Geometric Flows and Analysis on Metric Spaces

该项目的第一部分涉及两个非线性偏微分方程，这些方程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。 一个重要的特征是奇点的形成。 奇异性使解决方案能够对拓扑变化的情况进行建模，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，它创造了巨大的智力挑战，已经占领了数学家40多年。 拟议的研究旨在建立在过去几年的成功，以解决一些主要的开放问题。 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20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。 该项目还包括大量的博士生培训，该项目旨在研究热方程的两个非线性类似物：平均曲率的表面演化和汉密尔顿的里奇流。 几十年来，平均曲率演化一直是研究表面界面演化的自然模型。 利玛窦流描述了一种演化的几何，并被用于佩雷尔曼的庞加莱猜想的解决方案。 对这些方程的拟议研究的主要目标是研究与其解中奇点形成有关的问题，包括其结构和稳定性。 几何学和拓扑学的基本原理的解决是这项工作的产物。 该研究计划的另一个组成部分是使用在过去几年中开发的分析工具，对具有自相似或分形特征的空间进行调查。这里的目标之一是将空间变形为最佳形式，如果可能的话，以揭示隐藏的对称性，否则表明不存在隐藏的对称性。 这对于理解无限群的渐近形状非常有用，并且是过去15-20年中几个研究趋势的汇合的一部分。 类似思想的另一个应用是理论计算机科学中的嵌入问题：Cheeger，Naor和PI能够实质性地改进先前关于负型空间嵌入的最佳已知结果，与Goemans-Linial猜想的定量版本有关。 该项目具体涉及几何演化方程，嵌入问题，度量空间分析和几何群论。 工程中的演化方程为平均曲率流和Ricci流。度量空间分析的研究主要集中在三个方面：（1）bilipschitz嵌入问题及相关问题，（2）满足Poincare不等式的空间的结构，（3）Gromov双曲空间的边界结构。这三个领域的共同主题是满足庞加莱不等式的空间，以及导致奇异极限空间的重新缩放参数。该奖项反映了NSF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 first part of the project involves 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 simplify its shape as efficiently as possible over time. An important feature is the formation of singularities. Singularities are what 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 has led to numerous profound applications; on the other, it creates great intellectual challenges which have occupied mathematicians for more than 40 years. The proposed research aims to build on successes in the last few years, to address some of the main open problems. 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 20 years, due to new connections between different parts of mathematics, and applications to problems from computer science. The project also involves substantial training of PhD students.The project aims to study two nonlinear analogs of the heat equation: evolution of surfaces by mean curvature, and Hamilton's Ricci flow. Evolution by mean curvature has been studied for decades as a natural model for evolving surface interfaces. Ricci flow describes an evolving geometry, and was used in Perelman's solution of the Poincare conjecture. The primary objective of the proposed research on these equations is to study issues related to the formation of singularities in their solutions, including their structure and stability. The resolution of fundamental conjectures in geometry and topology is an outgrowth of this work. Another component of the research program is an investigation of spaces which have a self-similar or fractal character, using analytic tools that have been developed in the last few years. Here one of the goals is to deform the space into an optimal form, if possible, in order to reveal hidden symmetries, and otherwise show that no hidden symmetries exist. This is very useful for understanding the asymptotic shape of infinite groups, and is part of confluence of several research trends over the last 15-20 years. Another application of similar ideas is to embedding problems in theoretical computer science: Cheeger, Naor, and the PI were able to substantially improve the previous best known results on the embedding of spaces of negative type, in connection with the quantitative version of the Goemans-Linial conjecture. This project specifically addresses geometric evolution equations, embedding problems, analysis on metric spaces, and geometric group theory. The evolution equations in the project are mean curvature flow and Ricci flow. The research in analysis on metric spaces clusters in three areas: (1) bilipschitz embedding problems and related issues, (2) the structure of spaces satisfying Poincare inequalities, (3) the structure of boundaries of Gromov hyperbolic spaces. Common themes in all three areas are spaces satisfying Poincare inequalities, and rescaling arguments leading to singular limit spaces.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.