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## Biracks and Biquandles: Theory, applications, and new perspectives

Algebra and Representation Theory in the North (ARTIN) at Leeds:

### Organisers:

- Ilaria Colazzo (University of Leeds)
- Joao Faria Martins (University of Leeds)
- Fiona Torzewska (University of Bristol)
- Markus Szymik (University of Sheffield)

More information will be on the webpage soon.

In the meanwhile, you can register your interest by filling in the registration form.

A small registration fee (£ 50) will be charged. This will be a contribution towards the buffet lunches and coffee breaks.

We have limited funding to support PhD-students and young researchers. The deadline to apply is **30th September 2024**.

Please send an email to Ilaria Colazzo.

## Workshop description

Biracks and biquandles arise as set-theoretical solutions of the celebrated Yang-Baxter equation, $$f_{12} \circ f_{23}\circ f_{12}=f_{23}\circ f_{12}\circ f_{23},$$ for a linear map, $f \colon V\otimes V \to V\otimes V$, where $V$ is a vector space.

The Yang-Baxter equation arose in the early 1970s in the context of integrable models for statistical mechanics. During the 1980s, strong connections were found between the Yang-Baxter equation and quantised Lie theory, leading to the quantum groups revolution, where the Yang-Baxter equation was formulated in the context of $R$-matrices in quasi-triangular Hopf algebras. This synergy between integrable systems and Lie theory had a spectacular impact on low-dimensional topology, leading to the emerging field of quantum topology, of wide application to solving classification problems in low-dimensional topology, and moreover providing a crucial component of the mathematical framework for the recent developments in the mathematical modelling of (2+1)dimensional topological phases of matter and the ensuing paradigms for topological quantum computing.

In this workshop, we will focus on set-theoretical solutions of the Yang-Baxter equation, particularly those that are bi-invertible: biracks and biquandles. This field has developed rapidly in the last two decades due to its applications to knot theory and the theory of motion groups of loops in 3-dimensional space, and the recent breakthrough notion of skew braces. The latter provides a group-theoretical underpinning for the set-theoretical Yang-Baxter equation. Furthermore, in 2016, unexpected connections between skew braces and Galois theory were discovered, providing a generalisation of the classical Galois correspondence, “the Hopf-Galois correspondence”, and, on another topic relations were found between biquandles (derived from skew-braces) and discrete higher gauge theory (2017), which provided an algebraic model for loop-excitations in 3+1-dimensional topological phases with higher gauge symmetry.

This meeting aims to put together specialists in the field of “set theoretical solutions of the Yang-Baxter equation”, particularly the bi-invertible ones, and discuss some of their different perspectives and incarnations. We intend to gather a set of invited speakers from different fields (e.g. algebra, integrable systems, low dimensional topology / knot theory, and motion groups) and we will instruct speakers to ensure their talks are accessible for researchers in the wider field, facilitating communication and collaboration between participants.

## Funders

ARTIN (Algebra and Representation Theory in the North) meetings are sponsored by the London Mathematical Society, by the Glasgow Mathematical Journal Trust, and by the Isaac Newton Institute via the EPSRC grant (Ref: EP/V521929/1).

ARTIN at Leeds is furthermore funded by the School of Mathematics of the University of Leeds