Representations and Rigidity

三维流形是在我们生活的空间中局部建模的，但在全球范围内它可能完全不同。 这样的流形和它们的高维变体已经被证明是非常重要的，出现在数学和物理的许多分支中。最近，这些流形的研究已经看到了一个显着的转变时期，通过理解流形的对称性和相关的所谓的“覆盖空间”取得了进展。“这些都是由一个纯粹的代数对象被称为基本群简洁地编码。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之外的更一般的群。最近的工作集中在长期存在的问题，人们可能会认识到各种类别的群体从“本地数据”，并试图重建组作为global data.This项目的重点是表示，在各种设置中，出现在低维几何和拓扑结构的群体（如自由组，表面组，和Kleinian组）。该项目解决的问题，区分这些群体的设置有限生成的剩余有限群的profinite完成。值得注意的是，自由群体的案件仍然悬而未决，并继续为这方面的工作提供重点。该项目还涉及与数论，代数几何，映射类群，和高维双曲流形的拓扑结构的连接。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。

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.