Singular Ricci flow, Einstein flow and index theory

PI建议在微分几何和流形分析方面开展三个项目。这些都涉及几何分析的线性和非线性两个方面。几何流，如本提案中提到的那些，在许多科学分支中都有应用，包括材料科学和宇宙学。这一建议的研究将增进对这种流动的理论理解。利奇流是现代几何学的基本原理。大多数关于Ricci血流的工作都是通过外科手术来研究平滑的Ricci血流或Ricci血流。另一方面，许多偏微分方程有一类定义良好的奇异解。所提出的工作将探索三维Ricci流的一类值得注意的奇异解。同样，对于许多偏微分方程而言，都有很好的弱解概念。通过外科手术了解Ricci血流的极限，可能有助于阐明建立一个良好的弱Ricci血流概念的问题。爱因斯坦流是一种几何流动，具有与抛物线流动非常不同的特征。这项拟议的研究将阐明爱因斯坦流的一些一般特征。真空时空的长期行为在宇宙学中引起了明显的兴趣。找到一个基于无限维余循环的微分K-理论的框架将导致局部指数理论和相关泛函分析的新方向。在这个提议中有三个主要的主题：(1)奇异Ricci流。在与凯鹏华盈的联合工作中，PI已经表明，随着手术参数接近零，佩雷尔曼的利奇血流对手术的影响是有限的。PI建议研究这种极限的精细性质，更一般地研究奇异Ricci流。(2)爱因斯坦流。爱因斯坦流是一种描述具有常平均曲率空间超曲面分层的真空时空的长时间行为的流。PI将使用塌缩理论中的技术，以便在塌缩情况下理解爱因斯坦流的几何。(3)微分K理论。微分K-理论是K-理论与微分形式相结合的一种拓扑学理论。在与戈罗霍夫斯基的合作中，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.