Einstein Manifolds and Analysis on Metric Measure Spaces

摘要奖：DMS-0704404主要研究者：杰夫·奇格第一个主要目标（与G. Tian）是为了描述爱因斯坦度规如何退化。 在我们以前的工作中，我们对四维Einstein度规的坍缩给出了很好的局部描述，表明在远离一定数量的点处，坍缩是有界曲率的，而在极限处，坍缩是连续对称的。 我们希望将这种局部描述全球化，并尽可能地将其扩展到更高的维度。该项目的第二个主要目标，与B联合。Kleiner，研究了（Lipschitz）函数值在无限维Banach空间上的度量测度空间上的一阶微积分.在这种情况下，微分定理可以产生双Lipschitz非嵌入定理。 对于计算机科学中出现的应用程序，最有趣的目标是空间“Ell-One”。在这种情况下，Lipschitz函数不需要是可微的（任何地方），除非（正如Kleiner和我发现的那样）可微性在适当的扩展意义上被理解。这种感觉仍然很强烈，足以使我们能够给出一个反例来反驳戈尔曼-线性猜想。 本项目的一个重要方面（与Kleiner和A. Naor）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）。 我们的方法需要在几何测量理论的新技术的发展。研究某些特殊的（高维）光滑弯曲的物体称为爱因斯坦流形是重要的数学和现代物理学，例如，在连接相对论和弦理论。 特别是，我们想要一个理论，告诉我们什么是这些物体的“最扭曲”的例子。 我们的项目就是关注这个问题。 第二个主要的焦点与“度量空间”有关。 通过度量空间，人们可以理解任何两个物体之间存在距离的物体的集合。 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密切匹配。在这个主题的纯粹和应用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”度量空间是否可以被实现（也许以一种不明显的方式）为一个人们已经理解的“简单”度量空间的子集。在我们的项目中，开发了纯数学中的sophistocated工具，这些工具可以用于各种情况（纯和应用兴趣），以确定这种方法的可行性。

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.