Mathematical Sciences: The Structure of Cycle Spaces, Algebraic Manifolds, and Einstein Spaces

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A team of four geometers will investigate a variety of problems in complex differential geometry, Riemannian geometry, and related areas. The role of spaces of algebraic cycles in universal constructions in topology and homology-cohomology theory will be carefully studied. Work will progress on establishing the existence of Kahler-Einstein metrics on open manifolds, on the structure of p-Kahler spaces in Moishezon geometry, and on the structure of Riemannian 4-manifolds with boundary. The team of investigators will establish more of the ground work necessary for the understanding by physicists of our four dimensional space-time. The connections between the branch of mathematics broadly referred to as "differential geometry," and cosmological physics dates back to Einstein; the mathematical theory which Einstein applied goes back even further to Gauss and Riemann in the nineteenth century. The group being funded here maintains close connections with physicists, and steers their research in directions of current excitement both to mathematicians and physicists.