This project continues mathematical research on the existence and smoothness of solutions of overdetermined systems of first-order partial differential equations, both linear and nonlinear. It is striking that, in contrast to ordinary differential equations, there are simple linear partial differential equations possessing no solution. The analysis of this, and related basic questions, lies at the heart of this research. The work, phrased in terms of transformed equations known as pseudodifferential operators, seeks to obtain information about local solvability of systems of equations. Related work is going forward on the microlocal solvability of first order nonlinear equations through study of the singularities of the equations known as the analytic wave-front set. What has been discovered so far is that the singularities propagate exactly as in the linear case. A new line of investigation has also begun. This considers studies of systems of complex vector fields on a manifold. It mixes classical ideas of differential operators with cohomology theory on the manifold which can be used to define generalized characteristic classes. The first objective will be to examine the implications of the nonvanishing of those classes. Also to be considered are the geometrical properties absence of due to this nonvanishing. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.
