Abstract Proposal: DMS-9870164 Principal Investigator: Robert Bryant The principal investigator plans to apply the theory of differential systems and the method of equivalence to problems in differential geometry and mathematical physics that have resisted more traditional approaches, emphasizing two main problems. A Finsler structure on a manifold M assigns a notion of length to each tangent vector in M, leading to a notion of length for paths in M. Riemannian geometry is a special case where the length is derived from an inner product on tangent spaces. Finsler structures are essentially geometrized calculus of variation problems and the fundamental problems involve studying paths that are extremals of length (i.e., the geodesics), their stability properties, their computability, and so forth. As in the familiar Riemannian case, the geometric object that controls stability of geodesics is a sort of curvature tensor, called the flag curvature. Bryant plans to develop classification and global existence theorems for Finsler structures with constant flag curvature, using exterior differential systems techniques. The second main problem arises in certain models of super-symmetric string theory that require the construction on a smooth manifold of a connection with reduced holonomy, perhaps with torsion, out of a Riemannian metric and a three-form. The problem is to classify which pairs of metric and three-form will allow the physical theory to be super-symmetric. Bryant has already done the classification in various low dimensions and is ready to study the intermediate dimensions (six through twenty-six) that are of physical interest, using the techniques of exterior differential systems that contributed to the solution of the holonomy problem in the classical case (in which the three-form was identically zero). Bryant also plans to continue his collaboration with Griffiths and Hsu on the geometry of PDE and their conservation laws and to generalize his recent structure theorems for harmonic morphisms. Optimization is a central problem in mathematics, in which one tries to select the 'best' configuration in the space of possible configurations in a model for a physical system. An example is the problem of navigating on a body of water in which one must take water currents into account in planning the 'best' path from origin to destination, where 'best' is taken to mean 'shortest time of traverse'. A path that is optimal for a short period (a 'geodesic') might not remain optimal if pursued long enough. This is known as instability. (For example, in a river where the current is faster in midstream it turns out that downstream geodesics are stable, but that upstream geodesics are not.) The geometric quantity that measures this notion of stability is known as 'curvature', since it was first identified in studies of the curvature of the Earth. Bryant's work studies curvature and 'over-determined' systems of differential equations, and is relevant to optimization problems in motion planning, control theory, robotics, and string theory models in high energy physics.
