Representations and Rigidity

三维流形是局部模拟我们生活的空间，但从全局来看，它可能是完全不同的。这种流形及其高维变体已被证明是惊人的重要，出现在数学和物理的许多分支中。最近，流形的研究经历了一个显著的转变时期，随着对流形对称性和相关的所谓“覆盖空间”的理解取得了进展。它们被一个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，即人们如何从“本地数据”中识别出不同类别的群体，并试图将这些群体重构为全球数据。这个项目的重点是在各种环境下，在低维几何和拓扑中产生的群（例如自由群、曲面群和Kleinian群）的表示。该项目解决了在有限生成的剩余有限群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，自由组的案例仍然是开放的，并继续为这个方向的工作提供重点。该项目还涉及与数论、代数几何、映射类群和高维双曲流形拓扑的联系。在其他目标中，更好地理解结群的特征变异的正则分量的算法。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。

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.