Ancient Solutions and Singularity Analysis in Geometric Flows

几何流中的古代解和奇异性分析

• 批准号: 1811833
• 负责人: Natasa Sesum
• 金额: $ 18.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811833&HistoricalAwards=false
• 关键词: Ancient; Solutions; Singularity; Analysis; Geometric

The field of differential geometry is the study of curved objects, and this project focuses on the use of geometric evolution equations or `flows' to understand these objects. More specifically, these equations govern mathematically-defined processes that smoothly modify objects in a way that is driven by geometrically meaningful quantities such as length, area, volume or curvature. The equations that the PI studies have very nice regularity properties, in the sense that bumpy objects often become smoother as they evolve; in fact, it is reasonable to expect that the object will gain more symmetries as the flow proceeds, and this will help us better understand the class of geometric objects we started with. Unfortunately, these equations also have the potential to develop singularities in finite time, meaning that the solution may acquire sharp points and corners, and one cannot expect in general to have a smooth solution to the equation for a long time. In order for us to achieve our ultimate goal, and better understand the underlying geometric object using the flow techniques, we need to understand how to deal with singularities that may arise. These and similar topics have been the subject of several workshops in geometric analysis at Rutgers University organized by the PI (together with colleagues), attracting many graduate students and featuring both research talks and advanced graduate-course level lectures.One of the aims of this project is the classification of ancient solutions to nonlinear geometric flows, such as, the Ricci flow and the mean curvature flow. The PI proposes to combine the PDE techniques and geometric estimates to study the ancient solutions of such flows. Their classification is crucial for better understanding the singularities that occur in finite time. Ancient solutions to the two-dimensional Ricci flow describe trajectories of the renormalization group equations of certain asymptotically free local quantum field theories in the ultraviolet regime. The PI will classify ancient closed non-collapsed solutions to the three-dimensional Ricci flow, which would settle down a conjecture stated by Perelman in one of his papers. The PI shall also find minimal conditions that guarantee smooth existence of a solution to the Ricci flow and the mean curvature flow. The PI also wants to understand singularity formation in the following sense: what the stable singularity models in four dimensional Ricci flow are. This is related to understanding the singularity formation of a four-dimensional Ricci flow starting at generic initial data.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.

差异几何的领域是对弯曲对象的研究，该项目的重点是使用几何进化方程或“流”来理解这些对象。 更具体地说，这些方程式控制了数学定义的过程，这些过程以几何有意义的数量（例如长度，面积，体积或曲率）驱动的方式平滑修改对象。 PI研究具有非常好的规律性属性的方程式，从某种意义上说，随着颠簸的发展，它们通常会变得更加顺畅。实际上，可以合理地期望该对象随着流程的进行而获得更多的对称性，这将有助于我们更好地理解我们开始使用的几何对象类别。 不幸的是，这些方程在有限的时间内也有潜力发展奇异性，这意味着该解决方案可能会获得尖锐的点和弯道，并且通常不能期望长期以来对方程式具有平稳的解决方案。为了使我们实现我们的最终目标，并使用流动技术更好地理解了基本的几何对象，我们需要了解如何处理可能出现的奇异性。 These and similar topics have been the subject of several workshops in geometric analysis at Rutgers University organized by the PI (together with colleagues), attracting many graduate students and featuring both research talks and advanced graduate-course level lectures.One of the aims of this project is the classification of ancient solutions to nonlinear geometric flows, such as, the Ricci flow and the mean curvature flow. PI建议将PDE技术和几何估计值结合起来，以研究此类流的古老解决方案。他们的分类对于更好地理解有限时间内发生的奇异性至关重要。二维RICCI流的古老解决方案描述了紫外线范围内某些渐近均不含局部量子场理论的重新归一化组方程的轨迹。 PI将对三维RICCI流程将古代封闭的非封闭式解决方案分类，该解决方案将在他的一篇论文中解决一个猜想。 PI还应找到最小的条件，以确保将RICCI流和平均曲率流平滑地存在。 PI还希望从以下意义上理解奇异性形成：四维RICCI流中的稳定奇异性模型是什么。这与了解从通用初始数据开始的四维RICCI流的奇异性形成有关。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。