Schedule

Morning Session

Afternoon Session

Abstracts

Anastasia Doikou (Heriot-Watt University): Combinatorial Drinfel'd twists & the Yang-Baxter equation

We introduce the special set-theoretic Yang-Baxter algebra and show that it is a Hopf algebra subject to certain conditions. The associated universal R-matrix is also obtained via an admissible Drinfel'd twist. The structure of braces emerges naturally in this context by requiring the special set-theoretic Yang-Baxter algebra to be a Hopf algebra and a quasi-triangular bialgebra after twisting. The fundamental representation of the universal R-matrix yields the familiar involutive set-theoretic (combinatorial) solution of the Yang-Baxter equation. We also introduce  rack Hopf-like algebras and obtain rack and quandle solutions of the YBE. We show that the same combinatorial twist cane be used to produce non-involutive set-theoretic solutions.

Hal Simpson (University of Leeds): Bicyclic Biskew Braces

Skew braces are an algebraic structure with close historical and practical ties to the set-theoretic Yang-Baxter equation. We will discuss existing results about skew braces, covering gamma functions, the connection between skew braces and regular subgroups of the holomorph of a group, results about bi-skew braces, and ideals. We will use this to completely classify the finite bicyclic skew braces, and, among them, which are bi-skew.

George Altmann (University of Leeds): Peripheral Systems for a subclass of welded graphs

The peripheral system is an invariant of knots, consisting of a group W(D), along with two commuting elements m,l (the meridian and longitude) in W(D), for any knot diagram D. Considered up to group isomorphisms which send respective meridians and longitudes to each other, this is a complete invariant. We describe a generalisation of the peripheral system for the larger class of welded links, with additional information coming from a Z[W(D)]-module. By further defining this higher peripheral system on a certain subclass of welded graphs, we will explore the differences of this higher peripheral system to the classical case.

Ilaria Colazzo (University of Leeds): From Skew Braces to Set-Theoretic Solutions of the Yang–Baxter Equation

Skew braces offer a unifying lens for studying set-theoretic solutions of the Yang–Baxter equation. They connect familiar tools from group and ring theory to the combinatorial features of these solutions. In this talk I’ll give a friendly introduction to skew braces, explain their key structural ideas, and show how they can be used to build and analyse solutions—including a brief look at how simple solutions arise in this framework.

Joao Faria Martins (University of Leeds): Homotopy finite spaces, extended TQFTs, and representations of the braid and loop braid group

The Kontsevich-Quinn Finite Total Homotopy TQFT is a topological quantum field theory defined for any dimension $n$ of space and depending on the choice of a homotopy finite space $B$. (For instance, $B$ can be the classifying space of a finite group or of a finite 2-group.)

In this talk, I will report on recent joint work with Tim Porter on once-extended versions of this TQFT, taking values in the symmetric monoidal bicategory of groupoids, linear profunctors, and natural transformations. The construction works for all dimensions, in particular providing (1,2,3)- and (2,3,4)-extended TQFTs whenever a homotopy finite space $B$ is given.

I will show how to compute these once-extended TQFTs when $B$ is a homotopy 2-type (represented by a crossed module of groups), and then address the ensuing representations of the braid and loop braid groups.

Reference: Faria Martins J, Porter T: "A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids." arXiv:2301.02491 [math.CT]

Charlotte Roelants (Vrije Universiteit Brussel): Skew brace n-isoclinism

The notion of n-isoclinism of groups was originally defined by Hall (for n = 1) and later Bioch in order to classify groups of prime power order. Intuitively, we can view it as an expression of how similar the lower and upper central series of two groups are, starting from their n-th terms. In 2022, Letourmy and Vendramin introduced an equivalent of 1-isoclinism for skew braces. In this talk, we extend this notion to larger n, and study some of its basic properties. In particular, we cover the invariance of several nilpotency concepts under skew brace n-isoclinism. This talk is based on a paper in collaboration with Arpan Kanrar and Manoj Kumar Yadav.

Eric Jespers (Vrije Universiteit Brussel): Noetherian rings and the Yang-Baxter Equation

In this talk we focus on the following problems and the link with set-theoretic solutions of the Yang-Baxter equation: