Canonical Metrics, Geometric Flows and Formation of Singularities

近年来的新进展和新思想的涌入，揭示了代数几何中分析、黎曼几何、多势理论、经典几个复变量和最小模型规划之间深刻、丰富和统一的结构。提出的研究工作集中在一些开放的问题和开发程序，规范度量，几何流和复杂的蒙日-安培方程产生的几何和物理。拟议中的项目还旨在引入不同学科的数学研究和教学创新，并对罗格斯大学的研究生和本科生以及地区数学界产生直接的有益影响。PI还将组织和参与将提高国家教育水平的综合研究/教育计划和活动。此外，PI计划通过讲座和调查论文向广大受众传播几何，分析和代数界面的令人兴奋的前沿研究。PI将研究并继续在具有Ricci流的解析极小模型程序中取得进展。特别是，PI将研究代数变异上Kahler-Ricci流的有限时间和长时间奇点的形成。这种奇点的形成反映在等价于双域变换的典型几何/解析运算中，应该通过全局和局部度量均匀化来理解。PI还将研究奇异变体上的爱因斯坦型规范度量，特别是奇异度量的黎曼几何性质以及与弦理论应用相关的模问题，如几何转换和镜像对称。PI将采用L^2理论、非线性偏微分方程、佩雷尔曼的著作和Cheeger-Colding理论中的新理论和新技术。拟议研究的结果将开发新的工具，并对宇宙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.