Ancient Solutions and Singularity Analysis in Geometric Flows

微分几何领域是对弯曲物体的研究，本项目的重点是使用几何演化方程或“流”来理解这些物体。更具体地说，这些方程控制着数学定义的过程，这些过程以一种由具有几何意义的量(如长度、面积、体积或曲率)驱动的方式平滑地修改对象。PI研究的方程具有非常好的规律性，在某种意义上，凹凸不平的对象通常随着它们的演变而变得更加平滑；事实上，可以合理地预期随着流动的进行，对象将获得更多的对称性，这将帮助我们更好地理解我们开始时所使用的几何对象的类别。不幸的是，这些方程也有可能在有限时间内产生奇点，这意味着解可能会获得尖锐的点和角，人们不能期望方程在很长一段时间内有一个光滑的解。为了实现我们的最终目标，并使用流动技术更好地理解潜在的几何对象，我们需要了解如何处理可能出现的奇点。这些和类似的主题已经成为罗格斯大学几何分析的几个研讨会的主题，这些研讨会是由PI(与同事一起)组织的，吸引了许多研究生，并以研究讲座和高级研究生课程水平讲座为特色。这个项目的目标之一是对非线性几何流动的古代解进行分类，如Ricci流和平均曲率流。PI建议结合偏微分方程技术和几何估计来研究这种流动的古代解。它们的分类对于更好地理解发生在有限时间内的奇点至关重要。二维Ricci流的古老解描述了某些渐近自由的局域量子场论在紫外区的重整化群方程的轨迹。PI将对三维Ricci流的古代封闭非塌陷解进行分类，这将解决佩雷尔曼在他的一篇论文中提出的一个猜想。PI还应找到保证Ricci流和平均曲率流的解顺利存在的最小条件。PI还想从以下意义上理解奇点的形成：什么是四维Ricci流中的稳定奇点模型。这与从一般初始数据开始理解四维Ricci流的奇点形成有关。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。

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.