Singular Ricci flow, Einstein flow and index theory

PI提议开展微分几何和流形分析方面的三个项目。这些涉及几何分析的线性和非线性方面。几何流，如本提案中所述，在许多科学分支中都有应用，包括材料科学和宇宙学。本研究将有助于加深对这类流动的理论认识。里奇流是现代几何的基础。大部分关于Ricci流的工作都是在光滑Ricci流或带手术的Ricci流上进行的。另一方面，许多偏微分方程有一类定义明确的奇异解。所提出的工作将探索一类显着的奇异解的三维Ricci流。类似地，对于许多偏微分方程，也有很好的弱解概念。理解手术的Ricci流的限制可能会对弱Ricci流的概念的发展问题有所启发。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。所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具有明显的意义。找到一个基于无限维上循环的微分K-理论的框架，将为局部指标理论和相关的泛函分析带来新的方向。它也可能有拓扑的后果。有三个主要议题在这个建议。(1)奇异Ricci流在与Kleiner的联合工作中，PI已经表明，随着手术参数趋于零，Perelman的Ricci流存在极限。PI建议研究这种极限的精细性质，以及更一般的奇异Ricci流。(2)爱因斯坦流爱因斯坦流是一种描述真空时空的长时间行为的流，它具有由常平均曲率空间超曲面构成的叶理。PI将使用塌缩理论的技术来理解爱因斯坦流在塌缩情况下的几何形状。(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。

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.