Ricci Curvature and Ricci Flow

这项提议有两个主要主题。第一个是研究一般意义下具有非负Ricci曲率的度量空间。许多关于光滑空间的Ricci曲率的已知结果都是对非光滑空间的推广。然而，也有许多关于光滑空间的结果，它们的非光滑扩张是不清楚的。建议的研究将探索非光滑空间的曲率性质与光滑空间的曲率性质相似的程度。第二个主要话题是利玛窦的流动。李氏流是由R·汉密尔顿在20世纪80年代由S提出的，它是一种通过空间的利玛奇曲率来演化空间“形状”的方法。最近，Perelman惊人地利用Ricci流解决了三维拓扑学中最重要的问题。瑞奇流动的长期行为在很大程度上是未知的，将在提案中得到解决。此外，还将考虑将Perelman的结果推广到三维奥氏球上。总的来说，该建议涉及曲率的概念。在历史上，曲率首先被考虑为平面中的曲线，然后是三维空间中的曲线和曲面。Riemann展示了如何理解任意维光滑空间的曲率。事实上，有不同的曲率概念--由Riemann定义的截面曲率和Ricci曲率，后者是截面曲率的平均值。广义相对论的爱因斯坦方程是用Ricci曲率表示的。自从Riemann时代以来，人们就一直对理解非光滑空间的曲率感兴趣。Alexandrov很好地解释了非光滑空间具有非负截面曲率的含义。最近与塞德里克·维拉尼合作的工作给出了一个很好的概念，即非光滑空间具有非负Ricci曲率意味着什么。它的定义是“最优运输”，这是一门在应用数学方面有悠久历史的学科。提案中的部分研究使用“最优运输”来研究非光滑空间的几何形状。反过来，几何学的观点将被用来解决“最优交通”中的问题。

This proposal has two main topics. The first is to study measured metric spaces with nonnegative Ricci curvature in a general sense. Many of the known results about Ricci curvature for smooth spaces have extensions to nonsmooth spaces. However, there are also many results known for smooth spaces, for which the nonsmooth extension is unclear. The proposed research will explore the extent to which the curvature properties of nonsmooth spaces resemble those of smooth spaces. The second main topic is Ricci flow. The Ricci flow, introduced by R. Hamilton in the 1980's, is a way to evolve the "shape" of a space by means of its Ricci curvature. Recently, Perelman has made spectacular use of the Ricci flow to address the most important problems in three-dimensional topology. The long-time behavior of the Ricci flow is largely unknown and will be addressed in the proposal. In addition, the extension of Perelman's results to three-dimensional orbifolds will be considered.Overall, the proposal is concerned with the idea of curvature. Historically, curvature was first considered for curves in the plane, and then for curves and surfaces in three-dimensional space. Riemann showed how to make sense of the curvature of a smooth space of arbitrary dimension. In fact, there are various notions of curvature - the sectional curvature defined by Riemann and the Ricci curvature, which is an averaging of the sectional curvature.The Einstein equation of general relativity is phrased in terms of Ricci curvature. Ever since Riemann's time, there has been interest in making sense of the curvature of nonsmooth spaces. Alexandrov gave a good notion of what it means for a nonsmooth space to have nonnegative sectional curvature. Recent work, in collaboration with Cedric Villani, has given a good notion of what it means for a nonsmooth space to have nonnegative Ricci curvature. The definition is in terms of "optimal transport", a subject which has a history in applied mathematics. Part of the research in the proposal uses "optimal transport" to study the geometry of nonsmooth spaces. Going the other way, ideas from geometry will be used to address issues in "optimal transport".