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.
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 2021:
1/14/2021 Michael Brannan, Texas A&M University:
Quantum graphs and quantum CuntzKrieger algebras.
I will give a light introduction to the theory of quantum graphs and some related operator algebraic constructions. Quantum graphs are generalizations of directed graphs within the framework of noncommutative geometry, and they arise naturally in a surprising variety of areas including quantum information theory, representation theory, and in the theory of nonlocal games. I will review the wellknown construction of CuntzKrieger C*algebras from ordinary graphs and explain how one can generalize this construction to the setting of quantum graphs. Time permitting, I will also explain how quantum symmetries of quantum graphs can be used to shed some light on the structure of quantum CuntzKrieger algebras. (This is joint work with Kari Eifler, Christian Voigt, and Moritz Weber.)
1/21/2021 No talk this week (rescheduled for 3/4/2021).
1/28/2021 Victor Ostrik, University of Oregon:
Two dimensional topological field theories and partial fractions.
This talk is based on joint work with M.Khovanov and Y.Kononov. By evaluating a topological field theory in dimension 2 on surfaces of genus 0,1,2 etc we get a sequence. We investigate which sequences occur in this way depending on the assumptions on the target category.
2/4/2021 Georgia Benkart, University of Wisconsin  Madison:
Fusion rules for Hopf algebras.
The McKay matrix M_V records the result of tensoring the simple modules with a finitedimensional module V. In the case of finite groups, the eigenvectors for M_V are the columns of the character table, and the eigenvalues come from evaluating the character of V on conjugacy class representatives. In this talk, we will explore what can be said about such eigenvectors when the McKay matrix is determined by modules for a finitedimensional Hopf algebra. The quantum group u_q(sl_2), where q is a root of unity, provides an interesting example.
2/11/2021 (TALK AT 2PM US EASTERN) Xie Chen, California Institute of Technology:
Foliation structure in fracton models.
Fracton models are characterized by an exponentially increasing ground state degeneracy and point excitations with limited motion. In this talk, I will focus on a prototypical 3D fracton model  the Xcube model  and discuss how its ground state degeneracy can be understood from a foliation structure in the model. In particular, we show that there are hidden 2D topological layers in the 3D bulk. To calculate the ground state degeneracy, we can remove the layers until a minimal structure is reached. The ground state degeneracy comes from the combination of the degeneracy of the foliation layers and that associated with the minimal structure. We discuss explicitly how this works for Xcube model with periodic boundary condition, open boundary condition, as well as in the presence of screw dislocation defects.
2/18/2021 Theo JohnsonFreyd, Dalhousie University/Perimeter Institute:
Condensations and Components.
The 1categorical Schur's lemma, which says that a nonzero morphism between simple objects is an isomorphism, fails for semisimple ncategories when n≥2. Rather, when two simple objects are related by a nonzero morphism, they each arise as a condensation descendant of the other. Because of this, for many purposes the natural ncategorical version of "set of simple objects" is the set of components: the set of simples modulo condensation descent. I will explain this phenomenon and describe some conjectures, including conjectures about "higher categorical Smatrices" and, time permitting, about the image of the jhomomorphism in the homotopy groups of spheres.
2/25/2021 Noah Snyder, Indiana University, Bloomington:
Demystifying subfactor techniques for constructing tensor categories
The primary goal of this talk is to explain two techniques used for constructing tensor categories that were developed among the subfactor community by AsaedaHaagerup and JonesPeters in purely tensor categorical terms. The secondary goal is to explain how these techniques generalize using module categories. This talk is based on a part of our Extended Haagerup paper with GrossmanMorrisonPenneysPeters (which builds on work of De CommerYamashita) and on work in progress with PenneysPeters.
3/4/2021 Andy Manion, University of Southern California:
Higher representations and cornered Floer homology.
I will discuss recent work with Raphael Rouquier, focusing on a higher tensor product operation for 2representations of Khovanov's categorification of U(gl(11)^+), examples of such 2representations that arise as strands algebras in bordered and cornered Heegaard Floer homology, and a tensorproductbased gluing formula for these 2representations expanding on work of DouglasManolescu.
3/18/2021 Zhenghan Wang, University of California Santa Barbara:
Quantum algebras climb the dimension ladder.
Two interesting classes of quantum algebras are vertex operator algebras (VOAs) and modular tensor categories (MTCs) . The bulkedge correspondence of topological phases of matter make them into a unified theory of 2d and 3d. The mysterious 6d superconformal field theories from string theory suggest an inversion of dimensions: MTCs and VOAs should fit into a unified theory of 3d and 4d manifolds. I will mainly focus on a potential construction of MTCs from three manifolds in a recent joint work arXiv:2101.01674 with S. Cui and Y. Qiu. In the end, I will speculate how four manifolds with three manifold boundaries should give rise to VOAs that realize the boundary MTCs.
3/25/2021 Eric Rowell, Texas A&M University:
Torsion in the Witt group and higher central charges.
I will describe some recent joint work with Richard Ng, Yilong Wang and Qing Zhang in which we apply the theory of higher central charges to investigate the torsion subgroup of the Witt group for nondegenerate (and slightly degenerate) braided fusion categories. In particular we show that the Witt classes containing the Ising categories have infinitely many Witt inequivalent square roots.
4/1/2021 Jacob Bridgeman, Perimeter Institute:
Enriching topological codes: computing with defects.
Topological phases are a promising substrate for quantum computing due to their inherent error resistance. Unfortunately, there seems to be a tradeoff between how easily the codes can be realized in the lab, and whether a universal set of gates can be implemented. Including defects into the code can be used to boost the gate set. We use tube algebras to understand the properties of defects, and hopefully which gates can be implemented.
4/8/2021 Yilong Wang, Louisiana State University:
Classification of transitive modular categories.
The Galois group action on simple objects is one of the many interesting arithmetic properties of modular categories. This action is a powerful tool in the classification of modular categories by rank and by dimension. Therefore, it is natural to pursue a classification of modular categories purely by the properties of the Galois action.
In this talk, we introduce the notion of transitive modular categories, which are modular categories with a single Galois orbit. After giving some basic properties of such categories, we will explain how we use the representation theory of SL(2, Z/nZ) to obtain the full classification of transitive modular categories.
This talk is based on the joint work with SiuHung Ng and Qing Zhang.
4/15/2021 (Rescheduled to 5/20/2021)
4/22/2021 Radmila Sazdanovic, North Carolina State University
Bilinear pairings on twodimensional cobordisms and generalizations of the Deligne category.
The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter t while producing categories of representations of the symmetric group when modded out by the ideal of negligible morphisms when t is a nonnegative integer. The partition category may be interpreted, following Comes, via a particular linearization of the category of twodimensional oriented cobordisms. The Deligne category and its semisimple quotients admit similar interpretations. This viewpoint coupled to the universal construction of twodimensional topological theories leads to multiparameter monoidal generalizations of the partition and the Deligne categories, one for each rational function in one variable. This talk is based on joint work with M. Khovanov.
4/29/2021 Emily Riehl, Johns Hopkins University:
Elements of ∞Category Theory.
Confusingly for the uninitiated, experts in weak infinitedimensional category theory make use of different definitions of an ∞category, and theorems in the ∞categorical literature are often proven "analytically", in reference to the combinatorial specifications of a particular model. In this talk, we present a new point of view on the foundations of ∞category theory, which allows us to develop the basic theory of ∞categories  adjunctions, limits and colimits, co/cartesian fibrations, and pointwise Kan extensions  "synthetically" starting from axioms that describe an ∞cosmos, the infinitedimensional category in which ∞categories live as objects. We demonstrate that the theorems proven in this manner are "modelindependent", i.e., invariant under change of model. Moreover, there is a formal language with the feature that any statement about ∞categories that is expressible in that language is also invariant under change of model, regardless of whether it is proven through synthetic or analytic techniques. This is joint work with Dominic Verity.
5/6/2021 Tobias Osborne, Leibniz Universität Hannover:
Fusion categories and physics: a statistical physics approach.
In this talk I will describe how one can build statistical physics lattice models corresponding to braided fusion categories. Requiring that the resulting model is at a phase transition is a necessary (but probably not sufficient) condition for it to realize a conformal theory corresponding to the input category. I will describe how the condition of discrete preholomorphicity, as identified by Fendley, provides a simple constraint on the Boltzmann weights  for any braided fusion category  for the lattice model to be at a phase transition.
5/20/2021 Claudia Scheimbauer, Technische Universität München:
Higher Morita categories and extensions.
In this talk I will explain higher Morita categories of $E_n$algebras and bimodules and discuss dualizability therein. Important examples are a 3category of fusion categories and a 4category of modular tensor categories. Then we will discuss why these do not suffice for ReshetikinTuraev theories and I will give an outlook on workinprogress with Freed and Teleman on how to remedy this.
5/27/2021 Ehud Meir, University of Aberdeen:
Interpolations of symmetric monoidal categories and algebraic structures via invariant theory.
In this talk I will present a recent construction that enables one to interpolate algebraic structures. The construction also enables one to interpolate the categories of representations of the automorphism groups of these algebraic structures. I will explain how one can recover the constructions of Deligne for categories such as Rep(S_t), Rep(O_t) and Rep(Sp_t), the constructions of Knop for wreath products with S_t and GL_t(O_r), where O_r is a finite quotient of a DVR, and also the recent construction of the categories DCob_{\alpha} of Khovanov, Ostrik and Kononov.