Geometric group theory, analysis on metric spaces, and geometric flows

摘要本文主要研究几何演化方程、度量空间分析和几何群论。本文提出的演化方程为平均曲率流和Ricci流，问题涉及奇异集的结构、可纠偏性、切线唯一性、模型流唯一性等问题。度量空间分析的研究有两个重点：bilipschitz嵌入问题（及相关问题）和Gromov双曲空间的边界结构。几何群论项目是我早期工作的延续，受到Gromov论文“双曲群”和“无限群的渐近不变量”，刚性理论，3流形和几何映射/函数理论的影响；他们的主要目的是解决刚性和均匀化/几何化问题，通过分析其渐近结构的各种工具，从几何，分析，拓扑，动力学和组合。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。平均曲率演化作为表面界面演化的一种自然模型已经研究了几十年。里奇流描述了一种不断发展的几何，并被用在佩雷尔曼最近对庞加莱猜想的解中。研究这些方程的主要目的是研究奇异性，并证明它们具有非常特殊的形式。研究计划的另一个组成部分是对具有自相似或分形特征的空间进行调查，使用过去几年开发的分析工具。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是非常有用的，是近10-15年来几个研究趋势的融合。类似思想的另一个应用是在理论计算机科学中嵌入问题：Cheeger和PI能够为计算机科学中的Goemans-Linial猜想提供一个自然的、新的反例。

AbstractThe proposed research is in geometric evolution equations, analysis on metric spaces, and geometric group theory.The evolution equations in the proposal are mean curvature flow and Ricci flow, and the problems pertain to the structure of the singular set, and issues such as rectifiability, uniqueness of tangents, and uniqueness of model flows. The proposed research in analysis on metric spaces has two focal points: bilipschitz embedding problems (and related issues) and the structure of boundaries of Gromov hyperbolic spaces. The projects in geometric group theory are a continuation of my earlier work, which is influenced by Gromov's papers "Hyperbolic groups" and "Asymptotic invariants of infinite groups", rigidity theory, 3-manifolds, and geometric mapping/function theory; their principal aim is to address rigidity and uniformization/geometrization problems for groups by analyzing their asymptotic structure with a variety of tools from geometry, analysis, topology, dynamics and combinatorics.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 recent 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 and the PI were able to give a natural, new counterexample to the Goemans-Linial conjecture in computer science.