Work done on this project divides into two categories, problems arising from oscillatory integral theory and those arising from the theory of weights. Both are tied to the analysis of partial differential equations in various guises. The oscillatory integrals reflect the propagation of singularities of linear partial differential equations. Using the FBI-transformation, work will be done in determining regularity of complex vector fields, extending earlier results of Hormander. Other work will focus on Lebesgue and Sobolev estimates for Airy-type integrals applicable to boundary problems for wave equations. Further refinement of research on weighted integrals will be used to obtain estimates on nodal sets and eigenfunctions along lines initiated by Fefferman and Donnelly. It is expected that one can use Carleman-type inequalities to improve estimates on the eigenfunctions. Additional work will done on inverse boundary value problems to seek an existence counterexample for the case where the potential has a large weak type norm. 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 and validate methods for expressing solutions.
