement. The PI will recruit a diverse array of research students to tackle these problems. As a decorated instructor who has earned numerous teaching awards, the PI is able to inspire a broad range of students to pursue degrees in the mathematical sciences. This project will produce new mathematics and new mathematicians.In recent work, the PI and his collaborators used analytic tools familiar in operator theory, harmonic analysis, and random matrix theory to answer virtually all asymptotic questions about factorization lengths in numerical semigroups (staple objects in combinatorics). New probabilistic realizations of certain symmetric multivariate polynomials proved crucial in this work, and also provided proofs of classical positivity results while revealing broad generalizations. In turn, these results promise new applications in operator theory and operator algebras. For example, their combinatorial results suggest a statistical treatment of certain families of AF algebras. The PI also seeks physical and spectral interpretations of the peculiar probability measures that arise when one considers limits of certain combinatorial processes. These measures are of independent interest, for results from computer-aided design (spline theory) have already illuminated some of their properties. The connection between discrete geometry (lattices) and operator theory (Toeplitz determinants and frames) has also proven fruitful. For example, results of the PI and his collaborators on the asymptotics of Toeplitz matrices arose from problems inspired by lattice theory. In the other direction, frame-theoretic constructions and operator-related methods have been used to prove new results about lattices.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.