This award supports cooperative research on partial differ- ential equations and complex variables between Dr. Francois Treves of Rutgers University and Dr. P. D. Cordaro of the University of Sao Paulo, Sao Paulo, Brazil. The work will investigate the existence of local solutions in the differ- ential complexes originating from commuting systems of complex vector fields (on a smooth real manifold). Recently Cordaro and Treves have obtained general necessary conditions, at least when the rank of the characteristic set is . 1. Cordaro & Treves are currently investigating the possibility of proving sufficient conditions, in particular in the important case of Cauchy-Riemann structures. There is a natural conjecture that awaits proof, in the case of top-degree forms: local solvability ought to hold if the Levi form is not definite (positive or negative) at any point. This type of result is an essential part of the completion of the theory of hypo-analytic structures that has been building for the last ten years, through the efforts of Treves and his collaborators. The award will allow Cordaro & Treves to get together at regular intervals to work on this project. Cordaro has funds from CNPq NSF's corresponding agency in Brazil to visit Rutgers University. Treves and a graduate student will visit Sao Paulo in return.
