Mathematical Sciences:  Geometric and Analytic Theory       of Holomorphic Functions of Several Complex Variables
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Work focusing on the mathematical theory of several complex                variables will concentrate on four major directions.  The first                 concerns removable singularities for first order d-bar boundary                 operators.  These differential operators emulate conjugate                      differentiation on the boundary of domains, reducing holomorphic                boundary values to zero.  The ideas may be traced back to                       Riemann's celebrated discovery for the removable singularities of               bounded holomorphic functions of one variable.  Some sharp                      results have been obtained for the case of a ball in two complex                dimensions, but nothing for higher.  In addition, work will be                  done in comparing removable sets for bounded solutions of the d-                bar operator against sets removable for all solutions.                               A second thrust will seek to expand earlier work on                        polynomial convexity from totally real submanifolds to simply                   real manifolds.  The place to begin will be real manifolds                      passing through the origin.  The goal of such research is to                    provide geometric criteria which guarantee polynomial                           approximation of arbitrary continuous functions.                                     In general, real holomorphic functions on real analytic                    manifolds do not extend to be complex analytic.  If one reduces                 the requirement of analyticity to that of subharmonicity or even                pluriharmonicity, extensions are possible.  This work will seek                 to describe the obstructions to extension in terms of the                       geometry of the manifold.                                                            Finally, efforts will be made to obtain boundary uniqueness                results for analytic varieties of dimension at least two.  The                  source of this line of investigation is a simple observation:                   two analytic discs in a two (complex) dimensional ball, bounded                 by curves whose boundaries lie on the surface of the ball either                coincide or the boundaries intersect in very thin sets.  The                    proper statement of this result for higher varieties has been                   elusive, although recent work points to the concept of peak sets                as the proper context for the boundary intersections.
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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