Applications of the Ricci Flow in Differential Geometry

里奇流在微分几何中的应用

• 批准号: 1510401
• 负责人: Sean Paul
• 金额: $ 15.74万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510401&HistoricalAwards=false
• 关键词: Applications; Ricci; Flow; Differential; Geometry

Differential geometry is the branch of mathematics that studies the shapes of spaces through distances and angles. Mathematically these are measured by Riemannian metrics. This research project investigates problems centering around the Ricci flow, which is in turn a natural way to evolve Riemannian metrics in their underlying space. The concept of Ricci flow was invented by R. Hamilton in 1982 and has had wide applications in mathematics since then. For instance, the Ricci flow has had a profound and far-reaching impact on 3-dimensional geometry and topology, as evidenced by G. Perelman's solution to the Poincare conjecture in 2003. The investigator will study the applications of the Ricci flow in algebraic geometry and 4-dimensional topology. These applications will deepen our understanding of several related subjects, including partial differential equations, probability, complex analysis, and theoretical physics. By organizing conferences and seminars, the PI will motivate graduate students to join this research endeavor. Moreover, the PI will present the work to undergraduate students and the general public. The investigator will continue developing tools based on the Ricci flow. In particular, he will refine and localize the existing estimates in Ricci flow, for the purpose of applying Ricci flow in algebraic geometry and 4-dimension topology. He will generalize the pseudo-locality theorem of Perelman, and find the relationships between Ricci flow theory and the theory of metric spaces with lower bound of Ricci curvatures, i.e., the Cheeger-Colding theory. He will continue to develop the special estimates in the Kahler Ricci flow. Among other things, he will aim to combine the flow version of Chen-Lu inequality with other estimates. Based on the improved estimates, the PI will aim to understand the relationship between the minimal model program in algebraic geometry and the Kahler Ricci flow. He will also determine if some of the results in the Cheeger-Colding theory can be reproved using the Ricci flow method. Finally, the PI intends to study the expected deep connections between the Ricci flow and probability theory.

微分几何是数学的一个分支，它通过距离和角度来研究空间的形状。在数学上，这些是由黎曼度量来衡量的。这个研究项目调查的问题围绕里奇流，这反过来是一个自然的方式来发展黎曼度量在其基础空间。 Ricci流的概念是由R.汉密尔顿在1982年，并已广泛应用于数学。例如，Ricci流对三维几何和拓扑学产生了深刻而深远的影响，G。2003年佩雷尔曼对庞加莱猜想的解答。研究者将研究Ricci流在代数几何和四维拓扑中的应用。这些应用将加深我们对几个相关学科的理解，包括偏微分方程，概率论，复分析和理论物理。通过组织会议和研讨会，PI将激励研究生加入这项研究奋进。此外，PI将向本科生和公众展示这项工作。研究人员将继续开发基于Ricci流的工具。特别是，他将完善和本地化现有的估计在里奇流，应用里奇流在代数几何和4维拓扑的目的。他将推广Perelman的伪局部性定理，并找到Ricci流理论与具有Ricci曲率下界的度量空间理论之间的关系，即，奇格冷理论他将继续发展Kahler Ricci流中的特殊估计。除此之外，他的目标是将陈陆不等式的流量版本与其他估计相结合。基于改进的估计，PI将旨在了解代数几何中的最小模型程序与Kahler Ricci流之间的关系。他还将确定，如果一些结果在奇格冷理论可以用里奇流方法reprofied。最后，PI打算研究Ricci流和概率论之间预期的深层联系。