Mathematical Sciences: Harmonic Analysis on Reductive Lie   Groups
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The theory of Lie groups, named in honor of the Norwegian                  mathematician Sophus Lie, has been one of the major themes in                   twentieth century mathematics.  As the mathematical vehicle for                 exploiting the symmetries inherent in a system, Lie theory has                  had a profound impact upon mathematics itself and theoretical                   physics, especially quantum mechanics and elementary particle                   physics.                                                                             The representation theory of a class of such groups called                 semi-simple Lie groups, is one of the most important and                        difficult aspects of the subject, as it is interwoven throughout                the core of modern mathematics and mathematical physics.  At the                heart of this representation theory is the profound work of                     Harish-Chandra from 1955 to 1968, culminating in his Plancherel                 formula which described the harmonic analysis of square                         integrable functions.  That work left open the major analytic                   problem of the harmonic analysis for p between 0 and 2.  In                     technical terms this is the problem of characterizing the Fourier               transforms of tau-spherical functions for general p.  Professor                 Trombi is one of the few researchers in the world who is                        currently in pursuit of this difficult and important problem,                   fraught with imposing technical obstacles, and he has the best                  results to date.  In earlier work, he solved the case p=1, and                  more recently-under the previous award -- Professor Trombi                      completely characterized the Schwartz space for general p, a                    major piece of work,  His current investigations concern                        determining the intersection of these spaces, simplifying the                   Arthur-Campoli conditions in the Paley-Weiner theorem, and                      investigating weighted orbital integrals.  This is mathematical                 analysis of the most demanding technical nature.
{'FirstName': 'Peter', 'LastName': 'Trombi', 'PI_MID_INIT': 'C', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Peter C Trombi', 'EmailAddress': None, 'NSF_ID': '000107510', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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