CAREER: Random spaces and groups

PI将研究随机细胞复合物的拓扑结构和随机群的几何结构。相变是一个重要的主题——例如，当一些参数变化时，同调或同伦群的阈值消失。同样有趣的是更多的几何概念，如等周不等式和双曲。随机细胞复合体提供了扩展图的高维类似物，这是一个最近非常活跃的主题，与随机矩阵理论有联系。概率方法为生成具有极端几何性质的对象提供了一个强大而通用的框架，这一领域的进展将导致拓扑组合学和几何群论中有趣的新例子和反例子。这项研究利用数学的几个领域来研究随机生成形状的细微定性特征。这些相互作用是数学本身固有的兴趣所在，并且这项工作也正在着眼于拓扑数据分析这一新兴领域的实际应用。受资助的教育活动将自然地与研究相结合，PI将指导从事与资助相关项目的研究生和本科生研究人员。PI将继续参与各种外展活动：科罗拉多数学奥林匹克、加拿大/美国数学夏令营、数学圈、大学生数学俱乐部和问题解决研讨会。

The PI will investigate the topology of random cell complexes and the geometry of random groups. Phase transitions are one important theme -- vanishing thresholds for homology or homotopy groups as some parameter varies, for example. Also of interest are more geometric notions, such as isoperimetric inequalities and hyperbolicity. Random cell complexes provide high-dimensional analogues of expander graphs, a subject that has been a very active lately, with connections to random matrix theory. The probabilistic method provides a powerful and general framework for generating objects with extremal geometric properties, and progress in this area will lead to intriguing new examples and counterexamples in topological combinatorics and geometric group theory.This research draws on several areas of mathematics to study subtle qualitative features of randomly generated shapes. These interactions are of intrinsic interest within mathematics itself, and the work is also being done with a eye towards practical applications in the emerging area of topological data analysis. The educational activities supported by the grant will be naturally integrated with the research, and the PI will mentor graduate and undergraduate researchers working on projects related to the grant. The PI will continue to be involved with various outreach activities: Colorado Math Olympiad, Canada/USA Mathcamp, math circles, undergraduate math clubs, and problem solving seminars.