Ricci Curvature, Ricci Flow and Foliations

这个建议处理高维的弯曲空间。一个有趣的问题是如何测量一个非光滑空间的曲率。在截面曲率的情况下，这是由亚历山德罗夫在20世纪30年代提出的，他对奇异空间具有非负截面曲率的意义给出了一个很好的概念。该提议者最近的工作，与Cedric Villani联合，以及Karl-Theodor Sturm的相关工作，给出了奇异空间具有非负里奇曲率的一个很好的概念。提出的研究将探索非负里奇曲率空间的性质，或者更一般地说，里奇曲率有界以下。在另一个方向上，里奇曲率提供了一种使空间几何平滑的方法，通过里奇流。这种流动是由理查德·汉密尔顿在20世纪80年代引入的，他用它来描述具有非负里奇曲率的三维光滑空间的拓扑结构。最近，Perelman利用里奇流证明了三维拓扑学中的两个最大猜想，即庞加莱猜想和瑟斯顿的几何化猜想。尽管佩雷尔曼取得了巨大的成就，但在三维和高维空间中，关于里奇流仍有许多悬而未决的问题。拟议的研究将解决其中的一些问题。奇异空间产生的另一种方式是当一个高维空间被叶面化成低维空间。这种叶理的参数化空间几乎总是拓扑奇异的。研究的一部分是对这样的空间进行分析，或者更准确地说是证明一个横向指数定理。曲率的各种概念在三维空间的二维曲面的传统设置中是一致的。对于高维平滑空间，不一定是平面空间，这些不同的概念被称为截面曲率，里奇曲率和标量曲率。每一个都是前一个的平均，即里奇曲率是截面曲率的平均，标量曲率是里奇曲率的平均。里奇曲率是通过爱因斯坦的广义相对论方程进入物理学的。上述部分工作涉及“最佳运输”。这是对在弯曲空间中传输质量的最佳方式的研究。它是由蒙日在18世纪80年代发起的，最近几十年有了复兴，应用于偏微分方程和应用数学。我们已经证明了它在微分几何中也有应用。相反，微分几何中的概念也适用于最优运输。

This proposal deals with curved spaces of higher dimension.An interesting question is how to measure the curvature of a nonsmooth space. In the case of sectional curvature, this was initiated in the 1930's by Alexandrov, who gave a good notion of what it means for a singular space to have nonnegative sectional curvature.Recent work by the proposer, joint with Cedric Villani, together with related work by Karl-Theodor Sturm, has given a good notion of what it means for a singular space to have nonnegative Ricci curvature. The proposed research will explore properties of spaces with nonnegative Ricci curvature or, more generally, with Ricci curvature bounded below. In a different direction, the Ricci curvature gives a way to smooth out the geometry of a space, by means of the Ricci flow. This flow was introduced by Richard Hamilton in the 1980's, who used it to characterize the topology of three-dimensional smooth spaces with nonnegative Ricci curvature. Recently, Perelman has proved the biggest conjectures in three-dimensional topology, namely the Poincare Conjecture and Thurston's Geometrization Conjecture, using Ricci flow. Despite Perelman's great achievements, there are many open questions concerning the Ricci flow in dimension three and in higher dimensions. The proposed research will address some of these questions.Another way that singular spaces arise is when a higher-dimensional space is foliated into lower dimensional spaces. The parametrizing space for such a foliation is almost always topologically singular.Part of the proposed research is to do analysis on such spaces, or more precisely to prove a transverse index theorem.There arevarious notions of curvature, which coincide in the traditional setting of two-dimensional surfaces in three-dimensional space. For a higher dimensional smooth space, not necessarily living in a flat space, these different notions are called the sectional curvature, the Ricci curvature and the scalar curvature. Each one is an averaging of the previous one, i.e. the Ricci curvature is an averaging of sectional curvature and the scalar curvature is an averaging of Ricci curvature. The Ricci curvature enters in physics through Einstein's equations of general relativity.Part of the work described above concerns ``optimal transport''. This is the study of the optimal way to transport mass in a curved space. It was initiated by Monge in the 1780's and has had a revival in recent decades, with application to partial differential equations and applied mathematics.We have shown that it also has application in differential geometry. Conversely, concepts from differential geometry have application to optimal transport.