CAREER: Large scale geometry and negative curvature

空间的对称性及其与空间底层几何结构的关系使研究人员对代数结构和几何结构之间的联系有了深刻的认识。这个项目的重点是双曲空间及其结构在其相关对称群上诱导的粗糙几何。项目活动还包括对各级数学学生进行教学和指导的举措，重点是针对系里和本地区代表性不足的学生。资助的研究有三个主要目标：将粗糙负曲率的概念，特别是非圆柱形双曲，推广到广泛的拓扑群；为了建立层次双曲群商的一个强稳定性结果，得到了一类可以用双曲度量空间上的作用完全描述的群；并将组合方法从立方群扩展到更大的CAT(0)群，重点关注负曲率和边界方面。该研究计划的部分内容将被纳入学生项目，作为扩大PI所在机构本科生研究机会的一种方式。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。

The symmetries of a space and its relation to the underlying geometric structure of the space has led researchers to deep insights into the connections between algebraic and geometric structures. This project focuses on hyperbolic spaces and the coarse geometry their structure induces on their associated symmetry groups. The project activity also includes initiatives in teaching and mentoring mathematics students at all levels, with a focus on targeting students from under-represented populations both at the department and in the region. The funded research has three main goals: to generalize notions of coarse negative curvature, particularly acylindrical hyperbolicity, to a broad class of topological groups; to establish a strong stability result for quotients of hierarchically hyperbolic groups, a class of groups which can be completely described by their actions on hyperbolic metric spaces; and to extend combinatorial methods from cubical groups to the larger class of CAT(0) groups, with a focus on aspects of negative curvature and boundaries. Parts of this research program will be incorporated into student projects as a way to expand access to undergraduate research at the PI’s institution.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.