Differentiable structures on metric measure spaces, einstein spaces, quantitative behavior of singular sets

度量测度空间、爱因斯坦空间、奇异集的定量行为上的可微结构

• 批准号: 1406407
• 负责人: Jeff Cheeger
• 金额: $ 44.07万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406407&HistoricalAwards=false
• 关键词: Differentiable; structures; metric; measure; spaces

The project is part of an ongoing program involving the study of situations that are in some way singular i.e. the objects may have discontinuities, may not be everywhere smooth, may have subsets on which they become infinite, etc. One theme involves putting constraints on the size or nature of the singularities which can arise in the solutions to certain nonlinear partial differential equations (PDEs) which are of importance in mathematics and physics. The principle investigator and collaborators have introduced methodology which, for a significant class of such PDEs, provides better control on the singularities than was previously available. Another theme is to study spaces which can be quite wild (e.g. they can have fractional dimensions) but are nonetheless well enough behaved so that one can employ the methods of calculus, if differentiation is understood in a sufficiently generalized sense. Apart from their intrinsic interest, these methods have had a surprising application to a basic problem in theoretical computer science, the sparsest cut problem with general demands. The project focuses on three main areas: 1) The analytical and geometric structure of metric measure spaces with Lipschitz differentiable structure. 2) Degeneration of Einstein metrics. 3) Quantitative behavior of singular sets. Regarding 1), a basic question is to study the extent to which Lipschitz differentiability spaces are more general than spaces for which the measure satisfies a doubling condition and a Poincar\'e inequality holds in the sense of Heinonen-Koskela. Recent work with Bate characterizing Lipschitz differentiability spaces in terms of Alberti representations should play an important role. Regarding 2), a challenging question is whether for noncollapsed Gromov-Hausdorff limits of Einstein spaces with bounded Einstein constant, the singular set has Hausdorff codimension 4 (as conjectured by M. Anderson). Regarding 3), a goal is to extend to new cases, the techniques developed by the principle investigator, with A. Naber (and partly with R. Haslhofer), for studying the quantitative behavior of singular sets of certain elliptic and parabolic PDEs.

该项目是一个正在进行的计划的一部分，涉及研究在某种程度上是奇异的情况，即对象可能有不连续性，可能不是到处光滑，可能有子集，它们成为无限的，其中一个主题是对某些非线性偏微分方程（PDE）的解中可能出现的奇点的大小或性质施加约束这在数学和物理学中很重要。主要研究者和合作者介绍了一种方法，这种方法对于一类重要的偏微分方程，提供了比以前更好的奇点控制。另一个主题是研究空间，它可以是非常狂野的（例如，它们可以有分数维），但仍然表现得足够好，以便人们可以使用微积分的方法，如果微分在足够广义的意义上被理解的话。除了他们的内在利益，这些方法有一个令人惊讶的应用在理论计算机科学的一个基本问题，稀疏切割问题的一般要求。 该项目主要集中在三个方面：1）具有Lipschitz可微结构的度量测度空间的分析和几何结构。2)爱因斯坦度规的退化。3)奇异集的定量行为。关于1），一个基本的问题是研究Lipschitz可微空间比测度满足加倍条件的空间更一般的程度，并且Poincar\'e不等式在Heinonen-Kokorena意义下成立。最近的工作与贝特特征Lipschitz微分空间的Alberti表示应发挥重要作用。关于2），一个具有挑战性的问题是，对于具有有界爱因斯坦常数的爱因斯坦空间的非塌陷Gromov-Hausdorff极限，奇异集是否具有Hausdorff余维4（如M.安德森）。 关于3），目标是将主要研究者开发的技术扩展到新病例，其中A。Naber（部分与R。Haslhofer），研究了某些椭圆和抛物偏微分方程奇异集的定量行为。