Mathematical research undertaken in this project follows two related themes, both within the context of functions of several complex variables. The first is concerned with the question of removable singularities. There is a significant change in the phenomenon of singularities as soon as one goes from one to several variables. Points or sets where holomorphic functions may be extended as holomorphic functions are called removable singularities. One of the goals of this continuing work is to characterize removable singularities by means of the algebra of functions defined in neighborhoods of the singularities. In examples which have been worked out, when the singular set is convex with respect to the algebra, it is removable. A related question to be taken up concerns the characterization of smooth manifolds in the boundary of a domain which are removable. The second thrust of the project concerns the boundary values of holomorphic functions. Two points of view prevail. One is to determine whether a smooth function on the boundary continues holomorphically to the interior, the other is concerned with the question of the extent to which a holomorphic function on the interior extends to the boundary in some reasonable fashion. In the former case, work will concentrate on efforts to determine whether functions which can be continued holomorphically along lower dimensional manifolds, complex lines for example, are holomorphic in the large. Boundary considerations lead naturally to applications of geometric analysis, while the extension theory is closely tied with that of solving systems of partial differential equations.
