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

本文的研究内容涉及几何演化方程、度量空间分析和几何群论，演化方程为平均曲率流和Ricci流，涉及奇异集的结构、切线的唯一性和模型流的唯一性等问题。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。几何群论的项目是我早期工作的继续，这项工作受到了Gromov的论文《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；它们的主要目的是通过使用几何、分析、拓扑学、动力学和组合学的各种工具分析群的渐近结构来解决群的刚性和均匀化/几何化问题。该项目旨在研究热方程的两个非线性相似：平均曲率曲面的演化和汉密尔顿的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，并被用于Perelman最近求解Poincare猜想的过程中。建议对这些方程进行研究的主要目的是研究奇点，并证明它们具有非常特殊的形式。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。类似想法的另一个应用是在理论计算机科学中嵌入问题：Cheeger和PI能够给出一个自然的、新的反例来反驳计算机科学中的Goemans-Linial猜想。

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.