Evolutions Equations in Geometry

几何演化方程

• 批准号: 1812142
• 负责人: Tobias Colding
• 金额: $ 50.3万
• 依托单位: Massachusetts Institute of Technology
• 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: 2018-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812142&HistoricalAwards=false
• 关键词: Evolutions; Equations; Geometry

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.

演化方程是科学中的基本对象，描述自然现象如何随时间变化。 例如，对晶体生长和火焰传播等多种物理现象的建模涉及跟踪以曲率相关速度移动的前沿。当一个曲面以与曲率成比例的速度演化时，这导致一个经典的退化非线性微分方程，称为平均曲率流。 对于这样的非线性方程，解可能会形成尖锐的点和角，所以很自然地会问“解的正则性是什么？平均曲率流的最优规则性最近被PI和Minicozzi证明。 这个证明将分析和几何学结合在一起。预计许多新的成分和技术将导致PI将在本项目中研究的广泛方程的许多其他结果。主要是几何演化方程，如平均曲率流（MCF）和Ricci流。它涉及最优正则性和应用。 PI与Minicozzi一起解决了许多长期存在的平均曲率流的开放问题和问题，并期望开发的结果和想法也将对其他流有重要的应用，并计划继续进行。 普遍奇点的刚性和爆破的唯一性在平均曲率流中有着广泛的应用，从水平集方程的最优正则性到流的奇异集的最优估计。 PI计划研究Ricci流的类似结构。 该项目的第二部分将处理其他（非几何）的演化方程，这些方程是由社会科学和工程问题引起的。 一个特别的重点将是一个自然的进化方程，描述了一组人的意见如何演变，因为他们是相互影响的。这个奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的知识价值和更广泛的影响审查标准。