Colloquium-Distinguished Speculative First of April Talk (with Center for Quantum Computing, Information, Logic, and Topology): Skein modules and Quantum Computing in 3-manifolds

Abstract: The mathematical model of anyons (quasi-particles used in the theory of topological quantum computation) are Unitary Modular Tensor Categories. These are also used to construct topological quantum field theories (TQFT) in dimension 2+1, and in particular define invariants of framed links in 3-manifolds. The invariants can be thought of as quantum amplitudes of anyon world lines in a 3-dimensional spacetime. They also satisfy skein relations, which can be used to define skein modules of framed links in 3-manifolds. Thus quantum amplitudes, which we think of as the measurement results of “quantum computing in a 3-manifold,” are functionals on skein modules. These modules are hard to compute in general, and it is not very well-known how they relate to the geometric topology of 3-manifolds. This, of course, is also true for the quantum invariants. Thus it is interesting to study the relation between these concepts. We describe some results known around these problems and speculate about relations to finite type invariants (Kontsevich integral) and categorifications (link homologies).    

Short Bio: Dr. Uwe Kaiser received his PhD and habilitation from the University of Siegen in Germany. Additionally he also holds a K12-teaching certificate for Mathematics and Physics in Germany. Currently he is Associate Professor in the Mathematics Department at Boise State University, where he also serves as Associate Chair. His research interests are in Geometric and Algebraic Topology, more recently in Low Dimensional Topology and Quantum Topology. He is also working on several projects and grants related to STEM Education at Boise State University.