University Quantum Symmetries Lectures
( UQSL )
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 from 3:00 pm  4:00 pm (Eastern Time Zone). If you would like to attend, please email Corey Jones at to be added to the Google Group for weekly announcements of abstracts and Zoom links.
Below is a list of currently scheduled talks (which is updated frequently!):
8/27/2020 David Reutter, Max Planck Institute:
Semisimple topological field theories and exotic smooth structures.
A major open problem in quantum topology is the construction of an oriented 4dimensional topological quantum field theory (TQFT) in the sense of AtiyahSegal which is sensitive to exotic smooth structure. In this talk, I will sketch a proof that no semisimple field theory can achieve this goal and that such field theories are only sensitive to the homotopy types of simply connected 4manifolds. In this context, `semisimplicity' is a certain algebraic condition applying to all currently known examples of vectorspacevalued oriented 4dimensional TQFTs, including `unitary field theories' and `onceextended field theories' which assign algebras or linear categories to 2manifolds. If time permits, I will give a concrete expression for the value of a semisimple TQFT on a simply connected 4manifold and explain how the presence of `emergent fermions’ in a field theory is related to its potential sensitivity to more than the homotopy type of a nonsimply connected 4manifold. Throughout, I will use the CraneYetter field theory associated to a ribbon fusion category as a guiding example. This is based on arXiv:2001.02288.
9/3/2020 Andrew Schopieray, Pacific Institute for the Mathematical Sciences:
The importance of norm and trace for fusion categories.
An incredible amount of literature on fusion categories focuses on the weakly integral, i.e. fusion categories whose FrobeniusPerron dimension is an integer. This includes representation categories of finite groups and quasiHopf algebras, (weakly) grouptheoretical fusion categories, TambaraYamagami categories, etc. We will demonstrate how two rudimentary concepts from number theory (norm and trace) dictate the structure of fusion categories which lie far from the cozy world of rational integers. In particular, we will discuss the ongoing classification of fusion categories whose global dimension is a prime rational integer.
9/10/2020 Pavel Etingof, Massachusetts Institute of Technology: TBA
New incompressible symmetric tensor categories in positive characteristic.
Let $k$ be an algebraically closed field of characteristic $p>0$. The category of tilting modules for $SL_2(k)$ has a tensor ideal $I_n$ generated by the $n$th Steinberg module. I will explain that the quotient of the tilting category by $I_n$ admits an abelian envelope, a finite symmetric tensor category ${\rm Ver}_{p^n}$, which is not semisimple for $n>1$. This is a reduction to characteristic $p$ of the semisimplification of the category of tilting modules for the quantum group at a root of unity of order $p^n$. These categories are incompressible, i.e. do not admit fiber functors to smaller categories. For $p=1$, these categories were defined by S. Gelfand and D. Kazhdan and by G. Georgiev and O. Mathieu in early 1990s, but for $n>1$ they are new. I will describe these categories in detail and explain a conjectural formulation of Deligne's theorem in characteristic $p$ in which they appear. This is joint work with D. Benson and V. Ostrik.
9/17/2020 Florencia Orosz Hunziker, University of Colorado, Boulder:
Tensor categories arising from the Virasoro algebra.
In this talk we will discuss the tensor structure associated to certain representations of the Virasoro algebra. In particular, we will show that there is a braided tensor category structure on the category of C1cofinite modules for the Virasoro vertex operator algebras of arbitrary central charge. This talk is based on joint work with Jinwei Yang, Thomas Creutzig, Cuibo Jiang and David Ridout.
9/24/2020 Cain EdieMichell, Vanderbilt University:
Symmetries of modular categories and quantum subgroups
Since the problem was introduced by Ocneanu in the late 2000's, it has been a longstanding open problem to completely classify the quantum subgroups of the simple Lie algebras. This classification problem has received considerable attention, due to the correspondence between these quantum subgroups, and the extensions of WZW models in physics. A rich source of quantum subgroups can be constructed via symmetries of certain modular tensor categories constructed from Lie algebras. In this talk I will describe the construction of a large class of these symmetries. Many exceptional examples are found, which give rise to infinite families of new exceptional quantum subgroups.
10/1/2020 André Henriques, University of Oxford:
Extended chiral CFT and a new kind of anomaly
I will describe ongoing work with James Tener, in which we construct unitary chiral CFTs à la Segal, and I'll explain why it is in fact easier to directly construct extended chiral CFT as opposed to nonextended ones. On our way, we have found a new kind of anomaly which is related to unitarity: Hilbert spaces whose norm is only welldefined up to a positive scalar.
10/8/2020 Stuart White, University of Oxford:
Simple amenable operator algebras
The last decade has seen dramatic advances in the structure and classification of simple amenable C*algebras, driven by strong parallels with results for amenable von Neumann algebras in the 70's and 80's. In this survey talk, I'll try and explain where we are right now, how this parallels the von Neumann algebraic situation, and also why attendees of a quantum symmetries seminar might want to know about this. I won't assume any prior knowledge of C*algebras beyond their formal definition.
10/15/2020 Chelsea Walton, Rice University:
Universal Quantum Semigroupoids
In a recent paper (https://arxiv.org/abs/2008.00606), Hongdi Huang, Elizabeth Wicks, Robert Won, and I introduce the concept of a universal quantum linear semigroupoid (UQSGd). This is a weak bialgebra that coacts on a (not necessarily connected) graded algebra A. Our main result is that when A is the path algebra kQ of a finite quiver Q each of the various UQSGds introduced in our work is isomorphic to the face algebra attached to Q (an important weak bialgebra due to Hayashi). Most of the talk will be dedicated to setting up context and terminology towards the main result. So even if you’re new to weak bialgebras (as I was last year), you’ll be able to follow along.
10/22/2020 Fiona Burnell, University of Minnesota:
An introduction to fracton order
In recent years, a new type of order  reminiscent of, but qualitatively distinct from, the topological order that is described mathematically by modular tensor categories and topological quantum field theory  has emerged in condensed matter physics. This "fracton" order is decidedly geometrical in nature, but also describes physical phenomena, such as statistical interactions between particles and robust ground state degeneracies, that are usually associated with topologically ordered systems. Using a series of examples, I will describe a set of characteristics common to all fracton orders, and discuss some different variations of the concept of fracton order from the physics literature. I will then review some ideas for describing how fracton orders can be obtained by coupling many "layers" of topological orders.
10/29/2020 Dmitri Nikshych, University of New Hampshire:
Braided module categories
Let M be a module category over a braided fusion category C. A Cmodule braiding on M is an additional symmetry related
to the braiding of C and giving rise to representations of Artin braid groups of type B. I will show that braided Cmodule categories form a braided monoidal 2category equivalent to Z(Mod(C)), the 2center of the monoidal 2category Mod(C) of Cmodule categories. As an application, I will explain that braided extensions of C graded by a group A correspond to braided monoidal 2functors from A to the braided 2categorical Picard group of C, consisting of invertible braided Cmodule categories. Such functors can be explicitly described using the EilenbergMacLane abelian cohomology. I will also discuss a (conjectural) characterization of minimal modular embeddings of C in terms of Z(Mod(C)). This is a joint work with Alexei Davydov, based on arXiv:2006.08022.
11/5/2020 Guillermo Sanmarco, Universidad Nacional de Córdoba:
PreNichols algebras with finite GelfandKirillov dimension
Finite dimensional Nichols algebras of diagonal type are generalizations of (positive parts) of small quantum groups that play a crucial role in the classification of finite dimensional pointed Hopf algebras with abelian group of grouplikes. Some combinatorial tools of quantum groups were extended to this setting, which led to the classification of such Nichols algebras.
When one attempts to classify certain Hopf algebras with finite GelfandKirillov dimension Nichols algebras are not enough and one is naturally led to study preNichols algebras with the same restriction on the growth. In this talk we report our progress in the problem of determining all preNichols algebras of diagonal type with finite GKdim. We reduce the original problem to the one of constructing a preNichols algebra with finite GKdim that projects onto all other such algebras. Our main tools are the socalled distinguished preNichols algebras (which generalize De ConciniKacProcesi big quantum groups) and normal extensions of braided Hopf algebras.
11/12/2020 Christoph Schweigert, Universität Hamburg:
Topological field theories with boundaries: from tensor networks to FrobeniusSchur indicators
State sum models of TuraevViro type have various applications. Here, we explain their application to projected entangled pair states (PEPS) in tensor network models and to a deeper understanding of equivariant FrobeniusSchur indicators.
11/19/2020 César Galindo, Universidad de los Andes:
Braided Zesting and its applications.
In this talk, I will introduce a construction of new braided fusion categories from a given category known as zesting. This method has been used to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of supermodular categories. I will present a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.
The talk is based on the manuscript https://arxiv.org/abs/2005.05544 a joint work with Colleen Delaney, Julia Plavnik, Eric C. Rowell, and Qing Zhang.
12/3/2020 Colleen Delaney, Indiana University, Bloomington: TBA
12/10/2020 Cris Negron, University of North Carolina, Chapel Hill: TBA
1/14/2021 Michael Brannan, Texas A&M University: TBA
1/21/2021 Andy Manion, University of Southern California: TBA
1/28/2021 Victor Ostrik, University of Oregon: TBA
2/4/2021 Georgia Benkart, University of Wisconsin  Madison: TBA
2/11/2021 Xie Chen, California Institute of Technology: TBA
2/18/2021 Theo JohnsonFreyd, Dalhousie University/Perimeter Institute: TBA
2/25/2021 Noah Snyder, Indiana University, Bloomington: TBA
4/29/2021 Emily Riehl, Johns Hopkins University: TBA