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。我们现在有能力收集大量的数据，因此开发快速有效的方法来分析这些数据是一个中心问题。在许多情况下，我们可以赋予数据集中的点以距离和权重的概念，从而为我们提供数据的几何表示。然后可以尝试利用这个几何图形并利用这个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，以理解具有距离和（可能）重量概念的空间的几何和分析结构。该项目将解决以下一些基本问题：我能否在其他空间（例如欧几里得空间）中表示我的空间，以便高保真地保留几何形状？我的空间是否表现出低维行为，尽管生活在一个可能的高维（或无限）空间中？在我的空间里有微积分可以用来分析空间上的函数吗？这个项目将研究某些度量度量空间的解析和几何性质。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倍子集是否嵌入欧几里得空间，以及发展biLipschitz度量不变量来测量均匀光滑性。该项目的第二部分将重点关注度量度量空间中可校正行为的检测。可整流空间允许通过简单模型空间（通常是欧几里得空间）进行参数化，因此可以表现出低维行为。PI将研究卡诺可纠偏性的概念，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Cheeger引入的度量度量空间上的Lipschitz可微性的性质。该项目将确定这种微分是否不同于实值和RNP巴拿赫空间值函数。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已经在几何上描述了后者）。PI还将寻求将可微性理论扩展到非巴拿赫空间目标。

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.