Geometric flows and analysis on metric spaces

该项目旨在研究热方程的两个非线性类比：平均曲率曲面的演化和哈密尔顿的Ricci流。平均曲率演化作为一种自然的表面界面演化模型已经被研究了几十年。Ricci流描述了一种演化的几何，并被用在Perelman的Poincare猜想的解中。建议对这些方程进行研究的主要目的是研究奇点，并证明它们有一种非常特殊的形式。研究计划的另一个组成部分是利用最近几年发展起来的分析工具，对具有自相似或分形特征的空间进行调查。这里的目标之一是如果可能的话，将空间变形为最佳形式，以便揭示隐藏的对称性，否则就表明不存在隐藏的对称性。这对于理解无限群的渐近形状是非常有用的，也是过去10-15年来几个研究趋势汇合的一部分。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结合Goemans-Linial猜想的量化版本，实质上改进了先前关于负型空间嵌入的最已知结果。建议的研究涉及几何演化方程、嵌入问题、度量空间分析和几何群论。该方案中的演化方程为平均曲率流和Ricci流。拟在三个方面对度量空间簇进行分析：(1)bilipschitz嵌入问题及相关问题；(2)满足Poincare不等式的空间结构；(3)Gromov双曲空间的边界结构。这三个领域的共同主题是满足Poincare不等式的空间，以及重新缩放导致奇异极限空间的论点。

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 singularities and show that they have a very special form. 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 10-15 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.The proposed research is in geometric evolution equations, embedding problems, analysis on metric spaces, and geometric group theory. The evolution equations in the proposal are mean curvature flow and Ricci flow. The proposed 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.