Proposal:DMS-9801626 Principal Investigator: Joseph J. Kohn Abstract: Kohn will continue to do research in the theory of several complex variables, partial differential equations and related topics. In particular, he is interested in: global regularity, hypoellipticity, Holder and L-p estimates, domains with piecewise smooth boundaries, and embeddings of CR manifolds. Kohn is currently working on the subject of hypoellipticity both for boundary behavior in several complex variables and for differential and pseudodifferential operators. His interest is in finding qualitative (be they geometric or algebraic) conditions under which hypoellipticity holds. The technical part of much of this research is connected with estimates which do not "gain" smoothness, so the error terms are of the same magnitude as the object one is trying to estimate. Kohn is developing techniques to deal with this type of difficulty. Since Cardano's work on the cubic equation in the sixteenth century it has been known that complex numbers are needed, as an intermediary step, to find real solutions of certain problems even when such numbers appear neither in the statements nor in the solutions of the problems themselves. Thus, by the middle of the last century, it was recognized that complex analysis is deeply intertwined with many fundamental problems of mathematics, science, and technology. One of the most spectacular instances of this is the use of "Dirichlet's principle" (which arises in fluid mechanics) to understand the foundations of the theory of functions of one complex variable and, in turn, the use of complex analysis in the study of fluid flows. In recent times much of the research in complex analysis has focused on functions of several complex variables, where the counterpart of the Dirichlet principle appears in what is known as Hodge theory and in the so-called d-bar Neumann problem. Kohn is doing research in this area, research that also involves work in partial differential equations, harmonic a nalysis, and geometry. Kohn is currently working on problems of "smoothness" and "propagation of singularities" that arise in the study of partial differential equations.
