Abstracts 

Nicolas Faroß

Quantum automorphism groups of hypergraphs 

We propose a definition of quantum automorphism groups of hypergraphs, which will turn out to generalize the quantum automorphism group of Bichon for classical graphs. Further, we are able to show that our quantum automorphism groups act on hypergraph C*-algebras as recently defined by Weber. In particular, this action generalizes the one on graph C*-algebras by Schmidt-Weber in 2018. 

Malte Gerhold

 Cohomology of free unitary quantum groups 

In the talk, we will discuss the free unitary quantum groups (or "universal quantum groups") of Wang and van Daele from a (co)homological perspective. We find a \emph{free resolution of the counit}, a versatile tool which helps to compute cohomological data such as Hochschild cohomology or bialgebra cohomology of the associated Hopf algebras. For free orthogonal quantum groups, such resolutions have been found by Collins, Härtel, and Thom (in the Kac-case) and Bichon (in the general case), and they will serve as our starting point for finding resolutions for free unitary quantum groups.

Based on joint work with Isabelle Baraquin, Uwe Franz, Mariusz Tobolski and Anna Kula [arXiv:2309.07767]. 

Daniel Gromada

Quantum symmetries and diagrammatic categories

In the two talks, I would like to look on the notion of quantum symmetries from the viewpoint of diagrammatic categories. In the first talk, I will explain the correspondence between special Frobenius algebras and partition categories. The most interesting consequence of this correspondence is that, on one hand, all finite quantum spaces with a fixed categorical dimension are mutually quantum isomorphic (their quantum symmetries being described by the partition category) and, conversely, this provides a classification of fibre functors on the category of all partitions. The diagrammatic approach also allows an easy proof of the well-known fact that all finite spaces (of size at least four) have quantum symmetries. In the second talk, I would like to focus on my recent result on quantum symmetries of Hadamard matrices. Here, we use the same technique to show that all Hadamard matrices (of size at least four) have quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic. 

Paweł Kasprzak

The Plünnecke-Ruzsa theorem for compact (quantum) groups? 

The Plünnecke-Ruzsa inequality is a basic  theorem in additive combinatorics. Using  Pontryagin duality one can state this theorem in terms of irreducible representations of compact abelian groups. After sketching the modern proof of PR inequality due to George Petridis we shall  state a few  conjectures that can be viewed as a generalization of Plünnecke-Ruzsa inequality to compact (quantum) groups and prove or announce  a computer assisted check of them for a few finite groups and for SU(2). This is a joint work in stagnation with Paweł Lisiak and my  presentation will fit into the workshop format of the conference. 

Manish Kumar

Fullness of q-Araki-Woods factors

We discuss the factoriality and fullness questions of q-Araki-Woods von Neumann algebras constructed by Hiai, and our approach using conjugate variables. This is joint work with Adam Skalski and Mateusz Wasilewski, and with Simeng Wang.

Simon Schmidt

Solution groups and quantum automorphism groups of graphs

The two talks will focus on the connection of solution groups of linear constraint systems and quantum automorphism groups of graphs. In the first talk, we will construct graphs whose quantum automorphism groups are duals of homogeneous solution groups of linear constraint systems. We will look at particularly interesting cases of this construction in the second talk. For example, we will obtain a graph with quantum symmetry and finite quantum automorphism group in this way.

The talks are based on joint work with David Roberson and on-going joint work with Josse van Dobben de Bruyn and David Roberson. 

Piotr Sołtan

 Invariants of quantum groups related to the scaling group 

I will describe some of the results of a work in progress carried out by J. Krajczok an myself on some invariants of quantum groups. 

Ami Viselter

Lecture I: Invariant objects on quantum homogeneous spaces
Lecture II: Cocycles on quantum groups

The two independent talks will be devoted to "continuous-time random walks", i.e. Levy processes / convolution semigroups, on locally compact quantum groups and related objects.

The first talk will discuss several types of invariant objects, mostly ones related to convolution semigroups, on quantum homogeneous spaces and the relations between them. We will focus on homogeneous spaces coming from idempotent states.
The second talk will discuss the relations between convolution semigroups and cocycles, especially the problem of reconstructing the former from the latter. 

Marcel Wack

Computing (quantum) automorphism groups using OSCAR

An introduction to computational methods for (quantum) automorphism  groups of graphs and matroids in Oscar and Polymake. This talk is based on a work in progress by D. Corey, J. Schanz, M. Joswig, M.  Weber and myself. 

Hua Wang

A revisit of Woronowicz's Tannaka-Krein duality

Based on the previous work of Kac, Takesaki et al. and further studied later by Woronowicz and Sołtan, the work of Baaj & Skandalis raised and systematically developed the theory of multiplicative unitaries. This simple yet powerful framework plays a pivotal role for the Pontryagin duality of locally compact quantum groups. In this talk, I will present how one can recover the underlying multiplicative unitary directly from the representation theory of a compact quantum group, hence describing a parallel approach to Woronowicz's Tannaka-Krein reconstruction.

Mateusz Wasilewski

Quantum Cayley graphs

I will talk about a method of associating a quantum graph to a discrete quantum group together with a projection in its function algebra. These quantum graphs are analogues of Cayley graphs and they do not depend on the choice of a generating projection in the sense of metric geometry. Later I will show how they can help in finding examples of finite quantum groups having Frucht property, i.e. arising as quantum automorphism groups of quantum graphs.

Part of the talk will be based on an on-going joint work with Michael Brannan and Adam Skalski.