Ricci Flows through Singularities and Ricci Flows with Bounded Scalar Curvature

利玛窦流是一种几何过程，可用于将给定的几何形状改进为更均匀的几何形状。里奇流已成为深入研究的主题，因为它们已被用来证明各种长期存在的猜想，如3维的庞加莱猜想和几何化猜想。一般的期望是，里奇流在极限下产生几何形状，在某种意义上，这是拓扑固有的，即底层空间的松散构成。然而，里奇流通常在有限时间内发展出复杂的奇点。在第三维，这些奇点可以通过所谓的手术手工移除，流可以在奇点之外继续。最近，在三维空间中引入了一类新的“奇异Ricci流”。这些流动“在无穷小的尺度上自动通过奇点”，从而消除了有些不自然的手术过程。这个项目的目标是进一步理解这些流，并利用这种理解来研究某些度量和微分同构群空间的拓扑结构。此外，PI将研究更高维度的Ricci流，旨在理解它们的奇点形成，这可能会导致类似的手术或奇异流构造。这个研究项目分为两个项目。第一个项目是PI工作的延续（与Bruce Kleiner合作），研究奇异里奇流在奇点中的独特性和连续性。这个项目的总体目标是了解这项工作的几何、拓扑和分析应用。除此之外，PI有一个解决广义小猜想的策略，这将扩展PI和Kleiner之前的部分解决方案。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3维广义里奇流的研究。第二个项目是PI关于有界标量曲率的里奇流研究工作的延续。PI将研究在有界标量曲率假设下已被验证的几个猜想。如果去掉这个假设，这些猜想很可能仍然成立。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.