Differential Equations in Complex Riemannian Geometry

该研究项目集中于复杂几何中的几个公开问题，以及与几何和物理有关的几何流动。对这些问题的深入理解将有助于在几何和物理中研究由微分方程引起的解析奇点和几何奇点。该项目还旨在引入不同学科的数学研究和教学创新，并对罗格斯大学的本科生和研究生以及区域数学界产生直接的有益影响。国际和平研究所将继续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。PI将研究具有温和奇点的Kahler簇上的爱因斯坦类型的正则度量。特别是，PI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Kahler-Ricci流在Kahler簇上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。这种奇点的形成应该通过等同于正则几何手术和二元变换的全局和局部度规均匀化来理解。PI还旨在推广他在稳定和不稳定情况下关于复Hessian方程的Nakai-Moishezon判据的工作，在代数正性条件和非线性偏微分方程组之间建立联系。PI将使用几何L2理论、非线性偏微分方程组、切格-科尔丁理论和佩雷尔曼关于Ricci流的理论和技术。这项研究的成果将开发新的工具，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。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The research project focuses on several open questions in complex geometry and geometric flows in relation to geometry and physics. The deep understanding of these problems will help make fundamental progress in the study of analytic and geometric singularities arising from differential equations in geometry and physics. The project also aims to bring in research and teaching innovation in mathematics from various disciplines and has an immediate beneficial effect on undergraduate and graduate students at Rutgers as well as in the regional mathematical community. The PI will continue to organize and participate in the integrated research/education programs and activities that will promote the education level of the nation. The PI will investigate canonical metrics of Einstein type on Kahler varieties with mild singularities. In particular, the PI 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 finite-time and long-time formation of singularities of the Kahler-Ricci flow on Kahler varieties. Such singularity formation should be understood through global and local metric uniformization equivalent to canonical geometric surgeries and birational transformations. The PI also aims to extend his work on the Nakai-Moishezon criterion for complex Hessian equations in both stable and unstable cases, building connections between conditions of algebraic positivity and nonlinear PDEs. The PI will employ theories and techniques from geometric L2-theory, nonlinear PDEs, Cheeger-Colding theory and Perelman's work on Ricci flow. The outcome of the research will develop new tools and give profound insights and understanding of topological, geometric and algebraic structures of complex spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.