Department of Mathematics

Geometry and Topology

  •  Eric Samperton, UIUC
  •  Finite gauge groups, TQFT, and the computational complexity of 3-manifold invariants
  •  11/03/2020
  •  2:50 PM - 3:40 PM
  •  Online (virtual meeting)
  •  Honghao Gao (

I will give an overview of some relations between finite group theory, G-equivariant topological quantum field theory, and the computational complexity of invariants of 3-manifolds, both classical and quantum. We will start with one of the simplest kinds of invariants in knot theory: the coloring invariants, introduced by Fox when giving a talk to undergraduates in the 1950s. We will then build up to the idea of G-equivariant TQFT (aka homotopy QFT with target K(G,1)), which mathematically describes the topological order determined by a symmetry-enriched topological phase of matter. Physicists have studied these in part motivated by the search for new universal topological quantum computing architectures. Our goal will be to convey two complexity-theoretic lessons. First, when G is sufficiently complicated (nonabelian simple), the simple-to-define coloring invariants associated to G are, in fact, very difficult to compute, even on a quantum computer. Second, no matter what finite group G one uses, a 3-dimensional G-equivariant TQFT can not be used for universal topological quantum computation if the underlying non-equivariant theory is not already universal. This talk is based on joint works with Greg Kuperberg and Colleen Delaney.



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