Title: Reconstructing Spinor Categories

Abstract: Tensor categories whose fusion ring is isomorphic to the one of a spin group have a Z/2Z grading. The 0-part is known from the classification of orthogonal categories, assuming braiding. Its module action is then classified using a type B version of the so-called BMW algebra. This allows the reconstruction of the whole category.

Hans Wenzl's Website

Title: Introduction to Coherence in Monoidal Categories

Abstract: Categorical versions of notions such as associativity of a product often involve additional assumptions needed to obtain "coherence" results. I will explain the idea of coherence, and motivate the need for coherence theorems. I will then give a detailed account of the proof of a coherence theorem in the case of monoidal categories, with a focus on how the techniques involved can be adapted to other types of categories.

Title: Modular representation associated to modular categories

Abstract: We will show that modular categories give rise to projective representations of SL(2, Z) by graphical calculus. Time permitted, we will discuss the resolution of the anomaly and the Galois action on the modular data.

Title: On gauging symmetry of modular categories.

Abstract: Whenever there are group actions, interesting things happen. In this talk, we will focus on group actions on unitary modular categories, which are mathematical descriptions of symmetries of two dimensional topological phases of matter. We will introduce the concept of gauging on unitary modular categories following the paper by Cui, Galindo, Plavnik and Wang and work with simple examples.

Title: Exotic fusion categories: EH3 exists!

Abstract: Fusion categories generalize the representation categories of (quantum) groups, and we think of them as objects which encode quantum symmetry. All currently known fusion categories fit into 4 families: those coming from groups, those coming from quantum groups, quadratic categories, and those related to the extended Haagerup (EH) subfactor. First, I'll explain what I mean by the preceding sentence. We'll then discuss the extended Haagerup subfactor, along with the newly constructed EH3 fusion category (in joint work with Grossman, Izumi, Morrison, Peters, and Snyder), and the possibility of the existence of EH4. This might take 2 weeks to explain, so perhaps this will be part 1 of 2.

Title: Induced Representations of $U_q(\mathfrak{sl}_2)$ and $U_q(\mathfrak{sl}_3)$, and the Alexander Polynomial

Abstract: The Alexander polynomial arises by considering the knot invariant presented by Ohtsuki using a family of representations of $U_q(\mathfrak{sl}_2)$ at a root of unity. The multi-variable Alexander polynomial is a generalization of the Alexander polynomial to links where each strand is colored by a representation in a different parameter. In this talk, we will review the multi-variable construction and build the analogous $U_q(\mathfrak{sl}_3)$ representations from Verma modules. After noting some properties of these representations, we will explore their tensor product structure. More specifically, we provide the fusion rules for decomposable tensor products and discuss consequences of this decomposition.

Title: Enriched module categories and internal hom

Abstract: I will introduce the notion of a V-module category for a monoidal category V, and I will explain the basics of the adjunction calculations used in understanding such categories.

Title: Hopf Algebra Integrals and 3-Manifold Invariants

Abstract: I will review a few basic facts about integrals of Hopf algebras and sketch their use in the construction of 3-manifolds invariants. Particularly, the defining axiom of an integral can be directly understood as an algebraic translation of handle-slide moves both for Heegaard presentations (a la Kuperberg) and surgery presentations (a la Hennings). Other basic Hopf algebra relations correspond, for example, to handle cancellations.

Title: Introduction to the volume conjecture

Abstract: The volume conjecture is a conjecture that relates an algebraically defined knot invariant, the colored Jones polynomial, to a geometric knot invariant, the hyperbolic volume of the knot complement. We will define the hyperbolic volume of a knot complement and look at the example of the figure eight knot for which the volume conjecture has been proven.

Title: On Generalized Metaplectic Modular Categories.

Abstract: Metaplectic modular categories are modular tensor categories whose
fusion rules are given by the Verlinde fusing rules of Spin(n) at
level 2. One can generalize these fusion rules by replacing the cyclic
group of order n with an arbitrary finite abelian group A. I will
discuss the classification of modular categories with such fusion
rules in the case that A is of odd order. I will also discuss the
relation to twisted doubles of generalized dihedral groups.

Marcel Bischoff's website

Title: The structure of tensor categories via 3-dimensional topology

Abstract: Fusion tensor categories arise in many areas of mathematics: as representation categories for finite quantum groups, certain Hopf algebras, and loop groups; as the "basic invariants" of subfactors of von Neumann algebras in the theory of operator algebras; and also in the study of conformal field theory. Fusion tensor categories have a rich and fascinating structure. The goal of this talk will be to describe how 3-dimensional topology and topological field theory allow this structure to be understood and explained. This is joint work with Christopher Douglas and Noah Snyder.

Christopher Schommer-Pries's website

Title: Two constructions of the Jones polynomial, part 2

Abstract: In this talk, we will continue our discussion about the Jones polynomial following Peter Tingley's article. We will see how different choice of ribbon elements in $U_q(\mathfrak{sl}_2)$ will make a difference in sign in the construction of quantum invariants.

Reference: A minus sign that used to annoy me but now I know why it is there by Peter Tingley.

Title: Two constructions of the Jones polynomial

Abstract: In this talk, we will talk about two constructions of the Jones polynomial, one given by the Kauffman bracket and the other from the quantum group $U_q(\mathfrak{sl}_2)$. We will give a detailed description of $U_q(\mathfrak{sl}_2)$ as a ribbon Hopf algebra and discuss how a choice of ribbon element may affect the construction of the Jones polynomial by a sign.

Title: Colored Jones polynomial via quantum spin networks and recoupling theory

Abstract: The colored Jones polynomial can be defined by the Jones Wentzl idempotent. We will use properties of the Jones Wentzl idempotent to create colored trivalent graphs called quantum spin networks which will lead to methods for calculating the colored Jones polynomial.

Title: Introduction to Skein Theory and the colored Jones polynomial

Abstract: The Jones polynomial is the simplest example of a quantum knot invariant for knots in S^3. I will talk about generalizing this invariant to different 3-manifolds via Skein modules and skein algebras. I will also discuss a family of quantum knots invariants called the colored Jones polynomials (the original Jones polynomial being the first of which). This will be an introductory talk.

Title: A synoptic chart of tensor categories (Part 2)

Abstract: I'll discuss a synoptic chart of tensor categories, including the alphabet soup of properties and structures (rigid, pivotal, spherical, braided, balanced, ribbon, modular, etc), and the graphical calculus for each. This is an introductory talk with little to no prior knowledge of tensor categories assumed.

Title: A synoptic chart of tensor categories

Abstract: I'll discuss a synoptic chart of tensor categories, including the alphabet soup of properties and structures (rigid, pivotal, spherical, braided, balanced, ribbon, modular, etc), and the graphical calculus for each. This is an introductory talk with little to no prior knowledge of tensor categories assumed.

Louisiana State University Ng's website

Title: Arithmetic invariants of modular categories

Abstract: The study of Frobenius-Schur indicators has provided new insights on the arithmetic properties of spherical fusion categories. In particular, the congruence subgroup theorem, Cauchy theorem, and the conjectural congruence properties of modular categories were established via the generalized Frobenius-Schur indicators. These new results lead to a proof of the rank finiteness theorem of modular categories and allude to a new approach on the classification modular categories of small rank via the representations of SL(2,Z). In this talk, we will discussion these fundamental arithmetic theorems of modular tensor categories.

National Research University Higher School of Economics, Skolkovo Institute of Science and Technology (Sergey Lando's website)

Title: Quantum knot invariants:$\mathfrak{sl}(2)$ case study

Abstract: A construction due to D. Bar-Natan, M. Kontsevich, and E. Witten (around 1990) allows one to associate a knot invariant to any semisimple Lie algebra. The Lie algebra $\mathfrak{sl}(\mathbb{C};2)$ is the simplest such Lie algebra. The corresponding knot invariant is known to be theaggregate of colored Jones polynomials.

However, this class of knot invariants is far from being understood completely. In particular, a construction due to V. Vassiliev ascribes to a Lie algebra knot invariant a function on chord diagrams. Such functions are called weight systems. The weight system corresponding to the Lie algebra $\mathfrak{sl}(2)$ takes any chord diagram to a polynomialin a single variablec, the (quadratic) Casimir element of the Lie algebra. This weight system must admit a transparent description in terms of the combinatorics of the chord diagrams, but there is no such description at the moment.

The talk will explain to the audience the current state of the theory. No preliminary knowledge of topology of knots and their invariants is required.

Texas A&M University
Julia Plavnik's website

Title:Gauging the symmetry of modular categories

Abstract: Modular categories are interesting algebraic structures connected with a variety of mathematical subjects including topological quantum field theory, conformal field theory, representation theory of quantum
groups, von Neumann algebras, and vertex operator algebras. In addition to the mathematical interest, a motivation for pursuing a classification of modular categories comes from their application in condensed matter physics and quantum computing.

Gauging is a well-known theoretical tool used in physics to promote a global symmetry to a local gauge symmetry. This is an useful tool to construct
new modular categories from given ones.

In this talk, we will start by introducing some of the basic definitions and properties of modular categories, and we will also give some basic examples to have a better understanding of their structures. We will also present a mathematical formulation of gauging in terms of higher category formalism. We will show through concrete examples which are the ingredients involved in this process. In addition, if time allows, we will mention some classification results and open problems related with gauging.

References

• Classification of metaplectic modular categories
  by Eddy Ardonne, Meng Cheng, Eric Rowell, Zhenghan Wang
• Symmetry, Defects, and Gauging of Topological Phases
  by Maissam Barkeshli, Parsa Bonderson, Meng Cheng, Zhenghan Wang
• On Gauging Symmetry of Modular Categories
  by Shawn X. Cui, César Galindo, Julia Yael Plavnik, Zhenghan Wang

The University of Michigan
Diana Hubbard's website

Title: An annular refinement of the transverse element in Khovanov homology

Abstract: In 2006, Plamenevskaya proved that the homology class of a certain distinguished element in Khovanov homology is an invariant of transverse links. In this talk I will define an annular refinement of this element, kappa, and I will show that while kappa is not an invariant of transverse links, it is a conjugacy class invariant of braids. We will see applications of kappa relating to transverse links, braid destabilization, and the word problem in the braid group. This work is joint with Adam Saltz.

Texas A&M University
Eric Rowell's website

Title: Classifying Unitary Braided Fusion Categories

Abstract: The problem of classifying unitary braided fusion categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter (not to mention the joy it brings us). By de-equivariantization we can reduce the problem to studying modular categories (trivial Mueger center) or super-modular categories (Mueger center=sVec). Much is known about the former (i.e. Galois symmetries to rank-finiteness) whereas the latter are still somewhat mysterious: for example rank-finiteness is still open for super-modular categories, and it is not known if every super-modular category is the even half of a spin-modular category. I will give a panorama of the current state of affairs on these classification problems.

References

Slides used in the talk.

Papers that are relevant to the talk:
• Rank-finiteness for modular categories
  By Paul Bruillard, Siu-Hung Ng, Eric C. Rowell, Zhenghan Wang
• On the Classification of Weakly Integral Modular Categories
  By Paul Bruillard, César Galindo, Siu-Hung Ng, Julia Plavnik, Eric C. Rowell, Zhenghan Wang
• On classification of modular categories by rank
  By Paul Bruillard, Siu-Hung Ng, Eric C. Rowell, Zhenghan Wang
• Fermionic Modular Categories and the 16-fold Way
  By Paul Bruillard, Cesar Galindo, Tobias Hagge, Siu-Hung Ng, Julia Yael Plavnik, Eric C. Rowell, Zhenghan Wang
• Modular Categories of Dimension $p^{3}m$ with $m$ Square-Free
  By Paul Bruillard, Julia Yael Plavnik, Eric C. Rowell

A survey paper that has the relevant definitions in a concise format:
• From Quantum Groups to Unitary Modular Tensor Categories
  By Eric C. Rowell

Check out also Publications and Preprints of Eric C. Rowell

Louisiana State University
Patrick Gilmer's website

Title: An application of TQFT to modular representation theory

Abstrac: For $\color{black}{p>3}$ a prime, and $\color{black}{g>2}$ an integer, we use Topological Quantum Field Theory (TQFT) to study a family of $\color{black}{p-1}$ highest weight modules $\color{black}{L_p(\lambda)}$ for the symplectic group $\color{black}{\mathrm{Sp}(2g,K)}$ where $\color{black}{K}$ is an algebraically closed field of characteristic $\color{black}{p}$. This permits explicit formulae for the dimension and the formal character of $\color{black}{L_p(\lambda)}$ for these highest weights.

Reference

An application of TQFT to modular representation theory
by Patrick M. Gilmer, Gregor Masbaum

The Ohio State University

Title: Metaplectic modular categories and the associated TQFT

Abstract: Metaplectic modular categories are modular tensor categories that have same fusion rules as the categories of representations of the quantum groups U_q(so(2p+1)), where q is a root of unity and p is an integer. We will discuss the computation of the TQFT associated to these categories and present some recent results on the finiteness and integrality of the TQFT in genus 1 case.

References

Slides used in the talk.

• On Metaplectic Modular Categories and their applications
  By Matthew B. Hastings, Chetan Nayak, Zhenghan Wang
• Integral bases for TQFT modules and unimodular representations of mapping class groups
  By Patrick M. Gilmer, Gregor Masbaum, Paul van Wamelen
• Quantum Invariants of Knots and 3-Manifolds
  By Vladimir G. Turaev

The Ohio State University

Title: Tensor categories enriched in braided tensor categories.

Abstract: Fusion categories generalize the representation categories of quantum groups, and thus we think of fusion categories as objects which encode quantum symmetry. Recently, there has been a lot of interest in super fusion categories, which are enriched in super vector spaces. These objects are examples of tensor categories enriched in symmetric tensor categories. In this talk, I'll discuss an ongoing project with Morrison in which we study tensor categories enriched in a braided fusion category V, which is not assumed to be symmetric. We classify V-fusion categories in terms of oplax braided tensor functors from V to the centers of ordinary fusion categories. Under this correspondence, strong braided tensor functors correspond to V-complete V-fusion categories.

David Penneys' website

Vanderbilt University
Marcel Bischoff's website

Title: Extensions of modular tensor categories and subtheories in conformal field theory

Abstract: From a commutative algebra object in a (unitary) modular tensor category, one gets a new (smaller) modular tensor category of dyslectic modules. This process correspond to extensions in rational conformal field theory or condensation in topological phases of matter. In particular, one can study extensions in rational conformal field theory from a purely categorical point of view. Conversely, passing to a subtheory, one gets a bigger modular tensor category. But knowing only the modular tensor category $\color{black}{C}$ is not enough to determine the possible modular tensor categories of subtheories of a given theory realizing $\color{black}{C}$. Therefore on the categorical level one only gets necessary restrictions, whose structure I will discuss. Further, I will discuss examples coming from near group categories, which give the existence of interesting (often still hypothetical) subtheories of conformal field theories whose representation categories are pointed modular tensor categories.

References

Part of the talk can be found in appendix B in
Generalized Orbifold Construction for Conformal Nets By Marcel Bischoff

The Prerequisites are in:

• The Witt group of non-degenerate braided fusion categories By Alexei Davydov, Michael Mueger, Dmitri Nikshych, Victor Ostrik
• On the structure of the Witt group of braided fusion categories By Alexei Davydov, Dmitri Nikshych, Victor Ostrik

Ohio University

Alexei Davydov's website

Title: Commutative algebra in braided tensor categories

Abstract: Commutative separable algebras seem to play an important role in the emerging structure theory of finite braided tensor categories. The construction of the category of local modules can be seen as the procedure of “contracting” a commutative algebra. These contractions reduce the dimension of the category and eventually produce a category without commutative algebras (a completely anisotropic category). Deligne’s theorem implies that in characteristic zero there are only two completely anisotropic symmetric categories - the category of vector spaces Vect and its super analogue sVect. This has an implication for all braided fusion categories, grouping them into two classes - with symmetric centres contractible to either Vect or sVect. It happens that the completely anisotropic categories in each class form a group under the Deligne tensor product of categories. These are the Witt groups of non-degenerate and slightly degenerate braided fusion categories.

References

• On the structure of the Witt group of braided fusion categories
  by Alexei Davydov, Dmitri Nikshych, Victor Ostrik
• The Witt group of non-degenerate braided fusion categories
  by Alexei Davydov, Michael Mueger, Dmitri Nikshych, Victor Ostrik

The Ohio State University

Title: An invariant of link diagrams on surfaces via Hopf algebra bundles

Abstract: Hopf algebras have been used in the definition of several types of link invariants. One method created by Ruth Lawrence in 1989 involves "decorating" points on a link diagram with elements of a ribbon Hopf algebra. First we will introduce this type of link invariant. Then we will discuss how to extend this method to define an invariant of link diagrams on arbitrary surfaces. This will involve a special type of flat vector bundle, whose typical fiber is a ribbon Hopf algebra and whose transition maps respect the ribbon structure.

Indiana University

Title: Radford's theorem and the belt trick

Abstract: Topological field theories give a connection between topology and algebra. This connection can be exploited in both directions: using algebra to construct topological invariants, or using topology to prove algebraic theorems.
In this talk, I will explain an interesting example of the latter phenomena. Radford's theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a finite tensor category the quadruple dual functor is easy to understand.
It's somewhat mysterious that the double dual is hard to understand but the quadruple dual is easy. Using topological field theory, we show that Radford's theorem is exactly the consequence of the Dirac belt trick in topology. That is, the double dual corresponds to the generator of $\color{black}{\pi_1(\mathrm{SO}(3))}$ and so the quadruple dual is trivial in an appropriate sense exactly because $\color{black}{\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2}$.
This is part of a large project, joint with Chris Douglas and Chris Schommer-Pries, to understand local field theories with values in the 3-category of tensor categories via the cobordism hypothesis.

The Ohio State University

Title:Planar algebras in modular tensor categories

Abstract:　 I'll first give an introduction to Jones' planar algebras, which are a useful tool for the construction and classification of subfactors and fusion categories. A folklore theorem says that sufficiently nice planar algebras are equivalent to pivotal tensor categories together with a distinguished choice of generating object. I'll then discuss joint work with Henriques and Tener, which generalizes the notion of a planar algebra in the category of vector spaces to a planar algebra internal to a modular tensor category $\color{black}C$. We generalize the above theorem, showing that planar algebras internal to $\color{black}C$ are in one-to-one correspondence with module tensor categories $\color{black}M$ for $\color{black}C$, a functor from $\color{black}C$ to the Drinfel'd center $\color{black}{Z(M)}$, and a distinguished object in $\color{black}M$ which generates $\color{black}M$ as a $\color{black}C$-module.

David Penneys' website

References

• Categorified trace for module tensor categories over braided tensor categories, André Henriques, David Penneys, James Tener
• Planar algebras in braided tensor categories, André Henriques, David Penneys, James Tener

The Australian National University

Title: Classification of planar algebras by skein theory.　

Abstract: Planar algebras provide a framework for studying pivotal 2-categories. Planar algebras in turn can be described by generators and relations, which often lie in possible families called skein theories, usually based on a common evaluation algorithm. Given such a family, one is interested in classifying all planar algebras realizing these types of relations. Solving this problem has implications for the classification of planar algebras with low dimensional box spaces, an alternative to small-index classification. We will discuss the classification of Thurston-relation planar algebras, and applications to (subfactor) planar algebras singly generated by a 3-box.