This project will focus on several areas of mathematical research in the area of several complex variables. Work will continue on the problem of embeddability of three dimensional CR-manifolds, exploring connections between this problem and the Kuranishi construction of versal deformations for surface singularities. An integral part of this investigation is the development of computer software which facilitates the computation of these spaces. The ultimate goal is the construction of normal forms for CR-structures on three dimensional manifolds and the characteristics of embeddable structures. In addition, work will be done to apply pseudodifferential and blow-up methods to study the d-bar operator on domains with singular metrics. The applications include index theorems for Bergman type metrics and infinitesimal versions of the Runge theorem. 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.
