RUI: new frontiers in operators, functions, and matrices

Pomona College has been awarded $285,236 under a Project Grant from the National Science Foundation (NSF) Division of Mathematical Sciences (CFDA 47.049: Mathematical and Physical Sciences) for the period July 1, 2025, through June 30, 2028. The project, titled "New Frontiers in Operators, Functions, and Matrices," delivers research and educational products that integrate operator theory, linear algebra, combinatorics, and probability theory. The principal investigator and undergraduate student researchers will produce novel theoretical results addressing asymptotic questions in numerical semigroups, develop new probabilistic realizations of symmetric multivariate polynomials, and advance the understanding of approximately finite dimensional algebras through statistical methods. The work is designed to generate peer-reviewed publications, establish new connections between operator theory and combinatorics, and introduce innovative graph products and associated mathematical frameworks applicable to both pure and applied mathematical domains.

The award explicitly emphasizes undergraduate involvement and research training as a key deliverable. By recruiting and engaging students in addressing interdisciplinary mathematical problems, the project contributes to the NSF's broader impact mission of developing the next generation of mathematical scientists. No sub-awards are planned for this project. The research outcomes are expected to yield applications in operator algebras and provide novel proofs of classical positivity results with potential generalizations across multiple mathematical fields.

Abstract

This project concerns the dynamic interaction between operator theory (originally developed to provide a rigorous foundation for quantum mechanics), linear algebra (the language of data science), combinatorics (a field that often treats discrete enumeration problems), and probability theory (the study of chance and randomness). Many sub-projects are suitable for undergraduate involvement. The PI will recruit a number of students to tackle these problems. 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 novel proofs of classical positivity results while revealing broad generalizations. In turn, these results promise new applications in operator theory and operator algebras. For example, our new combinatorial results suggest a statistical treatment of certain families of approximately finite dimensional algebras. Certain multivariate symmetric polynomials that arise in this work gives rise to random vector norms, which connect traces of noncommutative *-polynomials to combinatorics and probability theory. Continuing the combinatorial theme, symmetrized tensor products of certain non-normal operators provide an exciting new sandbox for function-theoretic operator theorists. A natural byproduct of these investigations is a new graph product that suggests a host of open questions. 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.- subawards are not planned for this award.