Abstract：

I'll give a quick intro to spherical fusion categories and the Turaev-Viro-Barrett-Westbury construction, which associates an invariant of oriented 3-dimensional manifolds to each such category. Some of the simplest spherical fusion categories are the so-called Tambara-Yamagami categories, which depend on the data of a finite abelian group A, a choice of isomorphism between A and its dual, and a sign +1 or -1. Despite their fairly simple definition, these categories are known to give rise to TVBW invariants that are NP-hard to compute. I'll explain what this means, and then describe my recent work with Colleen Delaney and Clement Maria establishing an efficient algorithm for computing these invariants on 3-manifolds with bounded first Betti number. I will also try to say a few things about why such complexity/algorithm results are interesting in the context of 3-manifold topology and quantum computation.