Einstein Manifolds and Analysis on Metric Measure Spaces

[摘要]获奖：dms -0704404首席研究员：Jeff cheeger第一个主要目标（与G. Tian合作）是描述爱因斯坦度量可以简并。在我们之前的工作中，我们给出了爱因斯坦度量4维坍缩的很好的局部描述，表明在远离确定数量的点时，坍缩发生在有界曲率下，并且在极限下，产生具有连续对称性的爱因斯坦度量。我们希望将这个局部描述全球化，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. Kleiner联合，是研究具有无限维Banach空间值的（Lipschitz）函数的一阶计算度量度量空间。在这种情况下，微分定理可以得到双lipschitz非嵌入定理。对于计算机科学中的应用，最有趣的目标是空间“Ell-One”。在这种情况下，一个Lipschitz函数不一定是可微的（在任何地方），除非（正如Kleiner和我发现的）可微性被理解在一个合适的扩展意义上。这种感觉仍然很强烈，足以让我们给戈曼斯-利尼尔猜想一个反例。本项目（与Kleiner和A. Naor合作）的一个重要方面是使非嵌入定理量化（这是计算机科学家真正想要的）。我们的方法需要几何测量理论的新技术的发展。对某些特殊的（高维的）光滑弯曲物体的研究被称为爱因斯坦流形，这在数学和现代物理学中很重要，例如在相对论和弦理论中。特别是，我们想要一个理论，告诉我们什么是这类物体“最扭曲”的例子。我们的项目与这个问题有关。第二个重点与“度量空间”有关。通过度量空间，人们可以理解任何物体的集合，其中任何两个物体之间都有距离的概念。例如，物体可能是指纹，一个合适的距离概念将使计算机能够检测出哪个指纹与给定的指纹非常匹配。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“复杂”的度量空间，一个人想要理解，是否可以实现（也许以一种不明显的方式）作为一个“简单”的度量空间，一个人已经理解的子集。在我们的项目中，纯数学开发了复杂的工具，可以在各种情况下使用（纯粹的和应用的兴趣）来决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.