Canonical Metrics, Geometric Flows and Formation of Singularities

最近的进展和新思想的涌入揭开了分析、黎曼几何、多势理论、经典多复变量和代数几何中的最小模型程序之间的深刻、丰富和统一的结构。研究工作集中在一些公开问题和开发程序上，这些问题涉及几何和物理中的正则度量、几何流和复杂的Monge-Ampere方程。拟议的项目还旨在引入不同学科在数学方面的研究和教学创新，并对罗格斯大学的研究生和本科生以及区域数学界产生直接的有益影响。国际和平研究所还将组织和参与促进国家教育水平的综合研究/教育方案和活动。此外，PI计划通过讲座和调查论文向更广泛的受众传播几何、分析和代数界面上令人兴奋的前沿研究。PI将研究并继续在带有Ricci流的解析最小模型程序中取得进展。特别是，PI将研究代数簇上Kahler-Ricci流的有限时间和长时间奇点的形成。这种奇点的形成由等价于二次变换的正则几何/解析运算来反映，并且应该通过全局和局部度规统一来理解。PI还将研究奇异簇上的爱因斯坦类型的正则度量，特别是这种奇异度量的黎曼几何性质和相关的模问题，以及在弦理论中的应用，如几何跃迁和镜像对称。PI将运用L^2理论、非线性偏微分方程组、佩雷尔曼著作和契格-科尔丁理论中的新理论和新技术。拟议的研究成果将开发新的工具，并对宇宙的结构以及许多其他应用科学提供深刻的见解。

Recent progress and influx of new ideas have unraveled a deep, rich and unifying structure among analysis, Riemannian geometry, pluripotential theory, classical several complex variables and the minimal model program in algebraic geometry. The proposed research work focuses on a number of open problems and developing programs on canonical metrics, geometric flows and complex Monge-Ampere equations arising from geometry and physics. The proposed project also aims to bring in research and teaching innovation in mathematics from various disciplines and have an immediate beneficial effect on graduate and undergraduate students at Rutgers as well as in the regional community of mathematics. The PI will also organize and participate in the integrated research/education programs and activities that will promote the education level of the nation. Furthermore, the PI plans to disseminate the exciting frontier research at the interface of geometry, analysis and algebra to a broad audience through lectures and survey papers. The PI will investigate and continue to make progress in the analytic minimal model program with Ricci flow. In particular, the PI will study 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/analytic surgeries equivalent to birational transformations and should be understood through global and local metric uniformization. The PI will also investigate the canonical metrics of Einstein type on singular varieties, in particular, the Riemannian geometric properties of such singular metrics and related moduli problems with applications in string theory such as geometric transitions and mirror symmetry. The PI willy employ new theories and techniques from L^2-theory, nonlinear PDEs, Perelman's works and Cheeger-Colding theory. The outcome of the proposed research will develop new tools and give profound insights of the structure of the universe as well as many other applied sciences.