Mathematical Sciences: Ricci Curvature Equations and Kahler Geometry

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The principal investigator will study the geometry, topology, and moduli spaces of compact Kahler manifolds with semi-definite first Chern class. Fundamental questions concerning these manifolds will be solved using tools from the theory of Kahler-Einstein metrics. Specifically, the investigator will study a difficult problem posed by E. Calabi. The first task will be to investigate certain associated complex Monge-Ampere equations. The involved methods will include Gromov's compactness theorem, Uhlenbeck's curvature estimates for Yang-Mills connections, and some of Cheeger's work on injectivity radiuses of Riemannian manifolds. In recent years the ties between geometric partial differential equations and the gauge field theories of modern physics have grown strong. Inspired in part by physical problems, the principal investigator will study a variety of difficult questions related to differential geometry. Solution of these will have a profound effect on our deep understanding of the universe.