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

几何群论、度量空间分析和几何流

• 批准号: 0505610
• 负责人: Bruce Kleiner
• 金额: --
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2006-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505610&HistoricalAwards=false
• 关键词: Geometric; group; theory; analysis; metric

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.

摘要奖：DMS-0505610主要研究者：布鲁斯A.第一部分是关于两个几何演化方程：平均曲率流和汉密尔顿的里奇流。 主要目的是了解奇异集的结构和奇异集附近解的几何。研究计划的第二部分是由几何群论和刚性问题，特别是负弯曲（或Gromov双曲）空间的渐近结构激发的。这导致了使用近几年发展起来的分析工具对具有自相似特征的空间进行研究。 这里的目标是准对称变形的空间到一个最佳的形式，如果可能的话，为了揭示隐藏的对称性;显示没有隐藏的对称性存在;或显示，原来的空间真的是一个熟悉的东西变形。 这个项目的目的是研究热方程的两个非线性类似物：曲面的平均曲率演化和汉密尔顿的里奇流。平均曲率流是一个标准的（理想化的）物理过程模型，它涉及空间中两个区域之间的一个演化表面或界面。 Ricci流--一个控制随时间演化的弯曲空间几何的方程--最近成为头条新闻，因为它在佩雷尔曼关于100年前的庞加莱猜想的壮观工作中发挥了突出作用。 关于这两个方程的解，我们已经知道了很多，但仍有许多基本的问题没有解决，特别是那些与奇点的形成有关的问题，这在佩雷尔曼的工作中起着核心作用。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利用过去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研究。 这里的目标是将空间变形为最佳形式，如果可能的话，为了揭示隐藏的对称性，显示不存在隐藏的对称性，或者显示原始空间确实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。