Mathematical Sciences: Ricci Curvature Equations and Kahler Geometry

主要研究具有半定第一类的紧致Kahler流形的几何、拓扑和模空间。关于这些流形的基本问题将使用卡勒-爱因斯坦度规理论中的工具来解决。具体地说，调查员将研究E.Calabi提出的一个难题。第一个任务是研究某些相关的复Monge-Ampere方程。所涉及的方法包括Gromov的紧性定理，Uhlenbeck的杨-Mills联络的曲率估计，以及Cheeger关于黎曼流形的内射半径的一些工作。近年来，几何偏微分方程组和现代物理学的规范场理论之间的联系变得越来越紧密。在一定程度上受到物理问题的启发，首席研究员将研究与微分几何有关的各种难题。这些问题的解决将对我们对宇宙的深刻理解产生深远的影响。

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.