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: TBA
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 Stuart White, University of Oxford: TBA
10/8/2020 André Henriques, University of Oxford: TBA
10/15/2020 Chelsea Walton, Rice University: TBA
10/22/2020 Fiona Burnell, University of Minnesota: TBA
10/29/2020 Dmitri Nikshych, University of New Hampshire: TBA
11/5/2020 Guillermo Sanmarco, Universidad Nacional de Córdoba: TBA
11/12/2020 Christoph Schweigert, Universität Hamburg: TBA
11/19/2020 César Galindo, Universidad de los Andes: TBA
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
2/4/2021 Georgia Benkart, University of Wisconsin  Madison: TBA
2/11/2021 Xie Chen, California Institute of Technology: TBA