The study of Hilbert space operators links a wide variety of disciplines, ranging from control theory, approximation theory, quantum mechanics, function theory, signal processing, noncommutative geometry, and matrix theory, to name but a few. The PI will study complex symmetric operators, a broad class of Hilbert space operators which, while encompassing many of the well-known and useful classes, has not been adequately studied in generality until recently. Loosely put, a Hilbert space operator is called complex symmetric if it has a symmetric matrix representation (over the complex field) with respect to some orthonormal basis. This surprisingly large class includes all normal operators, truncated Toeplitz operators (including Jordan model operators and finite Toeplitz matrices), Hankel operators, and many non-normal integral and differential operators (including the classical Volterra operator and certain auxiliary operators produced by the complex scaling method for Schrodinger operators). The PI will examine these operators at the abstract level while also considering a number of specific questions that interface with function theory, matrix analysis, and other areas. For instance, connections to complex analysis have already engaged a number of researchers from both large institutions and small colleges. The PI will collaborate with colleagues old and new, as well as sponsor undergraduate research.
The emerging theory of complex symmetric operators has already proven fertile ground for undergraduate research. Many questions stemming from the proposed project are suitable for undergraduate research and the PI will recruit students from diverse backgrounds to work on them. The PI also plans to take his undergraduate researchers to conferences related to this proposal. The benefits for the students are many. For instance, they get to see mathematicians in their natural element, learn about cutting-edge research that is relevant to their own, interact with researchers in a social context, and speak candidly with graduate students ? something not available at an undergraduate institution. As part of their sustained research experience, student researchers will present posters and/or give talks in venues appropriate for undergraduate research.
The PI conducted research in operator theory, complex variables, functional analysis, and matrix analysis. These branches of mathematics are intimately tied to physics, quantum computing, statistics, and engineering. The PI studied several classes of operators which, while encompassing many of the classical and well-studied varieties, had not been studied in generality until quite recently. Complex symmetric operators are a broad class of operators whose development was spearheaded by the PI. 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 D. Sarason. Recent articles by the PI and his 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. The PIs work on these operators (complex symmetric operators and truncated Toeplitz operators) has been well received and earned him plenary lectures at conferences devoted to matrix analysis, operator algebras, and spectral theory. A substantial amount of energy was focused on mentoring undergraduate researchers and supervising undergraduate research. The PI recruited women and members of underrepresented groups and encouraged them to pursue graduate degrees in the mathematical sciences. Some of these students were involved in projects that developed new techniques in finite Fourier analysis that led to several publications in number theory (an exciting development). The work of several students was profiled in the Pomona College Magazine, the Harvey Mudd Bulletin, and the newsletter of the American Institute of Mathematics. Another project was recenty accepted for publication by the prestigious Notices of the American Mathematical Society.
