Mathematical Sciences: Overdetermined Systems Defined by Complex Vector Fields

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8903007 Treves Several factors contribute to the close connections which have evolved between the mathematical fields of several complex variables and partial differential equations. One certainly, was the discovery by Hans Lewy in 1957 of a simple, linear partial differential equation of first order which had no solutions whatsoever. The key to the example was the use of complex rather than real-valued coefficients. Part of the continuing work carried out in this project belongs to a long-term project to explain and understand this phenomenon. The context of the research is that of systems of complex vector fields representing overdetermined systems of partial differential equations. One says that such a system represents a differential complex. An algebraic characteristic of complexes is a quantity called the cohomology. Actually there are levels of cohomology; the one of interest here is the p-th cohomology. It is of interest because there is strong evidence to suggest that the vanishing of the p-th cohomology is related to the triviality of the homology, in dimension p-1, of the level sets of the solutions to the associated partial differential equations. A systematic study of this relationship is planned. A second line of investigation will be concerned with obstructions to local solvability of differential equations through the application of homotopy methods. Recent advances in this area indicate that homotopy formulas can be used on certain important differential operators even when the underlying domains do not satisfy even weak pseudo-convexity conditions on the boundary.