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.

UQSL- Fall 2020

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:

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- 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 non-extended ones. On our way, we have found a new kind of anomaly which is related to unitarity: Hilbert spaces whose norm is only well-defined 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 C-module 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 C-module categories form a braided monoidal 2-category equivalent to Z(Mod(C)), the 2-center of the monoidal 2-category Mod(C) of C-module categories. As an application, I will explain  that braided extensions of C graded by a group A correspond to braided monoidal 2-functors from A to the braided 2-categorical Picard group of C, consisting of invertible braided C-module categories. Such functors can be explicitly described using the Eilenberg-MacLane 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:

Pre-Nichols algebras with finite Gelfand-Kirillov 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 group-likes. 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 Gelfand-Kirillov dimension Nichols algebras are not enough and one is naturally led to study pre-Nichols algebras with the same restriction on the growth. In this talk we report our progress in the problem of determining all pre-Nichols algebras of diagonal type with finite GKdim. We reduce the original problem to the one of constructing a pre-Nichols algebra with finite GKdim that projects onto all other such algebras. Our main tools are the so-called distinguished pre-Nichols algebras (which generalize De Concini-Kac-Procesi 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 Frobenius-Schur indicators

State sum models of Turaev-Viro 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 Frobenius-Schur 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 super-modular 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:

Ribbon zesting can produce modular isotopes.

We observe that the modular categories of Mignard and Schauenburg can be related through the ribbon zesting construction.  Thus zesting is capable of producing different modular categories with the same modular data. We then use zesting to explain the phenomenology of topological invariants such as the W-matrix and Borromean tensor that go beyond the modular data, and offer a perspective for applying quantum knot theory to the study of (2+1)D topological order.

This talk in based on joint work in progress with Sung Kim and Julia Plavnik.

12/10/2020- Cris Negron, University of North Carolina, Chapel Hill:

Geometries for finite tensor categories.

I will describe three appearances of geometry in studies of finite tensor categories.  Or, more accurately, studies of areas of mathematics which take as inputs finite tensor categories.  Here we consider cohomology, support theory, and field theories.  I will describe how the relevant geometries provide an interesting point of intersection between these fields.  This talk will essentially be a survey, and I will try to focus on open questions.