Analysis and geometry of metric measure spaces

在过去的几十年里，几何和分析在现实世界的应用中变得越来越重要。我们现在能够收集海量的数据，因此开发快速有效的方法来分析这些数据是一个核心问题。在许多情况下，人们可以赋予数据集的点以距离和权重的概念，从而给出数据的几何表示。然后，可以尝试利用此几何图形并利用此信息来加快数据集的计算速度。这个项目寻求开发新的技术和定理来理解具有一定距离和(可能)重量概念的空间的几何和解析结构。该项目将解决的一些基本问题如下：我可以在其他空间(例如欧几里得空间)中表示我的空间，以便以高保真度保留几何图形吗？我的空间是否表现出低维行为，尽管我生活在一个可能的高维空间(或无限维空间)？我的空间有没有微积分可以用来分析空间上的函数？这个项目将研究某些类型的度量空间的解析和几何性质。在项目的第一部分，PI将研究有关度量嵌入的问题，包括是否将Hilbert空间的倍化子集嵌入到欧氏空间中，以及开发双Lipschitz度量不变量来度量一致光滑性。该项目的第二部分将专注于检测公制度量空间中的可纠正行为。可校正空间允许通过简单的模型空间(通常是欧几里德空间)进行参数化，因此可以表现出低维行为。PI将学习卡诺的可纠性概念，并将这些几何性质与奇异积分的有界性联系起来。最后，PI将继续研究由Cheeger提出的度量度量空间上的Lipschitz可微性的性质。该项目将确定实值函数和RNP Banach空间值函数之间的这种区别是否不同。如果这些概念最终被证明是不同的，那么PI将寻求发现前者的几何机制(PI已经被几何特征描述为后者)。PI还将寻求将可微性理论扩展到非Banach空间目标。

Geometry and analysis have become increasingly important to real world applications in the last few decades. We are now capable of gathering immense amounts of data, and so it is a central concern to develop quick and efficient ways of analyzing this data. In many instances, one can endow the points of a data set with a notion of distance and weight, thereby giving us a geometric representation of the data. One can then try to take advantage of this geometry and leverage this information to speed up computations on the data set. This project seeks to develop new techniques and theorems for understanding the geometric and analytic structure of spaces with some notion of distance and (possibly) weight. Some basic questions the project will address are the following: Can I represent my space in some other space (e.g. Euclidean space) so that the geometry is preserved with high fidelity? Does my space exhibit low dimensional behavior despite living in a possibly high (or infinite) dimensional space? Is there a calculus available on my space that I can use to analyze functions on the space?This project will study analytic and geometric properties of certain classes of metric measure spaces. In the first part of the project, the PI will study problems about metric embeddings including whether doubling subsets of Hilbert space embed into Euclidean spaces and developing biLipschitz metric invariants to measure uniform smoothness. The second part of the project will focus on detecting rectifiable behavior in metric measure spaces. Rectifiable spaces admit parameterizations by simple model spaces (usually Euclidean space) and so can exhibit low dimensional behavior. The PI will study Carnot notions of rectifiability and also link these geometric properties to boundedness of singular integrals. Finally, the PI will continue his study into the nature of Lipschitz differentiability on metric measure spaces as introduced by Cheeger. The project will determine whether such differentiation differs between real valued and RNP Banach space valued functions. If these notions turn out to be different, the PI will then seek to discover the geometric mechanisms for the former (the PI has already been geometrically characterized the latter). The PI will also seek to broaden the differentiability theory to non-Banach space target.