Mathematical Sciences: Complex Geometry and Representation  Theory of Lie Groups
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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 and elementary particle                   physics.  The abstract theory of representations of Lie groups                  provides a list of the minimal linear realizations -- or in                     technical jargon, the irreducible representations --  of the                    group.  These form the building blocks for all such realizations.               Often, Lie groups arise as motions of geometric objects, as for                 example, the orthogonal group is the group of distance preserving               motions of the sphere.  In order to study the fine structure of                 such representations, or of the group itself, it is necessary to                obtain realizations of the representation that reflect the                      inherent geometric structure.  This has been an area of                         fundamental research in noncommutative harmonic analysis for                    several decades.  Professor Rossi is an expert in this interface                of representation theory and geometry, especially complex                       geometry.                                                                               For a large class of Lie groups, the most accessible set of             irreducible representations is the holomorphic discrete series,                 discovered in the 1950's by Harish-Chandra.  However, there are                 many more representations of this type -- called highest weight                 representations -- that only recently have been realized                        concretely.  Some of these representations, said to lie in the                  analytic continuation, relate directly to elementary particle                   physics and also to the study of differential equations.                        Professor Rossi has been in the forefront of the discovery and                  investigation of such representations.  Recently, he observed                   that these geometric contexts would allow considerable                          generalization.  This point of view connects representation                     theory with analysis and geometry of several complex variables.                 In his present research, Professor Rossi intends to explore these               constructions and make precise which representations arise on                   vector bundles of forms on invariant domains in Grassmanians, and               as far as possible develop the analytic tools to make a complete                study of the structure of these representations.  For example,                  the representations which arise in this way are                                 subrepresentations of the tensor product of holomorphic discrete                series and adjoints, and they should be in the analytic                         continuation of non-holomorphic discrete series.
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{'Code': '1262', 'Text': 'MODERN ANALYSIS'}
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