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: 2054004
• 负责人: Guy David
• 金额: $ 10.27万
• 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: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054004&HistoricalAwards=false
• 关键词: Rigidity; Lipschitz; Related; Mappings; Metric

Classical calculus studies smoothly changing functions, curves, and surfaces inside the Euclidean space. In this project, the principal investigator will study calculus in arenas where the classical tools do not always apply. This includes non-smooth functions or sets, abstract geometries outside of the usual, Euclidean, geometry, and fractal spaces admitting complex behavior at many scales. The project aims to understand these objects by decomposing them into simpler pieces, embedding them into classical geometries, or approximating them by linear objects. Non-smooth analysis and geometry are important in many areas of pure and applied mathematics and computer science, since non-smooth problems arise in studying large data sets, in computational questions, and as limiting cases of smooth problems. The project will also emphasize results that are "quantitative": providing guaranteed estimates independent of the particular function or geometry being studied.More precisely, the project focuses on Lipschitz (and related) mappings. Building on his recent collaborative work, the principal investigator will study when Lipschitz mappings into metric spaces can be decomposed into or well-approximated by simpler mappings, like linear maps or maps that factor through trees. The principal investigator then plans to apply these techniques to provide new structural results on geometric objects defined through Lipschitz maps, like distance spheres or medial axes. The principal investigator will also investigate similar questions related to curves in metric spaces, continuing work on extensions of the so-called “Analyst’s Traveling Salesman Theorem” that links lengths of curves to quantitative flatness conditions. In addition, the principal investigator will investigate notions of differentiability of Lipschitz functions on non-smooth spaces, and how they constrain geometry. This is connected to the bi-Lipschitz embedding problem (“which spaces can be embedded in Euclidean space with bounded distortion?”), which is of major importance in geometry and computer science, and which will also be studied during the project. All these linked investigations will combine to improve our understanding of the analysis and geometry of non-smooth objects.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.

经典微积分研究欧几里得空间内平滑变化的函数、曲线和曲面。在这个项目中，主要研究者将研究经典工具并不总是适用的领域中的微积分，这包括非平滑函数或集合、抽象几何。通常的欧几里得几何和分形空间承认多种尺度的复杂行为，该项目旨在通过将它们分解为更简单的部分、将它们嵌入到经典几何中来理解这些对象。通过线性对象来近似它们，非光滑分析和几何在纯数学和应用数学以及计算机科学的许多领域中都很重要，因为非光滑问题出现在研究大数据集、计算问题以及光滑问题的极限情况中。该项目还将强调“定量”的结果：提供独立于所研究的特定函数或几何形状的有保证的估计。更准确地说，该项目的重点是利普希茨（和相关）映射，以他最近的合作工作为基础，首席研究员将。学习什么时候Lipschitz 映射到度量空间可以分解为更简单的映射或通过更简单的映射来很好地近似，例如线性映射或通过树分解的映射，然后首席研究员计划应用这些技术来为通过 Lipschitz 映射定义的几何对象提供新的结构结果。首席研究员还将研究与度量空间中的曲线相关的类似问题，继续研究链接的所谓“分析师旅行商定理”的扩展。此外，主要研究者将研究非光滑空间上 Lipschitz 函数的可微性概念，以及它们如何约束几何形状，这与双 Lipschitz 嵌入问题有关。嵌入到有界畸变的欧几里得空间中？”），这在几何和计算机科学中非常重要，并且也将在该项目期间进行研究，所有这些相关的研究将结合起来以提高我们对分析和几何的理解。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Lower bounds on mapping content and quantitative factorization through trees

通过树映射内容和定量分解的下限

• DOI: 10.1112/jlms.12595
• 发表时间: 2022
• 期刊: Journal of the London Mathematical Society
• 作者: David, Guy C.;Schul, Raanan
• 通讯作者: Schul, Raanan

Bi-Lipschitz embeddings of quasiconformal trees

拟共形树的 Bi-Lipschitz 嵌入

• DOI: 10.1090/proc/16252
• 发表时间: 2023
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: David, Guy;Eriksson-Bique, Sylvester;Vellis, Vyron
• 通讯作者: Vellis, Vyron

Quantitative decompositions of Lipschitz mappings into metric spaces

Lipschitz 映射到度量空间的定量分解

• DOI: 10.1090/tran/8930
• 发表时间: 2023
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: David, Guy;Schul, Raanan
• 通讯作者: Schul, Raanan

Bi-Lipschitz geometry of quasiconformal trees

拟共形树的 Bi-Lipschitz 几何

• DOI: 10.1215/00192082-9936324
• 发表时间: 2022
• 期刊: Illinois Journal of Mathematics
• 影响因子: 0.6
• 作者: David, Guy C.;Vellis, Vyron
• 通讯作者: Vellis, Vyron

A Non-injective Assouad-Type Theorem with Sharp Dimension

具有锐维数的非内射Assouad型定理

• DOI: 10.1007/s12220-023-01503-7
• 发表时间: 2024
• 期刊: The Journal of Geometric Analysis
• 作者: David, Guy C.