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

该项目是一个正在进行的项目的一部分，该项目涉及研究在某种程度上单一的情况，即物体可能具有不连续，可能不是处处光滑，可能具有无限的子集，等等。其中一个主题涉及对某些非线性偏微分方程（PDEs）的解中可能出现的奇点的大小或性质施加约束，这在数学和物理中很重要。主要研究者和合作者已经引入了一种方法，这种方法对于一类重要的pde来说，提供了比以前更好的奇点控制。另一个主题是研究空间，它可以是相当狂野的（例如，它们可以有分数维），但仍然表现得足够好，如果在一个足够广泛的意义上理解微分，人们可以使用微积分的方法。除了它们内在的兴趣之外，这些方法在理论计算机科学的一个基本问题上有一个惊人的应用，这个问题是具有一般需求的最稀疏的问题。本课题主要研究三个方面：1)具有Lipschitz可微结构的度量度量空间的解析结构和几何结构。2)爱因斯坦度量的退化。3)奇异集的数量行为。关于1)，一个基本问题是研究Lipschitz可微空间在多大程度上比测度满足加倍条件且在Heinonen-Koskela意义上庞加莱不等式成立的空间更一般。最近与Bate在Alberti表示方面刻画Lipschitz可微空间的工作应该发挥重要作用。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坍缩Gromov-Hausdorff极限，奇异集是否具有Hausdorff余维4（如M. Anderson所推测的）。关于3)，目标是将主要研究者与a . Naber（部分与R. Haslhofer）一起开发的技术扩展到新的情况下，用于研究某些椭圆和抛物偏微分方程的奇异集的定量行为。

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.