Einstein Manifolds and Analysis on Metric Measure Spaces

爱因斯坦流形和度量测度空间分析

• 批准号: 0704404
• 负责人: Jeff Cheeger
• 金额: $ 33.1万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0704404&HistoricalAwards=false
• 关键词: Einstein; Manifolds; Analysis; Metric; Measure

AbstractAward: DMS-0704404Principal Investigator: Jeff CheegerThe first main goal (joint with G. Tian) is to describe the howEinstein metrics can degenerate. In our previous work we gave agood local description of collapse of Einstein metrics indimension 4, showing that away from a definite number of pointscollapse occurs with bounded curvature and in the limit, givesrise to Einstein metrics with continous symmetries. We want toglobalize this local description and, in so far as possible,extend it to higher dimensions. A second main goal of theproject, joint with B. Kleiner, is to study first order calculuson metric measure spaces for (Lipschitz) functions with values ininfinite dimensional Banach spaces. In this context,differentiation theorems can give rise to bi-Lipschitznonembedding theorems. For applications arising in computerscience, the most interesting target is the space "Ell-One". Inthis case, a Lipschitz function need not be differentiable(anywhere) unless (as Kleiner and I discovered) differentiabilityis understood in a suitable extended sense. This sense was stillstrong enough to enable us give a counter example to theGoemans-Linial conjecture. An important aspect of the presentproject (joint with Kleiner and A. Naor) is to make thisnonembedding theorem quantitative (which is what computerscientists really want). Our approach requires the developementof new techniques in geometric measure theory.The study of certain special (higher dimensional) smoothly curvedobjects called Einstein manifolds is important in mathematics andin modern physics e.g. in connection with the theory ofrelativity and in string theory. In particular, we want a theorywhich tells us what can be the "most distorted" examples of suchobjects. Our project is concerned with this issue. A secondmain focus has to do with "metric spaces". By a metric space,one can understand any collection of objects where there is anotion of distance between any two of them. For example, theobjects might be finger prints and a suitable notion of distancewould enable a computer to detect which finger prints closelymatcheda given one. A key issue in both the pure and appliedaspects of the subject, is to be able to decide whether a"complicated" metric space which one wants to understand, can berealized (perhaps in a non-obvious way) as a subset of a"simpler" metrric space which one already understands. In ourproject, sophistocated tools from pure mathematics are developedwhich can be used in various cases (of pure and applied interest)to decide the feasibility of such an approach.

摘要奖：DMS-0704404首席研究员：Jeff Cheeger第一个主要目标(与G.Tian合作)是描述爱因斯坦度规如何简并。在我们以前的工作中，我们给出了4维爱因斯坦度规崩溃的一个很好的局部描述，表明在远离一定数量的点的情况下，崩溃是有界曲率的，并且在极限下，给出了具有连续对称性的爱因斯坦度规。我们希望将这一地方性描述全球化，并尽可能将其扩展到更高的维度。该项目的第二个主要目标是与B.Kleiner合作，研究取值于无限维Banach空间的(Lipschitz)函数在度量度量空间上的一阶微积分。在这种情况下，微分定理可以产生双Lipschitznon嵌入定理。对于计算机科学中出现的应用，最有趣的目标是“Ell-One”空间。在这种情况下，Lipschitz函数不必(在任何地方)可微，除非(正如Kleiner和我所发现的)在适当的扩展意义上理解可微性。这种感觉仍然很强烈，足以让我们给出一个反例来反驳戈曼斯-利尼亚尔猜想。目前的项目(与Kleiner和A.Naor合作)的一个重要方面是使这个非嵌入定理定量化(这是计算机科学家真正想要的)。我们的方法需要发展几何测度论中的新技术。对某些特殊的(高维)光滑的被称为爱因斯坦流形的物体的研究在数学和现代物理中是重要的，例如在相对论和弦理论中。特别是，我们想要一个理论，告诉我们这类物体的“最扭曲”的例子是什么。我们的项目与这个问题有关。第二个主要关注点与“度量空间”有关。通过度量空间，人们可以理解任何两个对象之间有距离的任何对象集合。例如，物体可能是指纹，适当的距离概念将使计算机能够检测出哪些指纹与给定的指纹最匹配。在这个主题的纯和应用两个方面的一个关键问题是，能够决定一个人们想要理解的“复杂的”度量空间是否可以(也许以一种不明显的方式)被理解为一个已经理解的“更简单的”度量空间的子集。在我们的项目中，纯粹的数学工具被开发出来，这些工具可以在各种情况下(纯粹的和应用的兴趣)使用，以确定这种方法的可行性。