Mathematical Sciences: Reciprocity for Fredholm 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 theory, etc. could be studied                 productively via the integral equations that model the phenomena.                    Thus, from the fertile minds of Hilbert, von Neumann, and                  other giants the subject of operator theory has grown to a                      central position in such investigations, and in core mathematics                as well.                                                                             At the heart of this methodology is the deep investigation                 of the spectrum of an operator.  For self-adjoint or normal                     operators, this theory is now a standard technique throughout                   analysis, and the spectral theorem provides the necessary                       building blocks for all such operators.  The current frontier,                  therefore, is the study of the structure of non-normal operators,               as for example in the Fredholm theory.  At this frontier,                       Professor Carey and his collaborator, Professor Pincus, are world               leaders with established reputations for deep, conceptually                     innovative, fundamental, interdisciplinary work.  Their theory of               the principal function, defined on the spectrum of certain                      operators, is a broad extension of the ordinary index.  This new                research collaboration has established connections between the                  principal function and the distribution of zeroes of functions                  associated with ergodic flows, the structure of subnormal                       operators and the theory of closed currents in the study of                     function algebras.  This work is particularly fertile as it                     directly combines operator theoretic ideas with other branches of               mathematics, notably commutative ring theory and algebraic                      geometry.  Currently, applications to the theory of commuting                   Toeplitz operators and linear and nonlinear partial differential                operators are envisaged, among other technical applications in                  operator theory.
{'FirstName': 'Richard', 'LastName': 'Carey', 'PI_MID_INIT': 'W', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Richard W Carey', 'EmailAddress': 'carey@ms.uky.edu', 'NSF_ID': '000172390', 'StartDate': '06/15/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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