Mathematical Sciences: Pure and Applied Operator Theory
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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 the                  same 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.  So it has been, that from the fertile minds of                      Hilbert, von Neumann, and other giants that the subject of                      operator theory has grown to a central position in such                         investigations, and in the 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, in the                    study of the structure of operators is in the non-normal theory.                At this frontier, Professor Pincus is a world leader with an                    established reputation for deep, conceptualy innovative                         fundamental, interdisciplinary work.                                                 In his early research career, while at Brookhaven                          Laboratories, Professor Pincus pioneered new ideas and                          techniques, both theoretical and computational, in medical                      tomography.  Notably, his use of the Radon transforms for image                 reconstruction appears to have been the precursor of the more                   well known work of others, for example the Nobel prize work of                  Cormack.  Subsequently, he began his investigation of the                       principal function, which has been one of the main themes of his                career in operator theory.  His theory of the principal function,               defined on the spectrum of certain operators, is a broad                        extension of the ordinary index.  This new object has its own                   topological and stability properties.  His penetrating                          investigations, in part in collaboration with his students and                  others, have established connections between the principal                      functions 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.                         These geometric measure theory techniques have recently led                Professor Pincus to introduce a new index associated with the                   generated operator algebras.  This new index has important                      properties connected with the Fredholm spectrum and its behavior                on singular subvarieties.  The investigation of this theory is                  continued in the current proposal.  Its focus is on the                         systematic introduction of techniques from algebraic geometry                   into operator theory.
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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