Mathematical Sciences: Cyclic Cohomology, Symbols, Local Index, Toeplitz-type Operators and Hyponormal Operators

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Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. At that time, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum mechanics, etc. could be studied productively via the integral equations that model the phenomena. Currently, operator theory has a strong interaction not only with physics, but also engineering and many branches of mathematics. Notable in the latter regard are K-theory, non-commutative differential geometry, index theory, and cohomology. Professor Xia is an expert in all of these aspects of operator theory, with deep and significant contributions to his credit. For many years, before taking residency in the United States, he was an active, prolific leader in Chinese mathematics. He has done outstanding work in function theory, quasi-invariant measures on infinite-dimensional spaces, and topological algebras. He has worked on scattering in quantum field theory, and in harmonic analysis he has collaborated with the great Soviet mathematician Gelfand. In recent years, he has concentrated on operator theory. This work has involved deep connections among hyponormal operators, contractions, and K-theory; cyclic cohomology and almost Lie algebras of operators; almost Lie groups of operators; and difference operators and mathematical physics. The proposed research focuses on three major areas of research. The first involves determining cyclic cohomology and relations among cocycles, symbols, and index. This relates to K- theory, non-commutative differential geometry, and quantum field theory. The second is the study of analytic models of hyponormal operators and subnormal operators, with relationship to invariant subspaces, cohomology, and algebraic geometry. Finally, Professor Xia will study Toeplitz operators and applications to differential equations and pseudo-differential operators.