Rigidity of Lipschitz and Related Mappings on Metric Spaces

Lipschitz 刚性及度量空间上的相关映射

• 批准号: 1758709
• 负责人: Guy David
• 金额: $ 10.47万
• 依托单位: Ball State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-18 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1758709&HistoricalAwards=false
• 关键词: Rigidity; Lipschitz; Related; Mappings; Metric

The most basic setting for differential calculus is the study of smoothly changing functions on the real line. In this project, more general, non-smooth, functions that are defined on geometric objects (metric spaces) that may be very different from the real line will be investigated. These geometries may be abstract objects with no reasonable embedding into any Euclidean geometry, or they may be fractal, or they may otherwise admit complicated behavior at all scales. Far from being a technical curiosity, non-smooth analysis and geometry have now become important tools in many areas of pure and applied mathematics and computer science, where non-Euclidean geometries may arise. For example, from discrete groups or dynamical systems, as limits of smooth objects, as large data sets, or in computational problems.In more precise terms, the goal of this project is to study the relationship between, on the one hand, the infinitesimal and global geometry of non-smooth spaces and, on the other hand, the analysis of Lipschitz and related classes of mappings defined on these spaces. One specific goal is to further understand the spaces which allow for differentiation of real-valued Lipschitz functions (in the sense of Cheeger): what topological and geometric properties can such spaces possess, and can we construct new examples? Another goal is to understand rigidity theorems for Lipschitz mappings and rectifiable curves in metric spaces. For example, when must Lipschitz mappings between metric spaces have more rigid (e.g., bi-Lipschitz) behavior on large subsets of their domains? Can we characterize rectifiable curves in metric spaces via local flatness conditions, as in the "Analyst's Traveling Salesman Theorem" of Jones in the plane? Understanding these questions involves combining techniques from classical geometric measure theory with those of the newer field of "analysis on metric spaces". Analytic investigations like these have also provided, and should continue to provide, insights into the geometry of the non-smooth.

差分微积分的最基本设置是研究真实线上平稳变化的功能。在这个项目中，将研究在几何对象（度量空间）上定义的更通用的，非平滑的功能，这些功能可能与实际线路大不相同。这些几何形状可以是抽象的对象，没有合理地嵌入任何欧几里得的几何形状，或者它们可能是分形的，或者它们在所有尺度上都可以接受复杂的行为。非平滑分析和几何形状远非技术好奇心，现在已经成为许多纯粹和应用数学和计算机科学领域的重要工具，在该领域可能会出现非欧国人的几何形状。 例如，从离散的组或动态系统，作为光滑对象的限制，作为大型数据集或计算问题。从更精确的术语中，该项目的目的是研究一方面，一方面，无限和全球的非平滑空间和全球几何形状之间的关系，另一方面，另一方面，另一方面，lipschitz的分析以及对这些麦克风的分析，以及这些分类的分析，并定义了这些定义的这些分类。一个具体的目标是进一步了解允许区分实价Lipschitz函数的空间（在Cheeger的意义上）：这些空间可以拥有哪些拓扑和几何特性，我们可以构建新的示例吗？另一个目标是了解Lipschitz映射的刚性定理和公制空间中可重新定义的曲线。例如，何时必须对公制空间之间的Lipschitz映射在其域的大子集上具有更多的刚性（例如Bi-Lipschitz）行为？我们可以通过局部平坦条件在公制空间中表征可整流的曲线，例如琼斯在飞机上的“分析师的推销员定理”中？理解这些问题涉及将经典几何测量理论与“公制空间分析”领域的技术结合在一起。 此类分析研究也提供了，并且应该继续提供对非平滑几何形状的见解。

项目成果

Lipschitz and Bi-Lipschitz maps from PI spaces to Carnot groups

Lipschitz 和 Bi-Lipschitz 从 PI 空间映射到卡诺群

• DOI: 10.1512/iumj.2020.69.8009
• 发表时间: 2020
• 期刊: Indiana University Mathematics Journal
• 影响因子: 1.1
• 作者: David, Guy;Kinneberg, Kyle
• 通讯作者: Kinneberg, Kyle

Rectifiability of planes and Alberti representations

平面的可修正性和 Alberti 表示

• DOI: 10.2422/2036-2145.201701_015
• 发表时间: 2019
• 期刊: Annali della Scuola normale superiore di Pisa. Classe di scienze
• 作者: David, Guy C.;Kleiner, Bruce
• 通讯作者: Kleiner, Bruce

Rigidity for convex-cocompact actions on rank-one symmetric spaces

• DOI: 10.2140/gt.2018.22.2757
• 发表时间: 2016-09
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Guy C. David;K. Kinneberg

A note on topological dimension, Hausdorff measure, and rectifiability

关于拓扑维数、豪斯多夫测度和可修正性的注释

• DOI: 10.1090/proc/15051
• 发表时间: 2020
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: David, Guy C.;Le Donne, Enrico
• 通讯作者: Le Donne, Enrico

A sharp necessary condition for rectifiable curves in metric spaces

度量空间中可矫正曲线的尖锐必要条件

• DOI: 10.4171/rmi/1216
• 发表时间: 2021
• 期刊: Revista Matemática Iberoamericana
• 作者: David, Guy;Schul, Raanan
• 通讯作者: Schul, Raanan