Mathematics Colloquia and Seminars

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Given d>0, and a transitive subgroup G of S_d, Hurwitz studied tuples of elements of G lying in prescribed conjugacy classes, having trivial product, and generating G. Despite the elementary nature of the problem, many open questions remain. In a completely elementary fashion, I will focus on two counting questions: what is the number of such tuples, and what is the number of braid group orbits of such tuples? These questions both have deep connections to geometry, and I will discuss some new results and directions for further research.