The principal investigator will complete the study of Kahler-Einstein metrics on compact Kahler manifolds. Compact algebraic manifolds with zero first Chern class will be classified. Stable vector bundles will be constructed over these manifolds. Complete Kahler-Einstein metrics whose scalar curvature is either negative or zero will be constructed. The investigators will work toward understanding non-compact complete Kahler-Einstein manifolds with zero Ricci curvature. These are being studied extensively by particle physicists who are interested in superstring theory. In a related investigation, non-collision singularities of the n-body problem from celestial mechanics will be studied. Super-hyperbolic and oscillatory solutions which arise from the final evolution of a gravitational system will be the focus of a special investigation. Two projects will focus on mathematics relevant to particle physics and cosmology. Einstein's dream of formulating a general field theory which encompasses all four basic forces of nature, from strong nuclear to gravitational, will come closer to realization upon completion of this project. Solutions to Newton's equations of planetary motion which predict orbits which go to infinity in finite time will be further investigated.
