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

欧几里得和非欧空间中的集合和测度的几何

• 批准号: 2154613
• 负责人: Raanan Schul
• 金额: $ 36.99万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154613&HistoricalAwards=false
• 关键词: Geometry; Sets; Measures; Euclidean; Non

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.

现代世界已经掌握了数据。可以用数学术语制定以有用和有序方式组织大量数据的任务。该项目调查了以下问题的数学类似物：我们期望以有用的方式组织多少数据？在此类组织的过程中出现了哪种类型的几何结构？如果我们允许自己无视一定数量的信息，并且可以量化这些答案吗？最后，是否有实施此类数据组织的实用算法？在几何语言中，数据自然存在于高维空间或距离概念与欧几里得概念完全不同的空间中。该项目旨在将精心研究和有效的工具转移到从低维欧几里得空间中分析到更高维度和更一般的环境，从而使高维数据可以在低维，结构化的环境中“可视化”。该项目将涉及研究生和博士后的培训和指导，并旨在开发可以导致纯数学家与数据科学界参与的工具。在许多应用程序中，为许多应用程序集提供了一个大数据集，代表了高维空间的一个子集，并试图忠实地代表大量较低尺寸的数据集中的大部分数据集。 “忠实地”在这里意味着基本的几何特征要么被保存或轻度扭曲。 Lipschitz的几何变换条件量化了数据点之间距离的失真。迄今为止，前面的任务已使用各种方法从计算机科学家和应用数学家那里得到了关注。该项目研究了植根于分析和几何形状的数学方法。一个关键点是，给定数据通常具有额外的几何结构，例如，它可能具有较小的Hausdorff尺寸或接近低维歧管的结合。这种增加的结构允许使用谐波分析和几何测量理论中的工具，尤其是可纠正性的理论。该理论的定量版本（称为统一的重新可及性）将在新颖的度量设置中进行探索。要考虑的其他主题包括低等级分解定理的定量改进，Lipschitz的度量度量空间的分解，Bi-Lipschitz mappings的低距离分解以及Lipschitz的高维空间的参数化，具有更大的参数奖励，该奖项大于一个范围。影响审查标准。