Representations and Rigidity

表述和刚性

• 批准号: 1812397
• 负责人: Alan Reid
• 金额: $ 25.8万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812397&HistoricalAwards=false
• 关键词: Representations; Rigidity

A three-dimensional manifold is locally modelled on the space in which we live, but globally it can be quite different. Such manifolds and their higher-dimensional variants have proved to be astonishingly important, arising in many branches of mathematics and physics. Recently, study of these manifolds has seen a period of remarkable transformation, with progress made by understanding symmetries of the manifolds and the related so-called "covering spaces." These are succinctly encoded by a purely algebraic object known as the fundamental group. The interplay between the fundamental group and the manifold is at the center of this research project. The project aims to better understand more general groups in addition to those arising from manifolds. Recent work has focused on the longstanding question of how one might recognize various classes of groups from "local data" and on trying to reconstitute the group as global data.This project is focused on representations, in various settings, of groups arising in low-dimensional geometry and topology (e.g. free groups, surface groups, and Kleinian groups). The project addresses questions of distinguishing such groups in the setting of finitely-generated residually finite groups by their profinite completions. Remarkably, the case of the free group is still open and continues to provide focus for work in this direction. The project also addresses connections with number theory, algebraic geometry, mapping class groups, and the topology of higher dimensional hyperbolic manifolds. Among other objectives is a better understanding of the arithmetic of canonical components of character varieties of knot groups.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.

三维流形在我们所居住的空间中局部建模，但在全局范围内它可能有很大不同。 事实证明，这种流形及其高维变体非常重要，出现在数学和物理学的许多分支中。最近，这些流形的研究经历了一段显着的转变，在理解流形的对称性和相关的所谓“覆盖空间”方面取得了进展。 这些由称为基本群的纯代数对象简洁地编码。基本群和流形之间的相互作用是该研究项目的中心。该项目旨在更好地理解除了流形产生的群之外的更一般的群。最近的工作集中在一个长期存在的问题上，即如何从“本地数据”中识别各种类型的群，并尝试将群重建为全局数据。该项目的重点是在各种设置中低维几何和拓扑中出现的群（例如自由群、表面群和克莱因群）的表示。该项目解决了在有限生成的残差有限群的设置中通过它们的有限完成来区分这些群的问题。值得注意的是，自由团体的案件仍然悬而未决，并继续为这一方向的工作提供焦点。该项目还解决了与数论、代数几何、映射类群和高维双曲流形拓扑的联系。其他目标之一是更好地理解结组字符变体的规范组成部分的算术。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

Sequences of high rank lattices with large systole containing a fixed genus surface group

包含固定属表面群的具有大收缩期的高阶晶格序列

• 发表时间: 2019
• 期刊: New York journal of mathematics
• 影响因子: 0.6
• 作者: Long, D.D;Reid, A. W.
• 通讯作者: Reid, A. W.