Differential Equations in Complex Riemannian Geometry

复杂黎曼几何中的微分方程

• 批准号: 2203607
• 负责人: Jian Song
• 金额: $ 18.55万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203607&HistoricalAwards=false
• 关键词: Differential; Equations; Complex; Riemannian; Geometry

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.

该研究项目侧重于相对于几何和物理学的复杂几何形状和几何流程中的几个开放问题。对这些问题的深刻理解将有助于在研究和物理学的微分方程引起的分析和几何奇异性研究中取得基本进步。该项目还旨在引入来自各个学科的数学研究和教学创新，并对Rutgers以及区域数学社区的本科生和研究生产生直接有益的影响。 PI将继续组织和参加将促进国家教育水平的综合研究/教育计划和活动。 PI将调查爱因斯坦类型的规范指标，这些品种具有轻度奇异性的Kahler品种。特别是，PI将研究Kahler-Einstein歧管的这种奇异指标和分析模量问题的Riemannian几何特性。 PI将通过研究Kahler-Irci流的奇异性在Kahler品种上的有限时间和长时间形成，并继续在RICCI流中取得进展。这种奇异性形成应通过全球和局部度量均匀化来理解，等效于规范的几何手术和异性转变。 PI还旨在扩展他在稳定和不稳定案例中针对复杂的Hessian方程的Nakai-Moishezon标准的工作，并在代数阳性和非线性PDE的条件之间建立联系。 PI将采用几何L2理论，非线性PDE，Cheeger-Colding理论以及Perelman关于RICCI流动的工作的理论和技术。该研究的结果将开发新的工具，并为复杂空间的拓扑，几何和代数结构提供深刻的见解和理解。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子评估来支持的，并具有更广泛的影响。