Ricci Curvature and Ricci Flow

该提案有两个主要议题。 第一个是在一般意义下研究具有非负Ricci曲率的度量空间。许多已知的关于光滑空间的Ricci曲率的结果都推广到了非光滑空间。 然而，也有许多已知的光滑空间的结果，其中的非光滑扩展是不清楚的。拟议的研究将探讨非光滑空间的曲率性质在多大程度上类似于光滑空间。第二个主题是Ricci流。 Ricci流是由R.汉密尔顿在20世纪80年代，是一种方式来发展的“形状”的空间，其里奇曲率的手段。 最近，佩雷尔曼利用里奇流来解决三维拓扑中最重要的问题。 里奇流的长期行为在很大程度上是未知的，将在提案中解决。此外，还将考虑把Perelman的结果推广到三维orbifolds。总的来说，这个建议是关于曲率的。 历史上，曲率首先被认为是平面中的曲线，然后是三维空间中的曲线和曲面。 黎曼证明了如何理解任意维的光滑空间的曲率。实际上，曲率有各种各样的概念--黎曼定义的截面曲率和里奇曲率，里奇曲率是截面曲率的平均值。广义相对论的爱因斯坦方程就是用里奇曲率来表述的。自Riemann时代以来，人们一直对理解非光滑空间的曲率感兴趣。 亚历山德罗夫给出了一个很好的概念，即非光滑空间具有非负截面曲率意味着什么。 最近与Cedric Villani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。 这个定义是根据“最佳运输”，一个在应用数学中有历史的主题。 提案中的部分研究使用“最优传输”来研究非光滑空间的几何形状。 另一方面，几何学的思想将被用来解决“最佳运输”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".