Analysis of Singularities of the Ricci Flow

微分几何是爱因斯坦相对论中的关键数学。事实上，爱因斯坦用流形上张量的语言写下了他著名的场方程，这些方程后来被数学家和物理学家们很好地研究了。当然，除了作为广义相对论的基础之外，微分几何在其他科学领域也非常有用，比如控制理论、计算机视觉、数据分析等。因此，理解流形的结构是科学中的一个基本问题。本项目将重点研究几何流在流形上的行为。几何流动的一个典型例子是热方程，它描述了热量在一个区域随时间的分布。这个提议研究了一种更高级的热方程形式，称为里奇流。流形上的黎曼度规告诉我们物体的形状，以及如何测量角度和距离。里奇流是黎曼度量的热型方程。人们希望，并且在某些情况下证实，里奇流将把流形上的给定度规演变为改进的度规，例如爱因斯坦度规。然而，这个理论和标准热方程的一个主要区别是里奇流是一个非线性方程，因此它通常在一段时间后出现奇点。当这些奇点被理解后，这个过程可以继续。这在证明长期存在的关于三维流形拓扑的庞加莱猜想中发挥了核心作用。本项目的主要目标是了解四维流形中的奇点，并研究我们的发现对四维流形结构的影响。由于Ricci流可以看作弦理论中的重整化群流，因此本研究在理论物理中还有其他可能的应用。其他相关的流，如平均曲率流，在其他领域有进一步显著的应用，例如在计算机可视化中，用于消除噪声，或在冶金中，用于金属的热处理。该项目的外联部分向公众传播研究成果，并促进青年人才的发展。里奇流是由理查德·汉密尔顿在八十年代早期提出的，他是在一本致力于理解正弯曲三维流形的基础著作中提出的。后来变得很清楚，如果一个人在给定流形上流动一个任意度规，流动通常会发展为奇点。人们需要理解这些奇点，以便继续流，并且不丢失任何关于空间的重要拓扑信息。Ricci流的奇异性用Ricci孤子来表示，Ricci孤子是流的不动点、模微分同态和标度。三维收缩利玛奇孤子通过汉密尔顿、艾维和佩雷尔曼的工作被分类。这对于理解里奇流在三维流形上的整形行为，以及庞加莱猜想的解决具有重要的意义。本课题的主要目标是对四维完全非紧Ricci孤子进行分类。这将通过对这些空间的渐近几何的完全理解和通过研究相应的刚性问题来实现。预计该项目将推进我们对四维里奇流行为的认识，这将使里奇流方法能够解决有关四维流形拓扑的一些重要问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Differential geometry is the key mathematics in Einstein's theory of relativity. Indeed, Einstein wrote his famous field equations in the language of tensors on manifolds, and these equations have since been well studied by mathematicians and physicists alike. Certainly, besides being fundamental in general relativity, differential geometry is also very useful in other fields of science, such as in control theory, in computer vision, data analysis, and many others. For this reason, understanding the structure of manifolds is a fundamental problem in science. This project will focus on the behavior of geometric flows on manifolds. A typical example of a geometric flow is the heat equation, which describes the distribution of heat in a region over time. This proposal studies a more advanced form of the heat equation, called the Ricci flow. A Riemannian metric on a manifold tells us about the shape of that object, how to measure angles and distances. The Ricci flow is a heat-type equation for Riemannian metrics. It is hoped, and confirmed in some cases, that the Ricci flow will evolve a given metric on a manifold to an improved one, such as an Einstein metric. However, a major difference between this theory and that of the standard heat equation is that the Ricci flow is a non-linear equation, and as such it usually develops singularities after some time. When such singularities are understood, the process may be continued. This has played a central role in the proof of the long-standing Poincare conjecture about the topology of three dimensional manifolds. The main goal of this project is to understand such singularities in dimension four, and to investigate the implications of our findings to the structure of four dimensional manifolds. Because Ricci flow can be seen as the renormalization group flow in string theory, there are other possible applications of this study to theoretical physics. Other related flows, like the mean curvature flow, have further remarkable applications to other fields, such as in computer visualization, for eliminating noise, or in metallurgy, for heat treatment of metals. The outreach components of this project disseminate the results to general public and contribute to the development of young talent.Ricci flow was introduced by Richard Hamilton in the early eighties, in a fundamental work devoted to understanding positively curved three dimensional manifolds. It became clear later that if one flows an arbitrary metric on a given manifold, the flow will generally develop singularities. One needs to understand these singularities in order to continue the flow, and to not lose any significant topological information about the space. The singularities of Ricci flow are modeled by Ricci solitons, which are fixed points of the flow, modulo diffeomorphisms and scaling. Three-dimensional shrinking Ricci solitons have been classified through the work of Hamilton, Ivey and Perelman. This has important consequences to understanding the behavior of Ricci flow with surgeries on three-dimensional manifolds, and indeed, for the resolution of the Poincare conjecture. The main goal of this project is to classify four-dimensional complete noncompact Ricci solitons. This will be achieved through a complete understanding of the asymptotic geometry of these spaces and through studying corresponding rigidity questions. It is expected that this project will advance our insight on the behavior of Ricci flow in dimension four, which will enable a Ricci flow approach to some important questions about the topology of four dimensional manifolds.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.