Mathematical Sciences: Boundary Behavior of Holomorphic Functions of Several Complex Variables

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9322326 Stout This award supports mathematical research on aspects of the theory of boundary behavior of holomorphic functions of several complex variables and related questions. One major direction of the work concerns removable singularities for the boundary values of holomorphic functions, that is, when can a function be extended to a point and a neighborhood as an holomorphic function? These points are called removable singularities. A related direction of the work focuses on the study of removable sets for holomorphic functions. These are sets where the natural boundary values of an holomorphic function agree with those of a continuous function. Work will also be done on domains which are not necessarily bounded by smooth manifolds, an area of several complex variables which has not received much attention to date. Finally, work will continue on the study of Cauchy-Riemann functions in terms of one-dimensional slices of their underlying domains. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms. ***