DMS-9424122 Shaw Shaw will continue her work on pde's arising from problems in several complex variables. In particular, she will study the Cauchy-Riemann equations and tangential Cauchy-Riemann equations on complex and CR manifolds. One of the central problems she will work on is to understand the local and global solvability and regularity of the tangential Cauchy-Riemann equations. She will also study the regularity of the Bergman projection and the boundary regularity of the d-bar on nonsmooth pseudoconvex 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.
