In this research, Dr. Shaw will continue her investigation of linear partial differential equations which arise from problems in several complex variables. In particular, she will study the solvability, regularity and embeddability of the tangential Cauchy-Riemann operators. These problems are of fundamental importance in the function theory of complex analysis. The global problem is an extension of her results on L2 existence and estimates for b on weakly pseudo-convex boundaries to Holder and Lp estimates. The local problem is to prove with kernel method the existence of a local solution for a system of complex vector fields satisfying the compatibility condition. The embeddability of abstract CR structures will also be addressed. This problem is closely related to the local solvability of b and the b - homotopy formula. Related problems on the range of b and the regularity of the Szego projection will also be studied. The study of these problems wil benefit from interactions with the Wisconsin group which is also actively working on related problems. Dr. Shaw will also teach a one-semester graduate course in Several Complex Variables. Additionally, she will organize a series of seminars by leading women mathematicians and top researchers in Several Complex Variables. This project furthers VPW program objectives which are (1) to provide opportunities for women to advance their careers in engineering and in the disciplines of science supported by NSF and (2) to encourage women to pursue careers in science and engineering by providing greater visibility for women scientists and engineers employed in industry, government, and academic institutions. By encouraging the participation of women in science, it is a valuable investment in the Nation's future scientific vitality.
