Singular Ricci flow, Einstein flow and index theory

奇异里奇流、爱因斯坦流和指数理论

• 批准号: 1510192
• 负责人: John Lott
• 金额: $ 38.63万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510192&HistoricalAwards=false
• 关键词: Singular; Ricci; flow; Einstein; index

The PI proposes to work on three projects in differential geometry and analysis on manifolds. These involve both linear and nonlinear aspects of geometric analysis. Geometric flows, such as those addressed in this proposal, have applications to many branches of science, including materials science and cosmology. The research in this proposal will enhance the theoretical understanding of such flows. The Ricci flow is fundamental in modern geometry. Most of the work on Ricci flow has been done on smooth Ricci flows or Ricci flows with surgery. On the other hand, many partial differential equations have a well-defined class of singular solutions. The proposed work will explore a remarkable class of singular solutions for the three-dimensional Ricci flow. Similarly, for many partial differential equations, there are good notions of weak solutions. Understanding limits of Ricci flow with surgery may shed light on the problem of developing a good notion of weak Ricci flow. The Einstein flow is a geometric flow with very different features than parabolic flows. The proposed research will elucidate some general features of the Einstein flow. The long-time behavior of vacuum spacetimes is of evident interest in cosmology. Finding a framework for differential K-theory based on infinite-dimensional cocycles will lead to new directions in local index theory and the associated functional analysis. It may also have topological consequences.There are three main topics in this proposal.(1) Singular Ricci flows. In joint work with Kleiner, the PI has shown that there is a limit of Perelman's Ricci flow with surgery, as the surgery parameter goes to zero. The PI proposes to study refined properties of such limits, and more generally of singular Ricci flows.(2) Einstein flow. The Einstein flow is a flow which describes the long-time behavior of a vacuum spacetime that has a foliation by constant mean curvature spatial hypersurfaces. The PI will use techniques from collapsing theory in order to understand the geometry of the Einstein flow in the collapsing case.(3) Differential K-theory. Differential K-theory is a topological theory that combines K-theory with differential forms. In joint work with Gorokhovsky, the PI will construct infinite-dimensional models of differential K-theory.

PI计划在微分几何和流形分析方面开展三个项目。这些涉及几何分析的线性和非线性方面。几何流，比如在这个提议中提到的，在许多科学分支中都有应用，包括材料科学和宇宙学。本课题的研究将增强对此类流动的理论认识。里奇流是现代几何学的基础。大多数关于利玛窦流的研究都是在平滑的利玛窦流或手术中的利玛窦流上完成的。另一方面，许多偏微分方程有一类定义良好的奇异解。提出的工作将探索一个非凡的类奇异解的三维里奇流。类似地，对于许多偏微分方程，也有弱解的好概念。通过外科手术了解利玛窦流的局限性，可以帮助我们更好地理解弱利玛窦流的概念。爱因斯坦流是一种几何流，与抛物线流有着非常不同的特征。提出的研究将阐明爱因斯坦流的一些一般特征。真空时空的长时间行为在宇宙学中引起了明显的兴趣。寻找基于无穷维环的微分k理论框架，将为局部指标理论和相关泛函分析开辟新的方向。它还可能产生拓扑结果。这项提案有三个主要主题。(1)奇异里奇流。在与Kleiner的联合工作中，PI已经表明，随着手术参数趋于零，佩雷尔曼里奇流有一个极限。PI建议研究这种极限的精细性质，更一般地说，是奇异里奇流。爱因斯坦流。爱因斯坦流是一种描述具有恒定平均曲率空间超曲面叶理的真空时空的长时间行为的流。PI将使用坍缩理论中的技术来理解坍缩情况下爱因斯坦流的几何形状。(3)微分k理论。微分k理论是一种将k理论与微分形式相结合的拓扑理论。在与Gorokhovsky的合作中，PI将构建微分k理论的无限维模型。