Asymptotic and quantitative geometry of groups and spaces

群和空间的渐近和定量几何

• 批准号: 1612061
• 负责人: Robert Young
• 金额: $ 12万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612061&HistoricalAwards=false
• 关键词: Asymptotic; quantitative; geometry; groups; spaces

AbstractAward: DMS 1612061, Principal Investigator: Robert YoungThis project proposes to study the geometry of surfaces in some spaces arising from mathematics and computer science. Surfaces are fundamental objects in geometry, and the shape of surfaces in a space provides insight into the space. The behavior of surfaces that are minimal or near-minimal is especially important. For example, minimal surfaces, like soap films, are known to be smooth under many conditions, but surfaces that are only close to minimal can be rough. The PI intends to develop new tools to study minimal and close-to-minimal surfaces in a variety of spaces and use them to study geometric and analytical problems. One proposed application of these tools is the study of the accuracy of the Goemans-Linial algorithm, an algorithm to approximate the best way of cutting an object into two roughly equal pieces. This algorithm is a key ingredient in many "divide-and-conquer" algorithms, which solve complex problems by decomposing them into simpler ones.The PI plans to develop new quantitative tools for studying the geometry of surfaces in groups and spaces and apply them to questions in geometric group theory, geometric measure theory, and theoretical computer science. First, the project aims to prove a conjecture of Gromov and Thurston on the filling functions of lattices in symmetric spaces, providing new understanding of the large-scale geometry of these spaces. Second, the project will explore decompositions of surfaces embedded in Euclidean space and try to use new tools from geometric measure theory to bound the geometry and topology of these surfaces. Third, the project will analyze surfaces of finite perimeter in the Heisenberg group and other nilpotent groups. If this last goal is successful, it would lead to sharp bounds on embeddings of the Heisenberg group into Banach spaces and sharp bounds on the accuracy of the best known approximate solution to the Sparsest Cut problem.

AbstractAward：DMS 1612061，首席研究员：Robert Young该项目旨在研究由数学和计算机科学产生的某些空间中的曲面几何。 曲面是几何学中的基本对象，空间中曲面的形状提供了对空间的洞察。 最小或接近最小的曲面的行为尤其重要。 例如，最小表面，如肥皂膜，已知在许多条件下是光滑的，但仅接近最小的表面可能是粗糙的。PI打算开发新的工具来研究各种空间中的极小和接近极小的曲面，并使用它们来研究几何和分析问题。 这些工具的一个拟议应用是研究Goemans-Linial算法的准确性，该算法近似于将物体切割成两个大致相等的部分的最佳方式。 该算法是许多"分而治之"算法的关键组成部分，这些算法通过将复杂问题分解为简单问题来解决复杂问题。PI计划开发新的定量工具，用于研究群和空间中的曲面几何，并将其应用于几何群论、几何测度论和理论计算机科学中的问题。 首先，该项目旨在证明Gromov和Thurston关于对称空间中格的填充函数的猜想，为这些空间的大规模几何提供新的理解。 其次，本计画将探讨嵌入欧几里德空间的曲面分解，并尝试使用几何测度理论的新工具来界定这些曲面的几何与拓扑。 第三，该项目将分析海森堡群和其他幂零群中有限周长的曲面。 如果这最后一个目标是成功的，它将导致尖锐的边界上嵌入的海森堡群到Banach空间和尖锐的边界上的准确性最知名的近似解决方案的Sparling-Cut问题。

项目成果

The integrality gap of the Goemans-Linial SDP relaxation for Sparsest Cut is at least a constant multiple of √ log n

Sparsest Cut 的 Goemans-Linial SDP 松弛的完整性差距至少是 �log n 的常数倍

• DOI: 10.1145/3055399.3055413
• 发表时间: 2017
• 期刊: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017
• 作者: Naor, Assaf;Young, Robert
• 通讯作者: Young, Robert

Dehn functions and Hölder extensions in asymptotic cones

渐近锥中的 Dehn 函数和 Hölder 扩展

• DOI: 10.1515/crelle-2018-0041
• 发表时间: 2019
• 期刊: Journal für die reine und angewandte Mathematik (Crelles Journal
• 作者: Lytchak, Alexander;Wenger, Stefan;Young, Robert
• 通讯作者: Young, Robert

The distortion dimension of $$\mathbb Q$$ Q -rank 1 lattices

$$mathbb Q$$ Q 阶 1 格子的畸变维数

• DOI: 10.1007/s10711-016-0189-6
• 发表时间: 2017
• 期刊: Geometriae Dedicata
• 影响因子: 0.5
• 作者: Leuzinger, Enrico;Young, Robert
• 通讯作者: Young, Robert

Vertical perimeter versus horizontal perimeter

• DOI: 10.4007/annals.2018.188.1.4
• 发表时间: 2018-06-01
• 期刊: ANNALS OF MATHEMATICS
• 影响因子: 4.9
• 作者: Naor, Assaf;Young, Robert
• 通讯作者: Young, Robert

Quantitative nonorientability of embedded cycles

嵌入式循环的定量非定向性

• DOI: 10.1215/00127094-2017-0035
• 发表时间: 2018
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Young, Robert