Ricci flow, optimal transport and index theory

项目编号：DMS 1207654，项目负责人：John W. lott，主要研究方向为微分几何和流形分析。这些涉及几何分析的线性和非线性方面。第一个项目是三维和四维Ricci流的长期行为。在三维空间中，主要的开放性问题是手术次数的有限性、截面曲率的衰减和几何的长时间渐近性。在四维空间中，目标是使用利玛窦流来识别正则几何形状。第二个项目涉及最优运输。我们将使用Wasserstein度规在概率测度空间上构造平行输运。我们还将研究具有正Ricci曲率的度量-度量空间的等周不等式和尖锐Sobolev不等式。第三个项目将发展具有黎曼叶状的流形上的横向狄拉克型算子的指标定理。首席研究员还将开发一个基于无限维矢量束的微分k理论模型。最优输运是对质量最优输运方式的研究。1781年，蒙日第一次考虑了这个问题。在过去的几十年里，在最优输运、偏微分方程和应用数学之间建立了联系。最近，在微分几何中也发现了联系。几何学的思想似乎可以应用于最优交通。里奇流在几何学和拓扑学上具有明显的重要性。正在提出的研究其长期行为的方法，特别是适用于坍塌流的单调量的使用，很可能适用于其他几何流。

AbstractAward: DMS 1207654, Principal Investigator: John W. LottThe principal investigator will work on three projects in differential geometry and analysis on manifolds. These involve both linear and nonlinear aspects of geometric analysis. The first project is the long-time behavior of the Ricci flow in three and four dimensions. In three dimensions, the main open problems are the finiteness of the number of surgeries, the decay of the sectional curvature and the long-time asymptotics of the geometry. In four dimensions, the goal is to use Ricci flow to identify canonical geometries. The second project concerns optimal transport. We will work on a construction of parallel transport on the space of probability measures, equipped with the Wasserstein metric. We will also work on isoperimetric inequalities and sharp Sobolev inequalities for metric-measure spaces with positive Ricci curvature. The third project will develop an index theorem for transverse Dirac-type operators on manifolds with Riemannian foliations. The principal investigator will also develop a model of differential K-theory based on infinite-dimensional vector bundles.Optimal transport is the study of the optimal way to transport mass. It was first considered by Monge in 1781. In the past few decades, links have been established between optimal transport, partial differential equations and applied mathematics. More recently, links have also been found with differential geometry. It seems likely that ideas from geometry will find applications to optimal transport. The Ricci flow has clear importance for geometry and topology. The methods that are being proposed to study its long-term behavior, in particular the use of monotonic quantities adapted to collapsing flows, may well have application to other geometric flows.