Miller 9505170 The proposed research deals with various aspects of geometric partial differential equations. Linear elliptic differential operators, gauge field equations, and Yang-Mills-Higgs equations are among the topics to be pursued. Methods from topology, geometry and analysis are to be employed: variational techniques, moduli space techniques, and index theory are to be applied to tackle these partial differential equations. The proposed research may have consequences in smooth manifold theory by uncovering new topological invariants of manifolds. Gauge fields arise naturally in theoretical physics as attempts to unify basic forces of nature. Yang-Mills-Higgs equations have applications in early universe physics and the theory of galactic formation.