Evolutions Equations in Geometry

演化方程是科学中的基本对象，描述了自然现象如何随着时间的推移而变化。例如，对一大类物理现象的建模，如晶体生长和火焰传播，涉及跟踪以曲率相关的速度移动的锋面。当曲面以与曲率成正比的速度演化时，这就产生了一个经典的退化非线性微分方程，称为平均曲率流。对于像这样的非线性方程，解可能会发展成尖锐的点和角，所以很自然地会问“解的正则性是什么？”平均曲率流的最优正则性最近被PI和Minicozzi证明。这个证明把分析和几何结合在一起。预计许多新的成分和技术将导致许多其他结果的广泛的方程，PI将在这个项目中调查。本项目分为两部分。主要涉及几何演化方程，如平均曲率流(MCF)和Ricci流。它涉及最优正则性及其应用。PI与Minicozzi一起解决了一些长期存在的关于平均曲率流的悬而未决的问题和猜想，并期望所开发的结果和想法也将对其他流具有重要的应用价值，并计划继续研究这些问题。普遍奇点的刚性和爆破的唯一性对于平均曲率流有着广泛的应用，从水平集方程的最优正则性到流的奇异集的最优估计。PI计划对Ricci流进行类似的猜想调查。该项目的第二部分将处理其他(非几何)演化方程，这些方程是由社会科学和工程学中的问题所驱动的。其中一个特别的焦点将是描述一群人的观点如何随着他们相互影响而演变的自然进化方程。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。

Evolution equations are basic objects in the sciences, describing how natural phenomena change over time. For instance, the modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, involves tracking fronts that move with curvature-dependent speed. When a surface evolves with speed that is proportional to the curvature, this results in one of the classic degenerate nonlinear differential equations called the mean curvature flow. For nonlinear equations like this, it is possible that solutions may develop sharp points and corners, so it is natural to ask "what is the regularity of solutions?'' Optimal regularity for mean curvature flow was recently proven by the PI and Minicozzi. The proof weaves together analysis and geometry. It is expected that many of the new ingredients and techniques should lead to many other results for a wide range of equations that the PI will investigate in this project.This project has two parts. The main part concerns geometric evolution equations, like mean curvature flow (MCF) and Ricci flow. It deals with optimal regularity and applications. The PI has, together with Minicozzi, settled a number of long-standing open problems and conjectures for mean curvature flow and expect that the results and ideas developed will have significant applications also to other flows and plan to pursue them. Rigidity of prevalent singularities and uniqueness of blow-ups has had a wide range of applications for mean curvature flow from optimal regularity of the level set equation to optimal estimates on the singular set of the flow. The PI plans on investigating similar conjectures for the Ricci flow. The second part of the project will deal with other (non-geometric) evolution equations that are motivated by questions in social science and engineering. One particular focus will be on a natural evolution equation that describes how the opinions of a group of people evolve as they are influenced by each other.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.