# ( 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 for this semester (which is updated frequently!):

Spring 2021:

1/14/2021- Michael Brannan, Texas A&M University:

Quantum graphs and quantum Cuntz-Krieger 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 non-commutative geometry, and they arise naturally in a surprising variety of areas including quantum information theory, representation theory, and in the theory of non-local games. I will review the well-known construction of Cuntz-Krieger 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 Cuntz-Krieger 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 finite-dimensional 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 finite-dimensional 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 X-cube 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 X-cube model with periodic boundary condition, open boundary condition, as well as in the presence of screw dislocation defects.

2/18/2021- Theo Johnson-Freyd, Dalhousie University/Perimeter Institute:

Condensations and Components.

The 1-categorical Schur's lemma, which says that a nonzero morphism between simple objects is an isomorphism, fails for semisimple n-categories 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 n-categorical 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 S-matrices" and, time permitting, about the image of the j-homomorphism 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 Asaeda--Haagerup and Jones--Peters 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 Grossman-Morrison-Penneys-Peters (which builds on work of De Commer-Yamashita) and on work in progress with Penneys-Peters.

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 2-representations of Khovanov's categorification of U(gl(1|1)^+), examples of such 2-representations that arise as strands algebras in bordered and cornered Heegaard Floer homology, and a tensor-product-based gluing formula for these 2-representations expanding on work of Douglas-Manolescu.

3/11/2021- No talk this week.

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 bulk-edge correspondence of topological phases of matter make them into a unified theory of 2d and 3d. The mysterious 6d super-conformal 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 non-degenerate (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 Siu-Hung Ng and Qing Zhang.

4/15/2021- (Rescheduled to 5/20/2021)

4/22/2021- Radmila Sazdanovic, North Carolina State University

Bilinear pairings on two-dimensional 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 non-negative integer. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. The Deligne category and its semisimple quotients admit similar interpretations. This viewpoint coupled to the universal construction of two-dimensional topological theories leads to multi-parameter 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 infinite-dimensional 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 infinite-dimensional category in which ∞-categories live as objects. We demonstrate that the theorems proven in this manner are "model-independent", 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 3-category of fusion categories and a 4-category of modular tensor categories. Then we will discuss why these do not suffice for Reshetikin-Turaev theories and I will give an outlook on work-in-progress 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.