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

摘要奖：DMS-0505610主要研究人员：布鲁斯·A·克莱纳该项目的第一部分涉及两个几何演化方程：平均曲率流和汉密尔顿的Ricci流。主要目的是了解奇异集的结构和在奇异集附近解的几何。第二部分研究程序的动机是几何群论和刚性问题，特别是负曲线(或Gromov双曲)空间的渐近结构。这导致了利用最近几年发展起来的分析工具来研究具有自相似性质的空间。这里的目标是如果可能的话，将空间准对称地变形成最佳形式，以揭示隐藏的对称性；表明不存在隐藏的对称性；或者表明原始空间实际上是熟悉的东西的变形。这与群论刚性问题直接相关。该项目旨在研究热方程的两个非线性类比：曲面的平均曲率演化和哈密尔顿的Ricci流。平均曲率流是一种标准的(理想化的)模型，它涉及空间中两个区域之间不断演化的表面或界面的任何物理过程。利奇流--一个控制随时间变化的弯曲空间几何的方程--最近因其在佩雷尔曼关于100年前的庞卡雷猜想的壮观工作中的突出作用而登上了头条。关于这两个方程的解已经知道很多，但仍然有许多基本的开放问题，特别是那些与奇点的形成有关的问题，奇点的形成在佩雷尔曼的工作中发挥着核心作用。该项目将解决其中的一些问题。研究计划的另一个组成部分是使用过去几年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。这里的目标是将空间变形为最佳形式，如果可能的话，以便揭示隐藏的对称性，表明不存在隐藏的对称性，或表明原始空间实际上是熟悉的东西的变形。这对于理解无限对称群的渐近形状是非常有用的。

AbstractAward: DMS-0505610Principal Investigator: Bruce A. KleinerThe first part of the project concerns two geometric evolution equations:mean curvature flow and Hamilton's Ricci flow. The main objectiveis to understand the structure of the singular set and the geometry ofthe solutions near the singular set.The second part of the research program is motivated by geometricgroup theory and rigidity questions, specifically the asymptoticstructure of negatively curved (or Gromov hyperbolic) spaces.This leads to an investigationof spaces which have a self-similar character using analytictools that have been developed in the last few years. Herethe goal is to quasisymmetrically deform the space into an optimal form, if possible,in order to reveal hidden symmetries; show thatno hidden symmetries exist; or show that the original spacewas really a deformation of something familiar. This is directlyrelated to group theoretic rigidity problems.The project aims to study two nonlinear analogs of the heat equation: theevolution of surfaces by their mean curvature, and Hamilton'sRicci flow. Mean curvature flow is a standard (idealized) model formany physical process which involve an evolving surface, or interface,between two regions in space. The Ricci flow -- an equationgoverning a curved space geometry which evolves with time -- has made headlines lately due to itsprominent role in the spectacular work of Perelman on the 100 year old Poincareconjecture. Much is known about the solutions of these two equations,but many basic open questions remain, especially those tied with theformation of singularities, which play a central role in Perelman's work.The project will address some of these questions.Another component of the research program is an investigationof spaces which have a self-similar, or fractal character, using analytictools that have been developed in the last few years. Herethe goal is to deform the space into an optimal form, if possible,in order to reveal hidden symmetries, show thatno hidden symmetries exist, or show that the original spacewas really a deformation of something familiar. This is very usefulfor understanding the asymptotic shape of infinite groups of symmetries.