RUI: New Frontiers in Operators, Functions, and Matrices

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.