Mathematical Sciences: Invariant Variational Problems and   Quantization
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{'SignBlockName': 'William Y. Velez', 'PO_EMAI': None, 'PO_PHON': None}
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, elementary particle                      physics, and relativity.  In particular, the structures and the                 symmetries are interwoven, as illustrated by the fundamental work               of Einstein, and later Yang-Mills, in which profoundly successful               nonlinear theories for the fundamental forces in nature have been               guided mathematically by the need for a large family of                         symmetries.  In a different direction, mathematicians and                       physicists, often independently, have developed a vast array of                 powerful techniques that classify the linear realizations of the                symmetries.  This is the subject of "representation theory".                            Professor Zuckerman has done pioneering creative work in                many facets of representation theory of Lie groups, and he is now               applying his work to fundamental physics.  Ten years ago he                     invented new ways to construct linear representations of Lie                    groups, his methods drawing from diverse, seemingly unrelated                   areas of mathematics.  Recently, Professor Zuckerman's efforts                  have been directed toward their adaptation to the exciting new                  study of "string theory" in physics.  The goal of this theory is                to provide a unified description of all the forces of nature.  In               the current proposal he will conduct research on both linear and                nonlinear manifestations of symmetry in string theory, as well as               other more conventional physical theories.  This includes the                   development of a new basis for the passage from classical to                    quantum physical theories, and resulting compatibility of both                  theories with Einstein's special and general relativity.                        Symmetries are realized nonlinearly at the classical level and                  linearly at the quantum level.  This dual role of symmetry is a                 fundamental aspect of his work.
{'FirstName': 'Gregg', 'LastName': 'Zuckerman', 'PI_MID_INIT': 'J', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Gregg J Zuckerman', 'EmailAddress': 'gregg.zuckerman@yale.edu', 'NSF_ID': '000438342', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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