Mathematical Sciences: Spectral Duality of Differential Operators Affiliated to von Neumann Algebras
Jingbo Xia Principal Investigator
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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 theory, etc. could be studied productively via the integral equations that model the phenomena. In the ensuing years, the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. Also central to Modern Analysis is the related discipline of operator algebras in which one studies collections of operators simultaneously. As the mathematical construct which best transfers the concepts of probability, measure theory, topology, and geometry to noncommutative contexts, operator algebras relate to a rich and important array of applications, from within mathematics itself to physics and microbiology. Professor Xia ia a young specialist in the area of operator theory. Although a recent Ph.D., he has already established an impressive publication record. His research successes include continuity results for the spectrum of periodic elliptic operators, for example Schrodinger operators. This work utilizes the action of the (commutative) n-dimensional vector group on the appropriate operators. He proposes now to extend his spectral duality theory to the action of noncommutative groups on compact spaces, and to the spectrum of the elliptic differential operators associated with the action. This is a difficult project that requires operator theory, operator algebras, geometry, harmonic analysis, and differential equations. However, significant results along the proposed lines would have important applications to the mathematical physics of disordered systems, the connection being through random differential operators.