Ricci Flows through Singularities and Ricci Flows with Bounded Scalar Curvature

Ricci流是一种几何过程，可以用来将给定的几何形状改进为更均匀的几何形状。利玛窦流已经成为深入研究的主题，因为它们已经被用来证明各种长期存在的猜想，如3维中的庞加莱猜想和几何化猜想。一般的期望是，里奇流在极限中产生一种几何，这种几何在某种意义上是拓扑所固有的，即底层空间的松散构成。然而，通常Ricci流在有限时间内发展出复杂的奇点。在三维空间中，这些奇点可以通过所谓的手术手动移除，并且流动可以继续超越它们。最近，一类新的“奇异Ricci流”被引入到三维空间。这些流动“以无穷小的尺度自动通过奇点”，从而消除了有点不自然的手术过程。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空间和同构群的拓扑。此外，PI还将研究更高维度的Ricci流，旨在了解它们的奇异性形成，这可能导致类似的手术或奇异流构造。该研究项目分为两个项目。第一个项目是继续PI的工作（与布鲁斯克莱纳合作）的独特性和连续性奇异里奇流通过奇点。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Smale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正标量曲率度量空间的拓扑和几何，以及三维中一般Ricci流的研究。第二个项目是PI关于有界标量曲率的Ricci流研究工作的延续。PI将研究几个已经在有界标量曲率假设下得到验证的模型。这个奖项反映了NSF的法定使命，并被认为是值得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的支持。

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.