RUI: Rational Schur Functions and their Applications

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This project lies at the intersection of operator theory and complex analysis, two areas of pure mathematics. Operator theory began with the study of matrices (arrays of numbers), and by the early twentieth century, had given much of the mathematical foundation for quantum mechanics. Since then, operator-theoretic techniques have had several applications, especially in constructing and studying a variety of useful functions, i.e., rules that encode specific types of information. For example, in control engineering, setups such as autopilot systems that accept inputs, receive feedback, and release outputs can be modeled with general matrices (or operators), while a system’s key information is often encoded in its transfer function. Recently, operator theory techniques have also been used to decompose signals into simple pieces, via something called a wavelet construction. In this project, the principal investigator will use operators to study problems related to a specific class of functions, called rational functions, which appear in such applications. Many of these problems have parts that are amenable to undergraduate research, and much of the research in this project will be explored alongside diverse groups of undergraduate researchers.More precisely, this project concerns rational functions in one and more variables that are bounded on certain domains. Such rational functions include finite Blaschke products (FBPs), which have played significant roles in factorization, interpolation, and approximation problems in complex analysis. Recently, FBPs have arisen in the context of a famous open problem in operator theory called Crouzeix’s conjecture, which basically says that the numerical range of a bounded operator on a Hilbert space is a 2-spectral set for that operator. The first goals of this project are to use FBPs to investigate and solve various cases of Crouzeix’s conjecture, use Crouzeix’s conjecture to ask and answer new questions about FBPs, and to explore related questions about spectral sets, compressed shifts, and truncated Toeplitz operators. The second topic of this project concerns multivariate (rational and non-rational) functions on the bidisk and polydisk. Specifically, the second main goal is to use model/realization theory to systematically characterize the structure and fine boundary regularity of bounded analytic functions on the bidisk and polydisk. This goal is motivated by recent work on both the structure of rational inner functions near boundary singularities and new canonical realization formulas for general bounded analytic functions.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.