Mathematical Sciences: Analysis of the Quasi-Regular        Representation of Lie Groups and Index Theory on Non-CompactManifolds
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One of the most fundamental developments in modern science has               been the theory of quantum mechanics.  The attempt to understand                both the physics and the underlying mathematics has led directly                to current frontiers in two important areas of Modern Analysis,                 representation theory and operator algebras.  Representation                    theory is a means of exploiting the inherent physical symmetries,               while the concept of operator algebras was invented to provide                  the correct framework for quantization.  Recently, there has been               a great expansion in our understanding of the mathematics                       underlying quantization.  This has come from deep interactions                  between physical ideas and mathematical constructs.  In                         particular, representation theory and operator algebras have been               brought together recently in a powerful theory that centers                     around the celebrated Atiyah-Singer index theorem.                                 Professor Fox is a young researcher who has mastered the broad               range of mathematics necessary to contribute to this field.  He                 is an expert in representation theory, operator algebras, and                   index theory.  He proposes to develop an index theorem for                      noncompact manifolds that generalizes Connes' extension of the                  Atiyah-Singer index theorem for compact manifolds.  This index                  theorem would be used to study the discrete spectrum of the                     quasi-regular representation of a semi-simple Lie group acting on               a locally symmetric space of finite volume.  In addition to its                 intrinsic interest in representation theory and operator                        algebras, a solution to this problem would lead to a better                     understanding of the various facets of quantization.  The                       investigator's prior work, in part collaborative, has laid a                    solid foundation for undertaking this study.
{'FirstName': 'Jeffrey', 'LastName': 'Fox', 'PI_MID_INIT': 'S', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Jeffrey S Fox', 'EmailAddress': 'jfox@euclid.colorado.edu', 'NSF_ID': '000448010', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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