February 26 | Eric Samperton (UCSB)
Khovanov homology: what it is and how to compute it
This talk will be half intro and half daydreaming. I'll briefly review the construction of the Khovanov homology of a knot and its connection to the Jones polynomial. We'll then consider the following question: if the Jones polynomial is related to quantum computing, how does Khovanov homology relate to quantum computing? In particular, Kuperberg exploited ideas from quantum complexity to show that approximating the Jones polynomial of a knot is #P-hard. As usual in topological quantum computing, his work exploits an analogy between knots and quantum circuits. Is there a categorified version of this analogy?
I'll argue that-surprisingly, perhaps-the physically "natural" way to try to compute (or compute *with*) Khovanov homology appears to be more closely related to adiabatic quantum computing instead of quantum circuits.
March 19 | Angus Gruen (Caltech)
February 12 | Colleen Delaney (UCSB)
A mathematical introduction to topological quantum computing
This is a talk aimed at advanced undergraduates and graduate students working outside the area of topological quantum computing. I will motivate the study of topological quantum computing and explain its mathematical formalism in an accessible way that does not presuppose exposure to category theory or quantum physics.
January 22 | Tian Yang (Texas A&M)
Some recent progress on the volume conjecture for the Turaev-Viro invariants
In 2015, Qingtao Chen and I conjectured that at the root of unity \(\exp(2\pi i/r)\) instead of the usually considered root \(\exp(\pi i/r)\), the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially with growth rates respectively the hyperbolic and the complex volume of the manifold. In this talk, I will present a recent joint work with Giulio Belletti, Renaud Detcherry and Effie Kalfagianni on an infinite family of cusped hyperbolic 3-manifolds, the fundamental shadow links complement, for which the conjecture is true.
January 8 | Eric Rowell (Texas A&M)
Metaplectic Modular Categories and Property F
I will discuss some progress on the property F conjecture, which states that the braid group representations associated with a simple object in a braided fusion category have finite images if and only if the objects FP-dimension is the square root of an integer. Metaplectic modular categories provide a large class of examples with such "weakly integral" objects, and verifying the conjecture here may provide some insight into the more general case.