This is an online seminar organized by Corey Jones, David Penneys, and Julia Plavnik. Topics of interest range widely over "quantum" mathematics, and include (but are certainly not limited to):

​

• Tensor categories

• Subfactors and operator algebras

• Hopf algebras and quantum groups

• Representation theory​

• Higher Categories

• TQFT and low dimensional topology

• Categorification

• Topological phases of matter

• Conformal field theory

• Algebraic quantum field theory

 

We typically meet Thursdays every other week from 2:00 pm - 3:00 pm, US Eastern Time Zone. If you would like to attend, please email Corey Jones at cormjones88@gmail.com to be added to the Google Group for announcements of abstracts and Zoom links.

​

Here's a link to our schedule from previous semesters.

​

Below is a list of currently scheduled talks for this semester (which is updated frequently!):

​

 

 

Spring 2023:

​

1/12/2023- David Rose, University of North Carolina, Chapel Hill:

​

A Kirby color for Khovanov homology

 

In the context of quantum link invariants, a Kirby color is a linear combination of cabling patterns with the property that the resulting colored link polynomial is invariant under the second (handle slide) Kirby move. In familiar cases (e.g. Witten--Reshetikhin--Turaev), these invariants admit a renormalization that is invariant under the first (blow-up) Kirby move and thus yield invariants of closed, orientable 3-manifolds.

 

We will discuss a Kirby color for Khovanov homology, an ind-object in the annular Bar-Natan category (the monoidal category associated to the annulus in the Khovanov theory), which is equipped with a natural handle slide isomorphism. Functoriality properties of Khovanov homology imply that Kirby-colored Khovanov homology is invariant under the handle slide Kirby move. Using work of Manolescu--Neithalath, Kirby-colored Khovanov homology agrees with the Morrison--Walker--Wedrich skein lasagna invariant, thus is an invariant of the 4-manifold resulting from 2-handle additions along the corresponding link in the 3-sphere.

​

​

​

1/26/2023- Moritz Weber, Saarland University: Classification of easy quantum groups

​

Easy quantum groups form a class of compact quantum groups (in the sense of Woronowicz) whose representation categories can be described combinatorially by partitions of sets, similar to Temperley-Lieb algebras, Brauer diagrams, planar partitions or related objects. We will briefly introduce easy quantum groups and then survey its classification. We will also mention graph categories as defined by Mancinska-Roberson in order to describe the representation categories of quantum automorphism groups of graphs.

​

​

​

2/9/2023- Terry Gannon, University of Alberta: The truth and more about quantum groups

​

The prototypical modular tensor categories are quantum groups at roots of unity, which correspond to the conformal nets associated to loop groups, or the vertex operator algebras which are associated to the affine Lie algebras at positive integer level. The classification of quantum subgroups is essentially the classification of conformal field theories associated to these quantum groups, or the classification of  module categories for those MTC, or countless other things. Although this question was first addressed by Witten in the 1980s, progress has been slow and physics with its limited attention span soon forgot about it. The subfactor community got interested in the 1990s and for the next decade injected many new ideas. Categorical interpretations followed in the early 2000's. But it wasn't until relatively recently when the problem was largely cracked. In a sense I'll explain, the problem is the analog of the classification of finite simple groups. In my talk I'll sketch the problem and the ideas going into its solution.

​

​

​

2/23/2023- Dylan Thurston, Indiana University, Bloomington: Title: Towards the quantum exceptional series

Abstract: Each of the two classical series of Lie groups, SL(n) and OSp(n), gives a two-parameter quantum invariant. There is also a conjectural third classical series, the exceptional series, containing all the exceptional Lie groups (and thus having only finitely many points). We look at the quantum version of this third series, and show that it satisfies a simple quantum Jacobi relation, giving a (conjectural) skein-theoretic description for a third two-parameter quantum exceptional polynomial invariant. We can unconditionally use these to compute previously out-of-reach knot polynomials (for particular exceptional groups) for all knots with 12 or fewer crossings.

This is joint work with Scott Morrison and Noah Snyder.

​

​

​

 

 

 

 

 

3/9/2023- Amrei Oswald, University of Washington: Twisted tensor products of bialgebras and Frobenius algebras

​

Abstract: We are interested in when the twisted tensor product of two bialgebras inherits a bialgebra structure. We begin with the comultiplication induced by the inverse of the twisting map, and show this only gives a bialgebra structure when the twisting map is trivial. However, we find that the twisted tensor product of two Frobenius algebras is always Frobenius when this gives a coalgebra. We also find that the twisted tensor product of special Frobenius algebras is special Frobenius and give a characterization of when the twisted tensor product of separable algebras is separable. All of the definitions and proofs are given diagrammatically, so these results hold in monoidal categories. This is joint work with Pablo S. Ocal.

​

​

​

3/23/2023- Jeongwan Haah, Microsoft Research: Invertible Subalgebras

​

An invertible subalgebra is a local operator algebra such that every local operator can be locally expressed by elements of the invertible subalgebra and those of the commutant within a full lattice local operator algebra. Every one-dimensional invertible subalgebra is trivial, but a two-dimensional invertible subalgebra can host a chiral anyon theory by a commuting Hamiltonian, which is believed not to be possible on any full local operator algebra. We prove that the stable equivalence classes of d-dimensional invertible subalgebras is an abelian group isomorphic to the group of all (d+1)-dimensional locality-preserving automorphisms of a full local operator algebra (QCA) modulo blending equivalence and shifts. Time permitting, I’ll discuss a metric topology on a superoperator space that includes all QCA, and show that this metric-completion contains all time evolutions by any sufficiently local Hamiltonians. This metric is meant to enable the study of topology of QCA with tails.

​

​

​

 

5/4/2023- Qing Zhang, Purdue University: TBA

​

​

​

​

​

​

​

​

​

​

 

​

​

​

​

​

​