Multiscale Methods in Quantitative Geometry

定量几何中的多尺度方法

• 批准号: 2005609
• 负责人: Robert Young
• 金额: $ 33.69万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005609&HistoricalAwards=false
• 关键词: Multiscale; Methods; Quantitative; Geometry

This NSF award provides funding for a project to develop new methods for working with objects with complicated geometry at many different scales. Coastlines, clouds, and leaves are examples of such objects that occur in nature. The famous coastline paradox states that since a coastline is so rough at so many different scales, it has no defined length. The PI plans to develop and study new methods to build such multiscale objects and to break them down into easily handled pieces. New ways to decompose objects often lead to great advances in mathematics, where simple problems can have complicated, multiscale solutions. The project’s three-prong approach includes studying ways to break down complex objects, build complex objects out of simple pieces, and to measure multiscale complexity. In addition to the research, the PI will train graduate students and postdocs in the techniques developed by this project through advising, seminars, minicourses, and reading groups and disseminate material on his webpage for public access.The project aims to study problems in metric geometry, geometric measure theory, and harmonic analysis related to maps and surfaces with multiscale structure, that is, objects like Lipschitz and Holder maps or fractals, that are hard to approximate by affine maps or by planes. One focus is the study of geometric problems that have non-smooth solutions but no smooth solutions, such as the Nash Embedding Theorem. Solutions to these problems often use multiscale or self-similar structure to break rules that smooth maps have to satisfy, and one aim of this project is to understand when and why this phenomenon occurs. Another focus is quantitative and uniform rectifiability. Uniform rectifiability has been a powerful tool for studying singular integrals and geometric measure theory in Euclidean space. Recent work has made it possible to find sharp bounds on the quantitative rectifiability of surfaces in the Heisenberg group, and this project will explore the possibility of extending notions of uniform rectifiability to the Heisenberg group and using them to solve problems in the geometry of Euclidean space and the Heisenberg group.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.

这个NSF奖为一个项目提供了资金，该项目旨在开发在许多不同尺度上处理复杂几何物体的新方法。海岸线、云彩和树叶都是自然界中出现的物体。著名的海岸线悖论指出，由于海岸线在许多不同的尺度上都很粗糙，所以它没有确定的长度。PI计划开发和研究制造这种多尺度物体的新方法，并将它们分解成易于处理的碎片。分解对象的新方法通常会带来数学上的巨大进步，简单的问题可以有复杂的、多尺度的解决方案。该项目的三管齐下的方法包括研究分解复杂物体的方法，用简单的碎片构建复杂物体的方法，以及测量多尺度复杂性的方法。除了研究之外，PI还将通过咨询、研讨会、迷你课程和阅读小组等方式，对研究生和博士后进行该项目的技术培训，并在其网页上发布资料，供公众查阅。该项目旨在研究与多尺度结构的地图和曲面相关的度量几何、几何测量理论和谐波分析问题，即像Lipschitz和Holder地图或分形这样难以用仿射映射或平面近似的对象。一个重点是研究具有非光滑解但没有光滑解的几何问题，如纳什嵌入定理。这些问题的解决方案通常使用多比例尺或自相似结构来打破光滑地图必须满足的规则，这个项目的一个目的是了解这种现象何时以及为什么会发生。另一个重点是定量和统一的可纠正性。一致可整直性是研究欧几里德空间奇异积分和几何测度理论的有力工具。最近的工作已经使得在海森堡群中找到表面的定量可矫正性的明确界限成为可能，本项目将探索将均匀可矫正性的概念扩展到海森堡群的可能性，并利用它们来解决欧几里得空间和海森堡群中的几何问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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

Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group

• DOI: 10.1007/s00526-022-02264-x
• 发表时间: 2022-08-01
• 期刊: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
• 影响因子: 2.1
• 作者: Young, Robert

Harmonic intrinsic graphs in the Heisenberg group

• DOI: 10.2422/2036-2145.202105_054
• 发表时间: 2020-12
• 期刊: ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
• 作者: Robert Young

Undistorted fillings in subsets of metric spaces

度量空间子集中的不失真填充

• DOI: 10.1016/j.aim.2023.109024
• 发表时间: 2023
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Basso, Giuliano;Wenger, Stefan;Young, Robert
• 通讯作者: Young, Robert