Mathematics Colloquia and Seminars

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I will discuss various theorems and work in progress that all have a common theme:  Given a large group G with a small subset S, the theorem or the goal is to show that S either generates G (if G is discrete) or generates a dense subgroup of G (if G is a Lie group).  If G is a Lie group densely generated by S, then a related goal is to show that S efficiently generates an ε-net of G.  Different versions of this question arise in the contexts of circuit compilation in quantum computing, chaos in braid group and mapping class group representations in quantum algebra, computational intractability of certain topological invariants in 3 dimensions, and the classification of branched coverings of surfaces.  Various ideas in group theory arise as well, including lower central series of free groups, arithmetic lattices, and a non-commutative generalization of the Chinese remainder theorem.