Geometric flows and analysis on metric spaces

该项目的重点是两个非线性偏微分方程，出现在一些不同的学科在科学和工程，以及从数学。 这些方程描述了弯曲表面或弯曲物体的运动，该运动随着时间的推移而演变，以便尽可能有效地简化其形状。 运动的一个重要特征是奇点的形成，使得解决方案能够模拟拓扑变化的情况，例如当肥皂泡伸长并分裂成两个气泡时。 一方面，这种灵活性导致了许多深刻的应用;另一方面，它带来了巨大的智力挑战。 PI将在最近取得的进展的基础上解决这一领域的一些主要未决问题。 该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。 在过去的25年里，由于数学不同部分之间的新联系以及计算机科学问题的应用，这一领域发展非常迅速。研究生将在这个项目中接受培训。本研究主要研究度量空间上的几何演化方程与分析。该提案中的演化方程是平均曲率流和Ricci流，项目侧重于不同方面的规律性。度量空间上的分析项目涉及到几何映射，如次黎曼背景下的bilipschitz映射、拟共形映射、Sobolev映射。该项目将在解决几何群论和几何映射理论的接口长期存在的问题，以及PDE的分析，特别是卡诺群接触系统弱解的规律性方面取得进展。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.