Ricci Flows through Singularities and Ricci Flows with Bounded Scalar Curvature

Ricci流是一个几何过程，它可以用来将给定的几何改进为更均匀的几何。Ricci流已经成为深入研究的对象，因为它们已经被用来证明各种长期存在的猜想，例如三维空间中的Poincare猜想和几何化猜想。一般的期望是，Ricci流产生的几何在某种意义上是基础空间的拓扑所固有的几何，即松散组成。然而，通常情况下，Ricci流在有限时间内会产生复杂的奇点。在维度3中，这些奇点可以通过所谓的手术手动移除，并且可以在它们之后继续流动。最近，在3维空间中引入了一类新的“奇异Ricci流”。这些流“在无限小的尺度上自动地流过奇点”，从而消除了有些不自然的手术过程。这个项目的目的是进一步了解这些流，并利用这种理解来研究某些度量空间和微分同胚群的拓扑。此外，PI将在更高的维度上研究Ricci流，旨在了解其奇点形成，这可能导致类似的手术或奇异流结构。第一个项目是PI(与Bruce Kleiner合作)关于通过奇点的奇异Ricci流的唯一性和连续性的工作的继续。本项目的总体目标是了解这项工作的几何、拓扑和分析应用。在其他方面，PI有一个解决广义斯梅尔猜想的策略，这将扩展PI和凯鹏华盈之前的部分解析。进一步的潜在应用涉及到正标量曲率度量空间的拓扑和几何，以及对三维一般Ricci流的研究。第二个项目是PI关于具有有界标量曲率的Ricci流研究的继续。PI将调查几个猜想，这些猜想已经在有界标量曲率的假设下得到验证。如果去掉这一假设，这些猜测很可能仍然是正确的。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A Ricci flow is a geometric process that may be used to improve a given geometry towards a more homogeneous one. Ricci flows have become the subject of intensive study, as they have been used to prove various long-standing conjectures, such as the Poincare and Geometrization Conjectures in dimension 3. The general expectation is that a Ricci flow produces a geometry in the limit that is in some sense inherent to the topology, i.e. the loose makeup, of the underlying space. However, usually a Ricci flow develops complicated singularities in finite time. In dimension 3 these singularities can be removed manually by so called surgeries and the flow can be continued beyond them. Recently, a new class of "singular Ricci flows" was introduced in dimension 3. These flows flow "automatically through singularities at an infinitesimal scale", thereby eliminating the somewhat unnatural surgery process. The goal of this project is to understand these flows further and to use this understanding to study the topology of certain spaces of metrics and diffeomorphism groups. In addition, the PI will work on Ricci flows in higher dimensions, aimed at understanding their singularity formation, which may result in a similar surgery or singular flow construction.The research project is split into two projects. The first project is a continuation of the PI's work (in collaboration with Bruce Kleiner) on the uniqueness and continuity of singular Ricci flows through singularities. The general goal of this project is to understand the geometric, topological and analytic applications of this work. Among other things, the PI has a strategy to resolve the Generalized Smale Conjecture, which would extend a previous partial resolution by the PI and Kleiner. Further potential applications concern the topology and geometry of the space of positive scalar curvature metrics, as well as the study of generic Ricci flows in dimension 3. The second project is a continuation of the PI's work on the study of Ricci flows with bounded scalar curvature. The PI will investigate several conjectures that have been verified under the assumption of bounded scalar curvature. These conjectures are likely to remain true if this assumption is removed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.