9401398 Bedford This award supports mathematical research on problems arising in the theory of several complex variables. The work will be carried out in three directions. In the first, work will be done on applications of the methods of complex analysis to the dynamics of polynomial mappings for two complex variables. The mappings are assumed to be univalent. Central to this study will be the use of pluri-potential theory analogous to the impressive use of potential theory in the study of one-dimensional dynamics. In the second direction, work will continue on the classification of domains with noncompact automorphism groups. Since it is trivial to construct nonsmooth domains with such automorphism groups, attention will be restricted to domains with smooth boundaries. The first two domains addressed will be those which are convex and those with real analytic boundaries. In the third direction, the use of Levi flat surfaces will be studied in connection with the corona theorem for bounded analytic functions. A central goal of the work is the generalization of Wolff's d-bar approach to the corona problem on simply connected domains to multiply connected domains and Riemann surfaces. 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. ***
