Representations and Rigidity

三维流形是以我们生活的空间为局部模型的，但在全球范围内，它可能会截然不同。这种流形和它们的高维变体已被证明是极其重要的，出现在数学和物理的许多分支中。最近，对这些流形的研究经历了一个显著的转变时期，在理解流形的对称性和相关的所谓的“覆盖空间”方面取得了进展。这些被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这个研究项目的中心。该项目的目的是更好地了解更一般的群体，而不是由流形产生的群体。最近的工作集中在一个长期存在的问题上，即如何从“局部数据”中识别不同类别的群，并试图将群重构为全局数据。该项目集中在低维几何和拓扑中出现的群的各种环境下的表示(例如自由群、表面群和Klein群)。该项目解决了在有限生成的剩余有限群的环境中通过它们的有限完备性来区分这些群的问题。值得注意的是，自由小组的情况仍然悬而未决，并继续为这一方向的工作提供重点。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的拓扑的联系。该奖项反映了NSF的法定使命，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。

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.