CAREER: Large scale geometry and negative curvature

空间的对称性及其与空间的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和粗糙的几何结构诱导其相关的对称群。 该项目活动还包括在各级教学和辅导数学学生的举措，重点是针对学生的代表性不足的人口都在部门和该地区。资助的研究有三个主要目标：将粗糙负曲率的概念，特别是非圆柱双曲性，推广到广泛的一类拓扑群;建立一个强稳定性的结果，为分层双曲群，一类群，可以完全描述他们的行动对双曲度量空间;并将组合方法从立方群扩展到更大类的CAT（0）群，重点是负曲率和边界方面。该研究项目的部分内容将被纳入学生项目，以扩大PI所在机构的本科生研究机会。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.