Geometry of Sets and Measures in Euclidean and Non-Euclidean Spaces

现代世界充斥着数据。以有用和有序的方式组织大量数据的任务可以用数学术语来表达。这个项目研究的问题，如以下数学类比：有多少数据，我们可以期望组织在一个有用的方式？在这样的组织过程中产生了什么样的几何结构？如果我们允许自己忽略一定数量的信息，这些答案会改变吗？这种选择的影响可以量化吗？最后，是否有实用的算法来实现这样的数据组织？在几何语言中，数据自然地驻留在高维空间或距离概念与欧几里德距离概念完全不同的空间中。该项目旨在将经过充分研究的有效分析工具从低维欧几里得空间转移到高维和更一般的环境中，使高维数据在低维结构化环境中“可视化”。该项目将涉及对研究生和博士后的培训和指导，旨在开发可以导致纯数学家和数据科学社区之间参与的工具。在许多应用中，人们被赋予一个大的数据集，表示为高维空间的子集，并且人们试图在低维空间中忠实地表示这个数据集的大部分。“忠实地”在这里意味着基本的几何特征要么被保留要么被轻微扭曲。几何变换的Lipschitz条件量化了数据点之间距离的失真。到目前为止，前面的任务已经得到了计算机科学家和应用数学家使用一系列方法的关注。这个项目研究植根于分析和几何的数学方法。一个关键点是，通常给定的数据具有额外的几何结构，例如，它可能具有小的Hausdorff维数或接近于低维流形的并集。这种增加的结构允许使用来自谐波分析和几何测量理论的工具，特别是可整流性理论。这个理论的定量版本，被称为均匀整流性，将探讨在新的度量设置。其他要考虑的主题包括低秩因子分解定理的定量改进，度量测度空间的Lipschitz分解，双Lipschitz映射的低失真因子分解，和Lipschitz参数化该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查进行评估来支持的搜索.

The modern world is awash in data. The task of organizing large amounts of data in useful and ordered ways can be formulated in mathematical terms. This project investigates mathematical analogs of questions such as the following: How much data can we expect to organize in a useful way? What type of geometric structures arise in the process of such organization? Do these answers change if we allow ourselves to disregard a certain amount of information, and can the impact of such a choice be quantified? Finally, are there practical algorithms to implement such data organization? In geometric language, data naturally resides in a high dimensional space or a space where the notion of distance is quite different from the Euclidean one. This project aims at transferring well studied and efficient tools for analysis from low-dimensional Euclidean spaces to higher-dimensional and more general settings, allowing high-dimensional data to be "visualized" in a lower-dimensional, structured environment. The project will involve the training and mentoring of graduate students and postdocs and aims to develop tools which can lead to engagement between pure mathematicians and the data science community.In many applications one is given a large data set, represented as a subset of a high-dimensional space, and one seeks to faithfully represent a large portion of this data set in a space of substantially lower dimension. "Faithfully" here means that essential geometric features are either preserved or mildly distorted. The Lipschitz condition for a geometric transformation quantifies the distortion of distances between data points. To date, the preceding task has received attention from computer scientists and applied mathematicians using a range of approaches. This project investigates mathematical approaches rooted in analysis and geometry. A key point is that often the given data has additional geometric structure, for example, it may have small Hausdorff dimension or be close to a union of low dimensional manifolds. Such added structure allows for the use of tools from harmonic analysis and geometric measure theory, especially, the theory of rectifiability. A quantitative version of this theory, known as uniform rectifiability, will be explored in novel metric settings. Other topics to be considered include quantitative improvements of low rank factorization theorems, Lipschitz decompositions of metric measure spaces, low-distortion factorization of bi-Lipschitz mappings, and Lipschitz parameterizations of high-dimensional spaces with parameterizing dimension greater than one.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.