Work will be done on geometric questions concerned with domains and functions of several complex variables. The principal investigator's work over the course of several grant periods has been concerned with a type of convexity related to the study of function algebras. The class of polynomials defined on a subset of n-dimensional complex space extend to a larger subset, the polynomial convex hull, in such a way that they do not increase in size. When the hull possesses analytic structure, related function algebras have a concrete realization. For example, if the originating set is connected, the hull is analytic (or void) but when the connectedness hypothesis is dropped, the conclusion may fail. But all examples of this type have infinite area. Work will be done in determining the relationship between area and the nature of the hull. There are at present two approaches which will be tried. Another line of investigation will pursue extensions of a remarkable result proved recently by the principal investigator: A holomorphic function of one complex variable, defined in a disc, whose global cluster set has finite linear measure, extends continuously to the boundary of the disc. What remains to be resolved is that of finding the optimal result. That, in itself, requires formulation. Recent work on hulls in the space of two complex variables in collaboration with John Wermer of Brown University will continue. They have had particular success analyzing hulls of sets which lie over one-dimensional discs. That is, the first coordinate belongs to some fixed disc. Sets with this structure fiber over the disc. When the fibers themselves have some geometric character, the hull can be described. For example, when the fibers are convex, the hull is built from graphs of bounded analytic functions. This result led to a new proof of the one-dimensional corona theorem. Thus, it is natural to consider how much of what has been done can be carried over to n-dimensional balls.
