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Quanlei Fang My research --- |
My primary mathematical interests lie in the areas of classical analysis, multivariable operator theory and their applications to systems and control theory. I am also interested in the applications of math in other fields.
The classic Nevanlinna-Pick problem, introduced independently by G. Pick in 1916 and R. Nevanlinna in 1919, considers the interpolation of prescribed values at finitely many prescribed points by a holomorphic function on the unit disk with supremum norm at most 1(a Schur class function). A necessary and sufficient condition for the existence of a solution, given by Pick, is the positive-semidefiniteness of the so-called Pick matrix. The sufficiency of this condition can now be formulated as the assertion that the Szeg\"o kernel is a complete Nevanlinna-Pick kernel. The problem is highly multifaceted and has inspired both pure and applied mathematicians for nearly a century. This is particularly so after its connections with operator theory and linear systems theory/ H-infinity control theory were discovered from 1970s through early 1980s. Recently, Nevanlinna-Pick interpolation theory has been generalized independently to a variety of different settings. The main thrust of my Ph.D. research was the extension of interpolation problems of Nevanlinna-Pick type to multivariable settings, in particular, the Drury-Arveson space on the commutative or noncommutative ball. In the meantime, since research in H-infinity control theory has moved on to the study of robust control for systems with structured uncertainties, and to various types of multidimensional systems, I will also study the connection between interpolation problems and multidimensional systems and control theory.
I have been expanding my research horizon into other parts of multivariable operator theory. On the one hand, I have interest in reproducing kernel Hilbert spaces other than the Drury-Arveson space, for example, Hardy spaces and Bergman spaces. On the other hand, I expect to explore more applications of multivariable operator theory in other fields (systems/control theory, mathematical finance etc.).
When I started my postdoc at SUNY-Buffalo, I began a joint research project with Jingbo Xia, studying Hankel operators on the unit sphere. We have succeeded in characterizing the Schatten class membership of Hankel operators on the unit sphere. Along the way we developed some very useful tools which have potential for solving other problems.
You can download my recent papers Here.
I found the following comics very interesting. The first one is from xkcd.com and the second one is from the book Robustness written by Hansen and Sargent. Mathematics can be very pure and can be very applied too! Enjoy!
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