Canonical Metrics, the Kahler-Ricci Flow, and Their Applica1ons

拟议的研究工作集中在Kahler几何、几何流和与几何和物理相关的复杂Monge-Ampere方程中的一些公开问题和开发程序。最近的进步和新思想的涌入揭开了分析、偏微分方程、复黎曼几何和代数几何之间深刻、丰富和统一的结构。该项目还旨在引入不同学科在数学方面的研究和教学创新，并对罗格斯大学的本科生和研究生以及区域数学界产生直接的有益影响。首席调查员将继续组织和参与将促进国家教育水平的综合研究/教育方案和活动。此外，首席研究人员计划通过讲座和调查论文向广大受众传播分析和几何界面上令人兴奋的前沿研究。这些项目将研究具有温和奇点的Kahler变种的爱因斯坦类型的正则度量。特别是，主要研究者将研究这类奇异度量和Kahler-Einstein流形的解析模问题的黎曼几何性质。PI将通过研究代数簇上Kahler-Ricci流的有限时间和长时间奇点的形成，继续在带有Ricci流的解析极小模型程序中取得进展。这种奇点的形成是由等价于双态变换的正则几何运算来反映的，应该通过全局和局部度规均匀化来理解。PI将采用L^2理论、非线性偏微分方程组和Cheeger-Colding理论中的新理论和新技术。这项研究将开发新的工具，并对复杂空间的拓扑、几何和代数结构提供深刻的见解和理解。

The proposed research work focuses on a number of open problems and developing programs on canonical metrics in Kahler geometry, geometric flows, and complex Monge-Ampere equations in relation to geometry and physics. Recent progress and influx of new ideas have unraveled a deep, rich, and unifying structure among analysis, partial differential equations, complex Riemannian geometry, and algebraic geometry. The project also aims to bring in research and teaching innovation in mathematics from various disciplines and have an immediate beneficial effect on undergraduate and graduate students at Rutgers as well as in the regional mathematical community. The principal investigator will continue to organize and participate in the integrated research/education programs and activities that will promote the education level of the nation. Furthermore, the principal investigator plans to disseminate the exciting frontier research at the interface of analysis and geometry to a broad audience through lectures and survey papers.These projects will investigate canonical metrics of Einstein type on Kahler varieties with mild singularities. In particular, the principal investigator will study the Riemannian geometric properties of such singular metrics and analytic moduli problems for Kahler-Einstein manifolds. The PI will continue to make progress in the analytic minimal model program with Ricci flow by studying both the finite time and long time formation of singularities of the Kahler-Ricci flow on algebraic varieties. Such singularity formation is reflected by canonical geometric surgeries equivalent to birational transformations and should be understood through global and local metric uniformization. The PI willy employ new theories and techniques from L^2-theory, nonlinear PDEs and Cheeger-Colding theory. The research will develop new tools and give profound insights and understanding of topological, geometric and algebraic structures of complex spaces.