Program and Abstracts

Schedule

Lectures

Corey Jones (North Carolina State University)

Title: Braided tensor categories in operator algebras

Abstract: Braided tensor categories have a long history of interaction with operator algebras. We will describe three topics of intersection: superselection sectors in algebraic quantum field theory, the asymptotic/symmetric enveloping inclusion from subfactor theory, and Connes' χ(M) and recent generalizations. Our goal is to highlight the similarities between these, which hint at a deeper connection between higher categories and operator algebras.

Background reading:

1. For tensor categories in Quantum field theory: Chapters 1,2, and 8 of Halvorson–Mueger, Algebraic Quantum field Theory, arXiv:math-ph/0602036

2. General reference on rigid C*-tensor categories: Chapter 2 of Neshveyev–Tuset, Compact quantum groups and their representation categories, Cours Spécialisés, vol. 20, Société Mathématique de France, Paris, 2013.

David Penneys (Ohio State University)

Title: Introduction to the classification program for subfactors

Abstract:　A factor is a von Neumann algebra with trivial center, and a subfactor is a unital inclusion of factors. Jones initiated the modern theory of subfactors with his famous Index Rigidity Theorem, in which he proved that the index of a II_1 subfactor, which is analogous to the index of a subgroup, has discrete and continuous ranges. The rich mathematical structure of a finite index subfactor led to the discovery of the Jones polynomial, leading to the mathematical field of quantum topology. Subfactors have also seen connections to many areas of mathematics and physics.

Subfactors are quantum mathematical objects which encode quantum symmetry. Classically, the symmetries of a mathematical object form a group. Groups act on objects by structure-preserving isomorphisms, i.e., a G-symmetry on a mathematical object X is a group homomorphism from G to End(X), where End is taken in some category. The symmetries of quantum mathematical objects, like C* and von Neumann algebras, are best described by unitary tensor categories (UTCs), as these quantum mathematical objects live in higher categories. For example, while von Neumann algebras and unital normal *-homomorphisms form a 1-category, subfactor theory naturally leads us to view them as objects of a 2-category whose 1-morphisms are Hilbert space bimodules, and whose 2-morphisms are bounded intertwiners. Fixing a von Neumann algebra M, End(M) = Bim(M), the W* tensor category of M-M bimodules. Thus a von Neumann algebra M admits richer symmetries given by a unitary tensor functor from a unitary tensor category C into Bim(M). Moreover, every unitary tensor category acts on some II_1 factor, and thus subfactors are universal hosts for quantum symmetries.

In this mini-course, we will focus on the classification program for finite index II_1 subfactors, focusing mainly on small index. This program has led to the discovery of many exotic examples of subfactors and unitary fusion categories. The 3 lectures will focus on the following topics:

1. Finite index II_1 subfactors and unitary tensor categories

2. Several equivalent notions of the standard invariant by example

3. Classification of small index standard invariants

Here are some notes from previous mini-courses I have given as introductions to subfactors: note 1 note 2

Note; Some other material includes: arXiv:1304.6141 (Jones-Morrison-Snyder to index 5) arXiv:1509.00038 (Afzaly-Morrison-Penneys to index 5+1/4)

Julia Plavnik (Indiana University)

Title: Introduction to modular tensor categories and their applications

Abstract: The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. These categories have also applications in different areas of mathematics like topological quantum field theory, von Neumann algebras, representation theory, and others.

In this mini-course, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories. We will give some concrete examples and introduce some important constructions, such as the Drinfeld center, to have a better understanding of their structures.

We will also present the main properties of modular categories, such as their connection with the modular group, and rank-finiteness, among others.

Then we will give an overview of the current situation of the classification program for modular categories. If time allows, we will present some open questions and conjectures in the area.

References:

1. Emily Riehl (Category theory in context: https://math.jhu.edu/~eriehl/context/)

2. Mac Lane (category theory for the working mathematician)

3. EGNO (https://math.mit.edu/~etingof/egnobookfinal.pdf)

Stefaan Vaes (K.U. Leuven)

Title: Property (T) discrete quantum groups associated with highly symmetric graphs

Abstract: After an introduction to Kazhdan’s property (T) for tensor categories and discrete quantum groups, I will present a number of genuinely quantum instances of property (T), all related to highly symmetric graphs. This includes the property (T) discrete quantum groups associated with Bruhat-Tits buildings that I defined in joint work with M. Valvekens. I will present a new and more systematic approach to this class of quantum groups. This is an ongoing joint work with L. Rollier in which we associate discrete quantum groups to (typically infinite) arbitrary vertex transitive locally finite graphs.

Background Reading:

Sergey Neshveyev and Lars Tuset, “Compact Quantum Groups and Their Representation Categories”, a textbook available for download at http://sergeyn.info/papers/CQGRC.pdf

Read Chapter 1, as well as the following sections of Chapter 2: 2.1, 2.2, 2.3 and 2.7.

Christian Voigt (University of Glasgow)

Title: The Drinfeld double construction and complex semisimple quantum groups.

Abstract: The Drinfeld double construction is a fundamental tool in the world of quantum groups and tensor categories, and has interesting links to representation theory, geometry, and analysis. In my lectures I will give an introduction to this construction, and illustrate some of the above connections in the case of q-deformations of compact Lie groups. The associated Drinfeld doubles can be viewed as deformations of complex Lie groups – which explains the second part of my title.

Background reading:

1. Arano, Unitary spherical representations of Drinfeld doubles, J. Reine Angew. Math. 742 (2018), 157–186.

2. Popa–Vaes, Representation theory for subfactors, λ-lattices and C*-tensor categories, Comm. Math. Phys. 340 (2015), no. 3, 1239–1280.

3. Voigt–Yuncken, Complex semisimple quantum groups and representation theory, Lecture Notes in Mathematics, vol. 2264, Springer, Cham, 2020.

Moritz Weber (University of Saarlandes)

Title: The classification program for “easy” quantum groups and tensor categories of partitions.

Abstract: In the 1980s Woronowicz defined compact matrix quantum groups and he proved a Tannaka–Krein type result: These quantum groups are completely determined by their representation categories. Building on that, Banica and Speicher defined the so called “easy” quantum groups in 2009, containing Sh. Wang’s free orthogonal quantum group ON+ and his free symmetric quantum group SN+ (defined in the 1990s) amongst others. The representation categories of “easy” quantum groups are given by categories of partitions (of sets). Hence, their representation theory can be studied by combinatorial means, using some diagrammatic calculus similar to the one of Temperley–Lieb algebras and Brauer diagrams. We will give an introduction to “easy” quantum groups, survey their classification program and mention links, applications and generalizations. Comments on some reading material: See [1] for the first definition of “easy” quantum groups; see [2,3] for an overview on the links to von Neumann algebras; see [4,5] for an introduction/overview on “easy” quantum groups.

Background reading:

1. Teodor Banica and Roland Speicher. “Liberation of orthogonal Lie groups”. Advances in Mathematics 222 (4 2009), pp. 1461–1501.

2. Michael Brannan. Approximation properties for free orthogonal and free unitary quantum groups. J. Reine Angew. Math. 672, 223–251, 2012.

3. Michael Brannan, Roland Vergnioux, Orthogonal free quantum group factors are strongly 1-bounded, Advances in Mathematics, Volume 329, 2018, Pages 133-156.

4. Moritz Weber, Easy quantum groups, 23 pages, in Free probability and operator algebras, ed. by Dan-V. Voiculescu, N. Stammeier, M. Weber, EMS, 2016.

5. Moritz Weber, Introduction to compact (matrix) quantum groups and Banica-Speicher (easy) quantum groups, Notes of a lecture series at IMSc Chennai, India, 2015, Indian Academy of Sciences. Proceedings. Mathematical Sciences, Vol. 127, Issue 5, pp 881-933, Nov 2017.

Research talks

Jason Crann (Caleton University)

Title: Hybrid quantum teleportation, subfactors and quantum chromatic numbers

Abstract: Motivated by recent activity in hybrid quantum error correction and the theory of local quantum operations, we generalize Werner’s teleportation schemes to the commuting operator framework. For a large class of inclusions N \subset M of finite von Neumann algebras, we obtain a correspondence between “tight” teleportation schemes for the relative commutant N’ \cap M and unitary Pimsner-Popa bases for M over N. When N is homogeneous and M is a finite-dimensional factor, we build on work of Brannan-Eifler-Voigt-Weber and establish an analogous correspondence between unitary Pimsner-Popa bases for M over N and “tight” representations of the linking algebra of the quantum automorphism groups of N’ and C^d (d=dim N’). Our techniques also allow us to generalize recent results of Brannan-Ganesan-Harris and Todorov-Turowska on chromatic numbers of complete quantum graphs. This is joint work with David Kribs and Rupert Levene.

Paramita Das (Indian Statistical Institute, Kolkata)

Title: The 2-category of connections

Abstract: We will discuss a certain generalization of `biunitary connections' introduced by Ocneanu, and obtain a 2-category out of it. We show that this 2-category has (fully faithful) realizations inside the 2-category of right correspondences over pairs of pre-C*-algebras. With some extra hypothesis, we show that this phenomenon extends to the level of von Neumann algebras. We also exhibit some examples. This is a joint work with Corey Jones, Shamindra Ghosh and Mainak Ghosh.

Yasuyuki Kawahigashi (University of Tokyo)

Title: A characterization of a finite-dimensional commuting square producing a subfactor of finite depth

Abstract: We give a characterization of a finite-dimensional commuting square with a normalized trace that produces a hyperfinite type II1 subfactor of finite index and finite depth in terms of Morita equivalence of fusion categories. This type of commuting squares were studied by N. Sato, and we show that a slight generalization of his construction covers the fully general case of such commuting squares. We also give a characterization of such a commuting square that produces a given hyperfinite type II1 subfactor of finite index and finite depth. These results also give a characterization of certain 4-tensors that appear in recent studies of matrix product operators in 2-dimensional topological order.

Mehrdad Kalantar (University of Houston)

Title: Boundary actions of discrete quantum groups

Abstract: We discuss various notions of boundary actions of discrete quantum groups, mainly focusing on the notion of topological boundaries, and in particular, the Furstenberg boundary. We motivate these concepts by first a quick review of the classical case and the applications of boundary actions of discrete groups in the structure theory of their operator algebras. We then present some of the recent developments in this theory in the quantum setting.

Brent Nelson (Michigan State University)

Title: Quantum Edge Correspondences

Abstract: A quantum graph is a triple $\mathcal{G}:=(B,\psi,A)$ consisting of a finite-dimensional C*-algebra $B$ with state $\psi$ and linear map $A\colon B\to B$ satisfying a quantized version of being an idempotent with respect to the Schur product. Every finite simple graph $(V,E)$ yields such a triple: $(\mathbb{C}^V, \frac{1}{|V|} \sum_{v\in V} \delta_v, A)$ where $A\in M_{|V|}(\{0,1\})$ is the adjacency matrix. Given a quantum graph $\mathcal{G}=(B,\psi,A)$, one can define a C*-correspondence $E_\mathcal{G}$ over $B$ called the quantum edge correspondence, which in the commutative case is simply the vector space $\mathbb{C}^E$ endowed with the natural left and right actions of $\mathbb{C}^V$. In this talk, I discuss how the Cuntz–Pimsner algebra $\mathcal{O}_{E_\mathcal{G}}$ is isomorphic to a universal C*-algebra defined in terms of linear maps on $B$ that respect the quantum graph structure. In particular, this implies $\mathcal{O}_{E_\mathcal{G}}$ is a quotient of the so-called quantum Cuntz–Krieger algebra $\mathcal{O}(\mathcal{G})$. This is based on joint work with Michael Brannan, Mitch Hamidi, Lara Ismert, and Mateusz Wasilewski.

Emily Peters (Loyola University of Chicago)

Canceled

Simon Schmidt (University of Copenhagen)

Title: On the quantum symmetry of distance-transitive graphs

Abstract: To capture the symmetry of a graph one studies its automorphism group. We will talk about a generalization of automorphism groups of finite graphs in the framework of Woronowicz’s compact matrix quantum groups. An important task is to see whether or not a graph has quantum symmetry, i.e. whether or not its quantum automorphism group is commutative. We will see that a graph has quantum symmetry if its automorphism group contains a certain pair of automorphisms. Then, focussing on distance-transitive graphs, we will discuss tools for proving that the generators of the quantum automorphism group commute and deduce that several families of distance-transitive graphs have no quantum symmetry.

Dimitri Shlyakhtenko (University of California, Los Angeles)

Title: An Inequality for Non-Microstates Free Entropy Dimension for Crossed Products by Finite Abelian Groups

Abstract: For certain generating sets of the subfactor pair M⊂M⋊G where G is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form M⋊(ℤ/2ℤ)⊕∞ for a large class of algebras M.

Roland Vergnioux (University of Caen)

Title: Hecke algebra and Schlichting completion for discrete quantum groups

Abstract: In recent joint work with Skalski and Voigt we construct and study the Hecke algebra and Hecke operators associated with an almost normal subgroup in a discrete quantum group. We also give in this framework a quantum version of the Schlichting completion, which yields an algebraic quantum group with a compact-open subgroup. We describe a class of examples arising from HNN extensions.