# ( 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 4-dimensional topological quantum field theory (TQFT) in the sense of Atiyah-Segal 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 4-manifolds. In this context, semisimplicity' is a certain algebraic condition applying to all currently known examples of vector-space-valued oriented 4-dimensional TQFTs, including unitary field theories' and once-extended field theories' which assign algebras or linear categories to 2-manifolds. If time permits, I will give a concrete expression for the value of a semisimple TQFT on a simply connected 4-manifold 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 non-simply connected 4-manifold. Throughout, I will use the Crane-Yetter 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 Frobenius-Perron dimension is an integer.  This includes representation categories of finite groups and quasi-Hopf algebras, (weakly) group-theoretical fusion categories, Tambara-Yamagami 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 C1-cofinite 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 Edie-Michell, 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 long-standing 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