Differential Equations in Complex Riemannian Geometry

复杂黎曼几何中的微分方程

基本信息

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

项目摘要

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.

该研究项目重点关注与几何和物理学相关的复杂几何和几何流中的几个开放性问题。对这些问题的深入理解将有助于几何和物理学中微分方程引起的解析奇点和几何奇点的研究取得根本性进展。该项目还旨在引入各个学科的数学研究和教学创新，并对罗格斯大学以及地区数学界的本科生和研究生产生直接的有益影响。 PI将继续组织和参与综合研究/教育计划和活动，以提高国家的教育水平。 PI 将研究具有轻微奇点的 Kahler 簇的爱因斯坦型规范度量。特别是，PI 将研究此类奇异度量的黎曼几何性质以及卡勒-爱因斯坦流形的解析模问题。 PI 将通过研究卡勒簇上 Kahler-Ricci 流奇点的有限时间和长期形成，继续在 Ricci 流解析最小模型程序方面取得进展。这种奇点的形成应该通过相当于规范几何手术和双有理变换的全局和局部度量均匀化来理解。 PI 还旨在扩展他在稳定和不稳定情况下复杂 Hessian 方程的 Nakai-Moishezon 准则方面的工作，在代数正性条件和非线性偏微分方程之间建立联系。 PI 将采用几何 L2 理论、非线性偏微分方程、Cheeger-Colding 理论和 Perelman 的 Ricci 流工作中的理论和技术。研究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。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。