The principal investigator will continue his research in spectral geometry on Riemannian manifolds. In particular he will consider families of Riemannian manifolds whose joint spectrum determines the metric. Professor Szabo will also continue his investigation of the Desarguesian spaces and their relationship with Hilbert's fourth problem. This award will support research in the general area of differential geometry and global analysis. Differential geometry is the study of the relationship between the geometry of a space and analytic concepts defined on the space. Global analysis is the study of the overall geometric and topological properties of a space by piecing together local information. Applications of these areas of mathematics in other sciences include the structure of complicated molecules, liquid-gas boundaries, and the large scale structure of the universe.