Mathematical Sciences: Spectral Duality of Differential     Operators Affiliated to von Neumann Algebras
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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.
{'FirstName': 'Jingbo', 'LastName': 'Xia', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Jingbo Xia', 'EmailAddress': 'jxia@acsu.buffalo.edu', 'NSF_ID': '000168012', 'StartDate': '07/15/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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