Analysis of Singularities of the Ricci Flow

微分几何是爱因斯坦相对论的核心数学。事实上，爱因斯坦用流形上的张量语言写下了他著名的场方程，这些方程从此被数学家和物理学家们很好地研究。 当然，微分几何除了是广义相对论的基础之外，在其他科学领域也非常有用，例如控制理论、计算机视觉、数据分析等等。因此，理解流形的结构是科学中的一个基本问题。这个项目将集中在流形上的几何流动的行为。几何流的一个典型例子是热方程，它描述了一个区域中热量随时间的分布。 这个提议研究了热方程的一种更高级的形式，称为里奇流。流形上的黎曼度量告诉我们物体的形状，如何测量角度和距离。Ricci流是黎曼度量的热型方程。人们希望，并在某些情况下证实，里奇流将发展一个给定的度量流形上的一个改进，如爱因斯坦度量。然而，这个理论与标准热方程的主要区别在于，里奇流是一个非线性方程，因此它通常在一段时间后发展出奇点。当这种奇异性被理解时，该过程可以继续。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。 这个项目的主要目标是理解四维空间中的奇异性，并研究我们的发现对四维流形结构的影响。 因为里奇流可以看作是弦理论中的重整化群流，所以这项研究在理论物理中还有其他可能的应用。其他相关的流动，如平均曲率流动，在其他领域也有显著的应用，如在计算机可视化中，用于消除噪音，或在冶金学中，用于金属的热处理。该项目的外联部分向公众传播成果，并促进青年人才的发展。里奇流介绍了理查德汉密尔顿在八十年代初，在一个基本的工作，致力于了解积极弯曲的三维流形。后来变得清楚的是，如果一个人在一个给定的流形上流动一个任意的度量，那么这个流动通常会发展出奇点。人们需要理解这些奇点，以便继续流动，并且不丢失任何关于空间的重要拓扑信息。Ricci流的奇异性由Ricci孤子来模拟，它们是流的不动点、模同态和标度。通过汉密尔顿，Ivey和Perelman的工作，对三维收缩Ricci孤子进行了分类。这对理解三维流形上的里奇流的行为有重要的影响，事实上，对于庞加莱猜想的解决也是如此。本项目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整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Ricci流行为的洞察，这将使Ricci流方法能够解决四维流形拓扑结构的一些重要问题。该奖项反映了NSF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.