**#26: Jan 29th**

**Speaker:** Alexander Frei (University of Copenhagen) **(12:00 US ET)**

**Title:** Cuntz-Pimsner algebras — covariances, the gauge theorem, the classification of ideals and their relation to Fell bundles

**Abstract: **We will start with a swift introduction to Dirac/matrix calculus, and highlight its use at a couple of central results from operator spaces. This will explain its usefulness in operator algebras similar to the use of Hilbert modules.

We then quickly move on to Cuntz-Pimsner algebras. From here we study the notion of covariances more deeply and connect the dots to categorical kernels. As a highlight we then arrive at the gauge-invariant uniqueness theorem, for all gauge-equivariant representations at once.

As a consequence we obtain – by systematic reduction – the classification of relative Cuntz-Pimsner algebras, originally obtained by Katsura.

*Teaser/spoiler:* We will find them classified precisely by kernel-covariance pairs.

Finally, as time permits, we explain their relation to Hilbert bimodules, dilations and their connection to Fell bundles as handled by Pimsner and further explored by Schweizer.

**Background material:
**If interested, for getting a little bit acquainted with today’s topic: The light-hearted section 2 from [Katsura04].

For further reading:

For an introduction to Hilbert modules [Lance].

For the treatment of covariances and the classification: under preparation.

For a step-by step treatment of their relation to Hilbert bimodules and dilations see [Schweizer].

For a comprehensive treatment of Fell bundles [Exel].

**Talk Notes:**Alex’s Notes

** **

**#27: Feb 5th**

**Speaker: **Kari Eifler (Texas A&M University) **(12:00 US ET)**

**Title: **Quantum Symmetries of Quantum Metric Spaces**
Abstract:** I will first give a light introduction to the theory of compact quantum groups. We will look at quantum metric spaces, which are a non-commutative analogue of finite metric spaces. Banica has defined the quantum symmetry group of a finite metric space, and I will talk about how to capture Banica’s definition using the Weaver-Kupperberg framework of quantum metric spaces. I will conclude by connecting this extension to the theory of non-local games.

**Non-local Games and Quantum Symmetries of Quantum Metric Spaces**

Background material:

Background material:

Talk Notes: eifler_QSSS

Talk Notes: eifler_QSSS

**#28: Feb 12th**

**Speaker: **Priyanga Ganesan (Texas A&M University) **(11:30 US ET)**

**Title: **Quantum Graph Homomorphisms**
Abstract: **Quantum graphs are an operator space generalization of classical graphs. In this talk, I will motivate the idea of a quantum graph and its significance in quantum communication. We will look at the different notions of quantum graphs that arise in operator systems theory, non-commutative topology and quantum information theory. I will then introduce a non-local game with quantum inputs and classical outputs, that generalizes the homomorphism game for classical graphs.

**1) Vern Paulsen’s lecture notes https://www.math.uwaterloo.ca/~vpaulsen/EntanglementAndNonlocality_LectureNotes_7.pdf**

Background material:

Background material:

2) The Quantum-to-Classical Graph Homomorphism Game https://arxiv.org/abs/2009.07229

**Talk Notes:**Priyanga Ganesan_talk

**#29: Feb 19th**

**Speaker: **Thibault Décoppet (University of Oxford) **(11:00 US ET)**

**Title: **Semisimple 2-Categories and Fusion 2-Categories

**Abstract:** I will review the notion of a 2-condensation monad introduced by Gaiotto-Johnson-Freyd. Using these, I will define semisimple 2-categories following Douglas-Reutter, and explain how finite semisimple 2-categories and multifusion categories are related. Examples of finite semisimple 2-categories will be given. Then, I will present the definition of fusion 2-categories, and give examples. Finally, I will mention some further results and conjectures on fusion 2-categories.**
Background material: **On the algebraic theory of fusion 2-categories (arxiv: 2012.15774)

**Talk Notes:**Thibault’s Notes

**#30: Feb 26th**

**Speaker: **Josh Edge (Denison University) **(UPDATED! 9:30 US ET)**

**Title: **Classification of symmetric Trivalent Categories**
Abstract: **A trivalent category is a planar algebra generated by a symmetric trivalent vertex. This planar algebra can be thought of as a quotient of the planar algebra of planar graphs containing only trivalent and univalent vertices. Trivalent categories of small dimension were classified by Morrison, Peters, and Snyder. In this talk, we will expand on this classification by classifying symmetric trivalent categories, planar algebras generated both by a symmetric trivalent vertex and a symmetric crossing. A symmetric trivalent category essentially removed the planarity requirement in previous work. We will discuss the relationship between planar algebras and monoidal categories. We will also classify which planar algebras have sub-braidings.

**Background material:**Emily Peter’s talks found here.

**Talk Notes:**Edge _ Symmetric_Trivalent_Categories_Presentation (5)

**#31: Mar 5th
**

**Speaker:**Sachin Valera (University of Bergen)

**(12:00 US ET)**

**Title:**Fusion Structure from Exchange Symmetry in (2+1)-Dimensions

**Abstract:**Exchange symmetry is a cornerstone of quantum mechanics, first appearing when we study systems of many identical particles. When we suppress some spatial dimensions, exchange symmetry indicates that effectively 2D quantum systems should support quasiparticles called anyons with exotic statistical behaviour.

Algebraically, anyonic systems are modelled using unitary ribbon fusion categories. Our goal will be to patch the gap between the fundamental principle of exchange symmetry and the categorical framework using a minimal prescription of postulates.

We will show that the superselection sectors (i.e. the possible topological charges) of anyonic systems form a fusion algebra, and emerge from a hierarchy of localised exchange symmetry mechanisms. Moreover, we prove that the superselection sectors of such systems are uniquely specified by the action of a special braid, which obeys some internal symmetries that characterise the behaviour of anyons.

**Background material: **Fusion Structure from Exchange Symmetry in (2+1)-Dimensions and vague familiarity with quantum mechanics.

**Talk Notes:** QSSS_valera_03.21, Observables vs self-adjoints

**#32: Mar 12th**

**Speaker: **Juan Orendain (Universidad Nacional Autonoma de México) **(12:00)**

**Title: **Remarks on the bicategory of von Neumann algebras

**Abstract: **C*-algebras, C*-correspondences and their unitary isomorphisms organize into a bicategory encoding Morita equivalence as weak isomorphisms, and in terms of which cocycle conditions for generalized Busby-Smith twisted actions and saturated Fell bundels admit natural formulations. The corresponding categorical object in the theory of von Neumann algebras is the bicategory of von Neumann algebras and von Neumann correspondences. I will review this categorical structure in detail and present a few remarks on it in the form of open questions, conjectures, and recent results relating it to cubical models of operator categories of second order.

**Background material:
**[1] A higher category approach to twisted actions on C*-algebras. A. Buss, R. Meyer, C. Zhu. Proc. Edinb. Math. Soc. (56) 2013, 387-426.

[2] Dualizability and index of subfactors. A. Bartels, C. L. Douglas, A. Hénriques. Quantum Topology (5) 2011, 289-345.

[3] Framed bicategories and monoidal fibrations. M. Shulman. Theory and Applications of Categories (20) 18, 2008, 650-738.

**Talk Notes:**Orendain 12 Mar 2021

**#33: Mar 19th**

**Speaker: **Rubén Martos Prieto (University of Copenhagen) **(11:00 US ET)**

**Title: **An overview of the Baum-Connes conjecture**
Abstract: **The Baum-Connes conjecture has been for the past few decades a formidable encouragement to understand the structure of group C*-algebras with diverse techniques coming from geometry, analysis, representation theory and, more recently, category theory. In this talk, we will have a glance at the conjecture since its origins until the most recent approaches. To do so, we will start by describing the main technical tools needed to state the conjecture. Next, we will mention some of the major positive results to the conjecture. Finally, we will explain a modern approach in terms of triangulated categories.

**Background material: 1)**“Introduction to K-theory for C*-algebras”, M. Rørdam, F. Larsen, N. Laustsen.

**2)**“K-theory for operator algebras”, B. Blackadar.

**3)**“Introduction to the Baum-Connes conjecture”, Alain Valette.

**4)**“The Baum-Connes conjecture via localisation of categories”, R. Meyer and R. Nest.

**Talk Notes:**Martos 19 Mar 2021

**#34: March 26th**

**Speaker:** Samuel Evington (Wilhelms Universität Münster) **(11:00 US ET)
Title: **Symmetries of Operator Algebras

**I will begin by discussing Cones’ classification of automorphisms of the hyperfinite II_1 factor R and its subsequent generalisations (e.g. group actions and actions of unitary fusion categories).**

Abstract:

Abstract:

I will then switch to the C*-setting and explain the new obstructions that occur (spoiler: algebraic K_1) and the result of adapting the von Neumann constructions to the C*-setting (spoiler: different descriptions of R lead to different classifiable C*-algebras).

The talk is based on joint work with Sergio Girón Pacheco.

**Talk Notes:**Sam‘s Notes

**#35: April 2nd**

**Speaker: **David Jekel (UCSD) **(12:00 US ET)
**

**Title:**Free Gibbs laws through free microstate entropy, (based on joint work with Wuchen Li and Dimitri Shlyakhtenko, arXiv:2101.06572, sections 3, 4, and 7)

**Abstract:**We explain the notions of free entropy and free Gibbs laws. The classical Gibbs law associated to some potential $V: \mathbb{R}^d \to \mathbb{R}$ is the measure $\mu_V$ that maximizes $h(\mu) – \int V\,d\mu$, where $h(\mu) = -\int \rho \log \rho\,dx$ is the differential entropy (here $\rho$ denotes the density of $\mu$). For the non-commutative version, we define spaces of smooth real-valued functions $V$ that can be evaluated on $d$-tuples of self-adjoint operators from a tracial von Neumann algebra. We describe non-commutative generalizations of probability measures (non-commutative laws) and differential entropy (Voiculescu’s free microstate entropy $\chi$). For certain functions $V$, we show that there exists a non-commutative law $\mu$ maximizing $\chi(\mu) – \mu(V)$, and that for generic $V$ this $\mu$ is unique. This is closely connected to the large $N$ asymptotic theory of certain random $N \times N$ matrix $d$-tuples.

**Background material:**

- Some familiarity with tracial von Neumann algebras is helpful, such as the material in Chapters 1, 2, 7, and 9 of Anantharaman and Popa’s book IIun.pdf (ucla.edu) . For beginners, I recommend first reading Jones’ notes VonNeumann2009.pdf (berkeley.edu) .
- The definition of non-commutative laws is very important; see section 5.2.2 of Anderson, Guionnet, and Zeitouni’s book cupbook.dvi (weizmann.ac.il)
- For a survey of free independence and free microstate entropy, read sections 1 and 2 of Voiculescu’s survey paper 0103168.pdf (arxiv.org)
- To become familiar with free Gibbs laws, see the introduction of On the statistical mechanics approach in the random matrix theory: Integrated density of states | SpringerLink
- These exercises are basic and review properties of the classical differential entropy, and of upper semi-continuous functions. By doing the exercises the week of the talk, people will be primed to understand the proofs better:
**Jekel prep_exercises**

**#36: April 9th**

**Speaker: **Luca Giorgetti (Vanderbilt University) **(12:00 US ET)**

**Title:** Realization of rigid C*-bicategories as bimodules over type II_1 von Neumann algebras**
Abstract: **Using Q-systems and standard solutions of the conjugate equations, we show that every rigid semisimple C*-bicategory can be realized as Connes’ bimodules over finite direct sums of II_1 factors.

The main point of our proof is a purely categorical bootstrap procedure that we shall present in the talk in the special case of multi-tensor C*-categories.

**Background material:**– [Yamagami 2004] Frobenius Algebras in Tensor Categories and Bimodule Extensions

– [Bischoff, Kawahigashi, Longo, Rehren 2015] Tensor categories and endomorphisms of von Neumann algebras (with apps to TQFT)

– [Giorgetti, Longo 2019] Minimal index and dimension for 2-C∗-categories with finite-dimensional centers

-[Giorgetti, Yuan 2020] Realization of rigid C*-bicategories as bimodules over type II1 von Neumann algebras

Joint work with Wei Yuan (Beijing, AMSS, CAS)

Supported by EU H2020-MSCA-IF-2017 grant beyondRCFT 795151

**Talk Notes:**Giorgetti _ QSSS

**#37: April 16th**

**Speaker: **Juan Camilo Arias Uribe (Universidad de los Andes)** (12:00 US ET)**

**Title: **Derived Fusion Categories**
Abstract: **Let C be an abelian spherical category. Call N_C the full subcategory of negligible objects in C, i.e., objects such that the trace of any of its endomorphisms vanishes. In this talk we propose a derived level definition of the fusion category and fusion ring for the category C. As an example, we show that the proposed definition gives the usual fusion rings in the case of the categories of representations of the big quantum group and the small quantum group.

Background material:

Background material:

- Derived counterparts of fusion cateogires of quantum groups – Arias
- Fusion categories arising from semisimple Lie algebras. Andersen – Paradowski – Communications in mathematical physics – 1995
- Representations of quantum algebras. Andersen – Polo – Kexin – Inventiones Mathematicae – 1991
- A guide to quantum groups. Chari – Pressley – 1994.
- Lectures on tensor categories and modular functors. Bakalov – Kirillov – 2001.

**Talk’s Notes: Arias Uribe _ Derived fusion categories**

**#38: April 23rd**

**Speaker: **Fabio Calderón Mateus (Universidad Nacional de Colombia) **(12:00 US ET)**

**Title: **Some algebraic properties of Universal Quantum Semigrupoids**
Abstract:** The study of quantum symmetries over a non-connected graded algebra leads to the concept of (co)actions of weak bialgebras and, in particular, the Universal Quantum Semigroupoids (UQSGs). For example, one remarkable set-up in which these UQSGs arise, is when the graded algebra is a path algebra over a finite quiver.

Last year, H. Huang, C. Walton, E. Wicks and R. Won proved two relevant results: 1. the associated UQSG of a path algebra is isomorphic to the Hayashi’s face algebra attached to the quiver, and 2. when the quiver is of extended Dynkin type, the associated UQSG of the preprojective algebra attached to the quiver is isomorphic to a certain quotient of Hayashi’s face algebra.

Based on those results, in joint work with Chelsea Walton, we study some ring-theoretic and homological properties of Hayashi’s face algebra. In this talk I will provide the motivation behind this research, the results already obtained and the following steps of our study. The talk will be self-contained, so only a basic familiarity with well-known concepts of algebra (Noetherian condition, primality, GKdim, etc.) is needed. No preliminaries on Hopf algebras or bialgebras are required.

**Background material:**

**–**https://arxiv.org/pdf/1103.2261.pdf for preliminaries on weak bialgebras.

**–**https://arxiv.org/pdf/2008.00606.pdf for preliminaries on UQSGs.

– https://www.cambridge.org/core/books/elements-of-the-representation-theory-of-associative-algebras/AA8066B5809D0F556A540400AD3A419C for preliminaries on quivers and path algebras.

**Talk Notes: Talk QSSS – algebraic properties of face algebras**