# Keynote Speakers

^{**}Due to unforeseen circumstances, Aner Shalev is no longer able to attend.

^{**}

The conference about "Groups, Rings and the Yang-Baxter equation 2023" will be held at Corsendonk Duinse Polders in the beautiful town of Blankenberge, Belgium. This international conference
is a sequel to the meetings held in 2017 and 2019 at Corsendonk Sol Cress, Spa, Belgium concentrating on recent developments in and the interplay between the areas of ring theory and group theory, with a focus on the methods involved in their study and applications to other areas, mainly related to the celebrated YBE.

There will be the opportunity for presenting a contributed talk (15 minutes) for interested participants. Please indicate during registration whether you would like to give such a talk, and in that case please send us an abstract of your talk before **April 17th, 2023**. We will inform all interested participants by May 1st whether we can schedule their proposed talk. There also will be a poster session to which participants can contribute.

The talks will begin early Monday morning, June 19th, and the last talk will be presented late afternoon on Friday, June 23th.
If you have any questions please do not hesitate to write to us: gryb2023@gmail.com

- A first option to enjoy on this free afternoon is a trip to Brugge. It is, locally, considered the Venice of Northern Europe due to its beautiful canals (Reien). Moreover, the market and tower is an amazing sight. You should be easily get there with a train from Blankenberge Station.
- A Second option is to visit Ostend (Oostende). It is one of the larger cities at the Belgian coast and has been graced with the funds of King Leopold II, who constructed several beautiful galleries and buildings. Moreover, the harbour contains several interesting old boats that one may visit, such as the Mercator.
- A third option is to go slightly deeper inland to Ghent (Gent). It is the capital of the province of East Flanders and has several interesting architectural marvels, a low-traffic center and can be easily reached via train from Blankenberge station
- A poster is available here.
- The event schedule is now online!
- The registration is now close.
- Please contact us at Contact Us
- The conference picture!

- Eli Aljadeff (Technion - Israel Institute of Technology, Haifa, Israel)
- Ariel Amsalem (Haifa University, Israel)
- Sigiswald Barbier (Ghent University, Belgium)
- Stefaan Caenepeel (Vrije Universiteit Brussel, Belgium)
- Peter Cameron (University of St Andrews, UK)
- Andrea Caranti (University of Trento, Italy)
- Francesco Catino (Università del Salento, Italy)
- Roberto Civino (Università dell'Aquila, Italy)
- Ilaria Colazzo (University of Exeter, UK)
- Asmaa Elsawy (Menoufia University, Egypt)
- Valerio Fedele (University of L'Aquila, Italy)
- Edouard Feingesicht (University of Caen, France)
- Andrew Darlignton (University of Exeter, UK)
- Ilaria Del Corso (Università di Pisa, Italy)
- Maria Ferrara (Univeristà della Campania, Italy)
- Davide Ferri (University of Pisa, Italy)
- Fabio Ferri (University of Exeter, UK)
- César Galindo (Universidad de los Andes, Colombia)
- Àngel García Blàzquez (Universidad de Murcia, Spain)
- Diego García Lucas (Universidad de Murcia, Spain)
- Tatiana Gateva-Ivanova (American University in Bulgaria, Bulgaria)
- Be'eri Greenfeld (University of California San Diego, US)
- Gurnoor Gujral (Cheentan, India)
- Roghayeh Hafezieh (GEBZE Technical University, Turkey)
- Harald Helfgott (University of Göttingen, Germany)
- Geoffrey Janssens (UCLouvain, Belgium)
- Arpan Kanrar (Harish-Chandra Research Institute, India)
- Rahul Kaushik (Indian Institute of science education and Research, Pune, India)
- Łukasz Kubat (University of Warsaw, Poland)
- Manoj Kumar (Harish-Chandra Research Institute, India)
- Victoria Lebed (University of Caen, France)
- Thomas Letourmy (Université libre de Bruxelles, Belgium)
- Neha Malik (Indian Institute of Science Education and Research Mohali, India)
- Pallabi Manna (National Institute of Technology, Rourkela, India)
- Isabel Martin-Lyons (Keele University, UK)
- Marzia Mazzotta (Università del Salento, Italy)
- Fatemeh Mohammadi (KU Leuven, Belgium)
- Juan Carlos Morales Parra (Heriot-Watt University, UK)
- Nishant Nishant (IISER Mohali, India)
- Jan Okniński (University of Warsaw, Poland)
- Pilar Paez-Guillan (University of Vienna, Austria)
- Ashutosh Pandey (University Delhi Kashmere Gate campus, India)
- Mani Shankar Pandey (The Institute of Mathematical Sciences, Chennai, India)
- Elena Pascucci (Sapienza Università di Roma, Italy)
- Vicent Pérez-Calabuig (University of Valencia, Spain)
- Kevin Piterman (Philipps-Universität Marburg, Germany)
- Julia Plavnik (Indiana University, USA)
- Ahmed Ratnani (UM6P, Morocco)
- Bernard Rybołowicz (Heriot-Watt University, UK)
- Raúl Sastriques Guardiola (University of Valencia, Spain)
- Ofir Schnabel (Braude college of engineering, Karmiel)
- Himanshu Setia (Indian Institute of Technology, India)
- Aner Shalev (The Hebrew University of Jerusalem, Israel)
- Promod Sharma (IITBHU Varanasi)
- Agata Smoktunowicz (University of Edinburgh, UK)
- Paola Stefanelli (Università del Salento, Italy)
- Lorenzo Stefanello (Università di Pisa, Italy)
- Pedro Tamaroff (Humboldt-Universität zu Berlin, Germany)
- Ghada Tharwat (Egypt)
- Senne Trappeniers (Vrije Universiteit Brussel, Belgium)
- Marco Trombetti (Università di Napoli, Italy)
- Paul Truman (Keele University, UK)
- Leandro Vendramin (Vrije Universiteit Brussel, Belgium)
- Arne Van Antwerpen (Vrije Universiteit Brussel, Belgium)
- Magdalena Wiertel (University of Warsaw, Poland)
- Simon Wood (Cardiff University, UK)

Ilaria Colazzo (University of Exeter, UK)

Fatemeh Mohammadi (KU Leuven, Belgium)

Arne Van Antwerpen (Vrije Universiteit Brussel, Belgium)

Leandro Vendramin (Vrije Universiteit Brussel, Belgium)

Nigel Byott (University of Exeter, UK)

Kenny De Commer (Vrije Universiteit Brussel, Belgium)

Anastasia Doikou (Heriot-Watt University, UK)

Fatemeh Mohammadi (KU Leuven, Belgium)

Jan Okninski (University of Warsaw, Poland)

Julia Plavnik (Indiana University in Bloomington, USA)

Leandro Vendramin (Vrije Universiteit Brussel, Belgium)

Ilaria Colazzo (University of Exeter, UK)

Carsten Dietzel (Vrije Universiteit Brussel, Belgium)

Thomas Letourmy (Universite Libre de Bruxelles, Belgium)

Kevin Piterman (University of Marburg, Germany)

Silvia Properzi (Vrije Universiteit Brussel, Belgium)

Senne Trappeniers (Vrije Universiteit Brussel, Belgium)

Arne Van Antwerpen (Vrije Universiteit Brussel, Belgium)

There will be a conference fee of 50 euros payable upon arrival.

The registration is now close. Please contact us at Contact Us

You can download the abstracts of all talks and posters at once in the abstract book below. A talk schedule is included on the last page and is also separately downloadable. There might still be some minor changes.

Abstract book (PDF)

Talk schedule (PDF)

Individual abstracts are available in the online schedule on the right.

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This talk is partially based on joint works with J. Bell and with E. Zelmanov.

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In this talk, based on joint work with M. Castelli, we focus on how the number of orbits, and hence the indecomposability, of a solution of the YBE is related to subsets of its permutation skew brace which generate it as a strong left ideal. We will then introduce related notions of generating sets of skew braces and discuss how, under certain conditions, these different notions coincide and tell us something about subsolutions of associated solutions. If time permits, we will discuss applications of the above to certain classes of solutions.

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We focus on the representations of the Hecke--Kiselman algebras. The radical is described in case the algebra $K[HK_{\theta}]$ satisfies a polynomial identity. The latter condition can be expressed in terms of the graph~$\theta$. All irreducible representations, and the corresponding maximal ideals, are then characterized for this case. Every representation either is one-dimensional and it comes from an idempotent in the Hecke--Kiselman monoid or it comes from certain semigroups of matrix type arising from $HK_{\theta}$. The result shows a surprising similarity to the classical theorems on the representations of finite semigroups.

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Among modular categories, the Verlinde Modular Categories—denoted as $\mathcal{C}(\mathfrak{g},q)$ and associated with a simple Lie algebra $\mathfrak{g}$ and a root of unity $q$—are perhaps the most significant due to their diverse applications in both low-dimensional topology and quantum computing, among other areas. These modular categories are constructed as the semisimplification of the tilting modules of the quantum group $U_q(\mathfrak{g})$.

The \emph{zesting construction} provides a way for generating new modular categories from an existing one. In this presentation, recent advancements in applying the zesting construction to Verlinde modular categories are discussed. We offer a comprehensive description of the zestings and the modular data related to the Zesting of Verlinde modular categories. This work is a collaboration with Giovanny Mora.

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We discuss the Isomorphism Problem in the case where $R=\mathbb{Q}$ and the groups are metacyclic. We explain the proof of the positive answer to the Isomorphism Problem in this case.

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Starting with an overview on the state of knowledge on general Isomorphism Problems and the modular one in particular, I will present a negative solution found rather recently, but also present positive structural results and several problems remaining open.

Joint work with Arne Van Antwerpen.

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Babai's conjecture states that, for any finite simple non-abelian group $G$, the diameter of $G$ is bounded by $(\log|G|)^{C}$, where $C$ is a constant. A series of results since (Helfgott, 2005-2008) has given us cases of Babai's conjecture for different families of groups. However, for linear algebraic groups $G$, the dependence of $C$ on the rank of $G$ has been very poor (exponential-tower).

(a) How much can one improve the bound, while keeping the general inductive idea in Larsen-Pink (1998-2011) (which they used to classify subgroups in $SL_n$; generalized for use in this context by Breuillard-Green-Tao, 2010-2011) or in (Pyber-Szabo, 2010-2016)?

(b) Can one change the strategy and prove a yet better bound?

On (a), we will show the main ideas that have allowed us to make C exponential on the rank, while keeping an inductive argument that resembles those used before. (Actually keeping the Larsen-Pink inductive process gives a worse but still exponential bound, if combined with our other improvements.)

We will also discuss (b): changing the strategy - from general dimensional bounds as in Larsen-Pink - we have been able to make $C$ polynomial on the rank.

In this talk, we present the algebraic structure of \textit{dual weak brace} that is a triple $(S, +, \circ)$ having $(S, +)$ and $(S, \circ)$ as Clifford semigroups and satisfying the relations \begin{align*} a\circ (b+c)=a \circ b-a+a \circ c \qquad\& \qquad a \circ a^-=-a+a, \end{align*} for all $a,b,c \in S$, where $-a$ and $a^-$ denote the inverses of $a$ with respect to $+$ and $\circ$, respectively. In particular, if $(S, +)$ and $(S, \circ)$ are groups, then $(S,+, \circ)$ is a skew brace. Every dual weak brace gives $S$ rise to a solution $r_S$ close to being bijective and non-degenerate which we prove is a strong semilattice of bijective and non-degenerate solutions $r_\alpha$ coming from specific skew braces $B_\alpha$. Such skew braces are those realizing $S$, since we show that it is a strong semilattice $[Y, B_\alpha, \phi_{\alpha, \beta}]$ of skew braces $B_\alpha$, with $\alpha \in Y$.

This talk is based on a work in progress with F. Catino and P. Stefanelli.

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In 1978, Daniel Quillen conjectured that the poset of non-trivial $p$-subgroups of a finite group $G$ is contractible if and only if $G$ has non-trivial $p$-core. Quillen established the conjecture for solvable groups and some families of groups of Lie type. The major step towards the resolution of the conjecture was done by Michael Aschbacher and Stephen D. Smith at the beginning of the nineties. They roughly proved that if $p>5$ and $G$ is a group of minimal order failing the conjecture, then $G$ contains a simple component PSU$(n,q^2)$ failing a certain homological condition denoted by $(\mathcal{QD})$ (namely, the top-degree homology group of its $p$-subgroup poset does not vanish).

In this talk, I will present recent advances in the conjecture, with a particular focus on the prime $p=2$, which was not covered by the methods developed by Aschbacher-Smith. In particular, we show that the study of the conjecture for the prime $p=2$ basically reduces to studying $(\mathcal{QD})$ on the poset of $p$-subgroups of certain families of classical groups. Part of this work is in collaboration with S.D. Smith.

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Since then, many other graphs have been defined on groups (including the power graph and generating graph), and similar definitions have been made for rings (including the zero-divisor graph). I have been involved in this with a large group of mathematicians, mostly in India, as a result of an on-line research discussion group run during the pandemic.

The main features of this particular interaction between graphs and algebraic structures are:

- Like Brauer and Fowler, we can use graphs to understand groups better.
- Some of the graphs that arise are interesting for graph theory and its applications.
- Interesting classes of groups can be defined by graphs, either by requiring that two particular graphs on a group (such as the power graph and commuting graph) are equal, or by asking when the graph belongs to a particular class (such as cographs or perfect graphs).

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The aim of this talk is to introduce the audience to a recent joint work with Marco Trombetti (Università degli Studi di Napoli Federico II) and Frank O. Wagner (Université de Lyon) in which we tried to understand the first-order theory of skew braces (and related nilpotency concepts) with particular attention to $\omega$-categorical, stable skew braces.

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In 1940 G. Higman discovered a perfect analogue of Dirichlet's Unit Theorem for a group ring $\mathbb{Z}[T]$ where $T$ is a finite abelian group: $(\mathbb{Z}[T])^{\ast}=\pm T\times \mathbb{Z}^g$ for a suitable explicit constant $g=g(T)$. In 1960 Fuchs in [Abelian Groups, (Pergamon, Oxford, 1960); Problem 72] posed the following problem.

*Characterize the groups which are the groups of all units in a commutative and
associative ring with identity.*

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An open problem in Hopf-Galois theory regards the surjectivity of the Hopf-Galois correspondence; the well-know bijective Galois correspondence between subgroups of the Galois group and intermediate fields can be generalised to Hopf–Galois structures, but the correspondence one obtains is injective but not necessarily surjective. Very few examples where this correspondence is surjective are known; it is of interest to find new ones, and to understand more in general why it should be (or not be) surjective.

The goal of this talk it to approach this problem using the point of view of skew braces. We show that using a new version of connection between skew braces and Hopf–Galois structures, recently developed with S. Trappeniers, we can translate the problem of the surjectivity of the Hopf–Galois correspondence in a natural problem in skew brace theory, thereby deriving several new examples of Hopf–Galois structures for which the Hopf–Galois correspondence is surjective.

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We will delve into the intricate web of connections among different algebraic structures, such as Lie rings, groups, Jordan algebras, braces, pre-Lie algebras, and braided groups.

Throughout our exploration, we will draw upon the contributions of esteemed researchers who have made significant strides in this field. Notable figures such as Lazard, Magnus, Kostrykin, Zelmanov, Shalev, Rump, and Gateva-Ivanova have made invaluable contributions to uncovering these connections, and their work serves as the foundation for our discussion.

Moreover, we will introduce some fresh insights and discoveries that have emerged from recent research. Specifically, we will focus on the intriguing connections between braces and pre-Lie rings. Additionally, we will address some open questions that currently remain unresolved.

Part of this talk is related to a joint work with Aner Shalev.

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A group $H$ of order $n$ induces a natural unique elementary crossed product grading class of the matrix algebra $A=M_n(\mathbb{C})$. This class admits a representative determined by any tuple of distinct elements in $H$, $(h_1,h_2,\ldots ,h_n)$, where the grading is givenby $A_h=\text{span}\{E_{ij}| h=h_ih_j^{-1}\}$. On the other hand, any simple twisted group algebra $\mathbb{C} ^{\alpha}G$ which corresponds to a group of central type $G$ of order $n^2$ and a nondegenerate cocycle $\alpha \in Z^2(G,\mathbb{C} ^*)$ is also equipped with a natural $G$ grading of $A$. We prove that such an $H$-elementary crossed product grading class is a quotient grading class of such twisted group algebra if an only if $H$ is an IYB group, that is a multiplicative group of a brace.

Joint work with Yuval Ginosar.

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It is well known (and easy to show) that the finite dimensional algebra $A$ is not determined by its $T$-ideal of identities (similarly the finite the dimensional super algebra $A_{2}$). The purpose of this lecture is to present the following result: The semisimple part of $A$ (resp. of $A_{2}$) is "basically" determined by the ideal of identities. Of course I'll explain what "basically" means here. These results may be extended to the group graded setting.

Joint work with Karasik.

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The aim of this talk is to explain the intriguing relationship between ring-theoretical and homological properties of algebras $\mathcal{A}_K(X,r)$ and properties of finite non-degenerate solutions $(X,r)$ of the Yang--Baxter equation. The main focus is on when such algebras are Noetherian, (semi)prime and representable.

The talk is based on a joint work with I. Colazzo, E. Jespers and A. Van Antwerpen.

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We shall give an overview on the theory of braided quivers and why it is relevant, and we shall survey some results on the DYBE that generalise known results on the YBE. In particular, we introduce the theory of

Finally, we sketch the interplay between DYBE and Garside theory, discussing the dynamical analogue of a famous result by F. Chouraqui.

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This is based on a joint work with Antonio Giambruno and Ernesto Spinelli.

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The situation in case of a Garside group $G$ with a modular lattice structure turns out to be quite similar - each such group contains a canonical distributive subgroup $\mathcal{D}(G) \leq G$ - the \emph{distributive scaffold} - whose lattice-decomposition $\mathcal{D}(G) \cong \prod_{i=1}^k \mathbb{Z}$ is induced by a lattice decomposition $G \cong \prod_{i=1}^k \beth_i$ into primary lattices $\beth_i$ which are called the \emph{beams} of $G$. In this sense, each modular Garside group contains the structure group of an involutive solution whose lattice-structure also controls the decomposition into beams.

In this talk, I give an outline of the architecture of modular Garside groups, starting with the decomposition of a modular Garside group into beams and ending with a characterization of the beams of \emph{dimension} $\geq 4$.

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There is a limited number of rooms available in the conference center. Please register for the conference first. Afterwards you will receive contact details and instructions on how to book a room at the conference venue.

The rates for full board are

- 126.45 euros (studio for single use)
- 111.85 euros (double room for single use)
- 98.35 euros (shared double room)

per person per day.

You can get to the hotel by public transport. Tram stop Blankenberge Duinse Polders is right in front of the hotel. The coastal tram De Panne - Oostende - Knokke (line 0) stops at this stop: De Lijn). Also, Blankenberge station is only 2 km away from Corsendonk Duinse Polders (more info: www.belgiantrain.be).

Coordinates of the conference site:

**Corsendonk Duinse Polders**

A. Ruzettelaan 195, 8370 Blankenberge

Below Brussels Airport is a railway station.

Take the train to Bruxelles Midi (Brussels South), where you change train with end destination Blankenberge. From here you can take the coastal tram in the direction to "Knokke". Get off at the stop "Duinse Polders", this is only 2 stops.

**There is also an airport called 'Brussels South', which is located close to the city of Charleroi.
Note that it is more difficult and it takes significantly longer to arrive from this airport to the conference site.
See the relative item. **

Take the train to Bruxelles Midi (Brussels South), where you change train with end destination Blankenberge. From here you can take the coastal tram in the direction to "Knokke". Get off at the stop "Duinse Polders", this is only 2 stops.

This is the terminal for Eurostar trains and Thalys trains:

Take the train to Blankenberge.
From here you can take the coastal tram in the direction to "Knokke".
Get off at the stop "Duinse Polders", this is only 2 stops.

Either take one of the Flibco busses just outside the airport to Bruxelles Midi (Brussels South) railway station, or take a local bus (company TEC) to "Charleroi Gare" and take a train to "Bruxelles Midi" (Brussels South), several trains have this as end destination. Change in Bruxelles Midi to the IC-train with end destination "Blankenberge". From here you can take the coastal tram in the direction to "Knokke". Get off at the stop "Duinse Polders", this is only 2 stops.