RUI: Opportunities in Operator Theory

Pomona College, Claremont CA

Investigators

Abstract

This project involves several interrelated threads that have all born fruit in recent years. The most novel and dynamic aspect of the project is the exciting interplay between the geometric properties of certain discrete objects and the study of continuous transformations between the type of infinite-dimensional spaces that form the natural setting for quantum mechanics. The project will also study the properties and ramifications of certain transformations that enjoy special symmetries. Potential applications are to signal processing, control engineering, and quantum mechanics. The proposed research will build bridges between different areas of the mathematical sciences. To do this, the principal investigator will collaborate with colleagues old and new, as well as sponsor undergraduate research. Indeed, many questions stemming from this project are suitable for undergraduate research and the principal investigator will recruit a diverse array of students to work on them. These students will be equipped with the skills, expertise, confidence, and passion necessary to pursue careers in the mathematical sciences. This project will create not only new mathematics, but also the new mathematicians necessary to enhance and enrich the nation's infrastructure for research and education. A promising aspect of recent research concerns new links between discrete geometry (properties of lattices) and the asymptotic properties of structured matrices (such as Toeplitz matrices). In the other direction, the principal investigator and his collaborators have developed frame-theoretic constructions and operator-related methods to prove new results about lattices. Together with collaborators, the principal investigator will also study complex symmetric operators and truncated Toeplitz operators. Complex symmetric operators are a broad class of operators that has not been adequately studied in generality until recently. The study of truncated Toeplitz operators, a rapidly growing branch of function-theoretic operator theory, has undergone spirited development stemming from a seminal 2007 paper of Sarason. A recent series of articles by the principal investigator and his growing list of collaborators have unearthed surprising links between these two classes. Research on this subject has the potential to be transformative, having relevance to a number of fields. For instance, connections to function theory and matrix analysis have already engaged researchers from both large institutions and small colleges. Moreover, the study of complex symmetric and truncated Toeplitz operators has already proven to be fertile ground for undergraduate research. 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.