CAREER: Random spaces and groups

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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.