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.