Analysis and geometry of metric measure spaces

在过去的几十年里，几何和分析对真实的世界的应用变得越来越重要。 我们现在能够收集大量的数据，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，可以赋予数据集的点距离和权重的概念，从而为我们提供数据的几何表示。 然后可以尝试利用这种几何形状并利用这些信息来加速数据集的计算。 该项目旨在开发新的技术和定理，用于理解具有距离和（可能）重量概念的空间的几何和分析结构。 该项目将解决的一些基本问题如下：我能否在其他空间（例如欧几里得空间）中表示我的空间，以便保持几何形状的高保真度？ 尽管我生活在一个可能的高维（或无限维）空间中，我的空间是否表现出低维行为？ 在我的空间上是否有可用的微积分，我可以用它来分析空间上的函数？本专题将研究某些类度量测度空间的分析和几何性质。 在项目的第一部分，PI将研究度量嵌入的问题，包括希尔伯特空间的加倍子集是否嵌入到欧几里得空间中，以及开发biLipschitz度量不变量来度量均匀光滑性。该项目的第二部分将集中在度量空间中检测可纠正的行为。 可求长空间允许简单模型空间（通常是欧几里得空间）的参数化，因此可以表现出低维行为。PI将研究卡诺的可求长性概念，并将这些几何性质与奇异积分的有界性联系起来。 最后，PI将继续他的研究性质的Lipschitz微分度量测度空间介绍Cheeger。 该项目将确定这种差异是否不同的真实的值和RNP Banach空间值的功能。如果这些概念被证明是不同的，那么PI将寻求发现前者的几何机制（PI已经从几何上描述了后者）。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.