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Group Actions on Manifolds and Related Spaces: Regularity, Structure, and Complexity

流形及相关空间的群作用：规则性、结构和复杂性

基本信息

批准号：
2002596
负责人：
Thomas Koberda
金额：
$ 17.31万
依托单位：
University of Virginia Main Campus
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：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2002596&HistoricalAwards=false
关键词：
Group Actions Manifolds Related Spaces

项目摘要

Manifolds are fundamental objects in mathematics which generalize and systematize space as we perceive it, and as such have been of great interests to mathematicians for a long time. One fruitful way in which manifolds can be studied is through their symmetries, which taken together form an algebraic object called a group. The focus of this project is the interaction between the structure of a manifold, both local (up close) and global (as an ensemble), and its symmetries, or group actions on manifolds for short. The project aims to resolve some basic questions in this area and investigate some new phenomena. Among the problems of particular interest are those concerning smoothness of group actions, in the sense of calculus. A resolution of some of these problems has the potential to yield insight into old open questions about exotic spheres, which are certain manifolds which can be deformed into standard spheres, but only in a highly singular way. Groups acting on manifolds are a broad class of objects which have received a large amount of attention, and their theory has developed rapidly in recent years. Planned broader impacts include research experiences for undergraduates, conference organization, and a book on right-angled Artin groups.This project will contribute to these advances in several ways. One area of focus for the project is critical regularity of group actions. Previous work with Kim will be extended in order to compute the critical regularity in one dimension of right-angled Artin groups, which would then be the first natural class of non-nilpotent groups whose precise one-dimensional critical regularity is both finite and known. Moreover, the principal investigator intends to develop critical regularity in higher dimensions, with the goal of encoding the diffeomorphism type of a compact manifold by a set of finitely generated groups. Such a result would give an algebraic approach to the 4-dimensional smooth Poincare conjecture. Another focus of the project is to develop a structure theory for thin subgroups of semisimple Lie groups, with the goal of resolving a conjecture of Shalom about the discreteness of their commensurators. The final main focus of the project is in the area of mapping class groups of surfaces, where the PI has outlined a program to prove the nonlinearity of mapping class groups, and to introduce logic into low dimensional topology by investigating the model theory of the curve graph of a surface.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.

流形是数学中的基本对象，它将我们所感知的空间进行了概括和系统化，因此长期以来一直受到数学家的极大兴趣。流形研究的一个富有成效的方法是通过它们的对称性，它们一起形成了一个称为群的代数对象。这个项目的重点是流形的结构之间的相互作用，包括局部（近距离）和全局（作为系综），以及它的对称性，或简称流形上的群作用。该项目旨在解决这一领域的一些基本问题，并调查一些新现象。其中特别感兴趣的问题是那些有关光滑的群体行动，在微积分的意义。其中一些问题的解决方案有可能深入了解有关奇异球的旧的开放问题，奇异球是某些可以变形为标准球的流形，但只能以高度奇异的方式变形。作用于流形上的群是一类广受关注的对象，近年来其理论发展迅速。计划的更广泛的影响包括本科生的研究经验，会议组织，和一本书的直角阿廷集团。这个项目将有助于这些进步在几个方面。该项目的一个重点领域是群体行动的关键规律性。以前的工作与金将被扩展，以计算临界正则性在一维直角阿廷群，这将是第一个自然类的非幂零群的精确一维临界正则性是有限的和已知的。此外，主要研究者打算在更高的维度上发展临界正则性，目标是通过一组非线性生成群来编码紧致流形的非线性同态类型。这样的结果将给出四维光滑庞加莱猜想的一个代数方法。该项目的另一个重点是发展半单李群的薄子群的结构理论，目的是解决Shalom关于其离散子的猜想。该项目的最后一个主要重点是在映射类组的表面，其中PI概述了一个程序，以证明映射类组的非线性，通过研究曲面曲线图的模型理论，将逻辑引入低维拓扑。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。

项目成果

期刊论文数量（8）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

An algebraic characterization of $k$–colorability

$k$–可着色性的代数表征

DOI：
10.1090/proc/15391
发表时间：
2021
期刊：
Proceedings of the American Mathematical Society
影响因子：
1
作者：
Flores, Ramón;Kahrobaei, Delaram;Koberda, Thomas
通讯作者：
Koberda, Thomas

Shapes of hyperbolic triangles and once-punctured torus groups

双曲三角形和一次穿孔环面群的形状

DOI：
10.1007/s00209-021-02745-3
发表时间：
2021
期刊：
Mathematische Zeitschrift
影响因子：
0.8
作者：
Kim, Sang-hyun;Koberda, Thomas;Lee, Jaejeong;Ohshika, Ken’ichi;Tan, Ser Peow;Gao, Xinghua
通讯作者：
Gao, Xinghua

Small C1 actions of semidirect products oncompact manifolds

紧流形上半直积的小 C1 作用