Geometric flows and analysis on metric spaces

该项目侧重于两个非线性偏微分方程，它们出现在科学和工程的许多不同学科中，以及数学中。这些方程描述了一个曲面或弯曲物体的运动，随着时间的推移，它的形状会尽可能地简单。运动的一个重要特征是奇点的形成，奇点的形成使解决方案能够模拟拓扑变化的情况，例如当一个肥皂泡拉长并分裂成两个气泡时。一方面，这种灵活性带来了许多深刻的应用；另一方面，它带来了巨大的智力挑战。PI将以最近的进展为基础，解决这一领域的一些主要开放性问题。项目的第二部分应用几何和分析的思想来研究粗糙物体的结构，包括分形。由于数学不同部分之间的新联系，以及对计算机科学问题的应用，这个领域在过去的25年里发展得非常迅速。研究生将在这个项目中接受培训。本文主要研究几何演化方程和度量空间的分析。本文提出的演化方程为平均曲率流和Ricci流，项目关注的是不同方面的规律性。度量空间分析中的课题涉及几何映射，如次黎曼环境下的bilipschitz、拟共形、Sobolev映射等。本项目将在解决几何群论与几何映射理论的交叉领域长期存在的问题，分析偏微分方程，特别是卡诺群群中接触系统弱解的正则性方面取得进展。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

The project focusses on two nonlinear partial differential equations that arise in a number of different disciplines in science and engineering, as well as from within mathematics. These equations describe the motion of a curved surface or curved object which evolves so as to simply its shape as efficiently as possible over time. An important feature of the motion is the formation of singularities that enable the solutions to model situations where topology changes, for example when a soap bubble elongates and splits into two bubbles. On the one hand, this flexibility leads to numerous profound applications; on the other, it creates great intellectual challenges. The PI will build on recent progress to address some of the main open problems in this area. The second part of the project applies ideas from geometry and analysis to study the structure of rough objects, including fractals. This area has been developing very rapidly in the last 25 years, due to new connections between different parts of mathematics, and applications to problems from computer science. Graduate students will be trained in this project. The proposed research studies geometric evolution equations and analysis on metric spaces. The evolution equations in the proposal are mean curvature flow and Ricci flow, and the projects focus on different aspects of regularity. The projects in analysis on metric spaces are concerned with geometric mappings such as bilipschitz, quasi-conformal, Sobolev mappings in the sub-Riemannian setting. The project will make progress in solving longstanding problems at the interface of geometric group theory and geometric mapping theory, and analysis of PDE, especially the regularity of weak solutions to the contact system in Carnot group groups.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.