Skew braces and applications

The proposal concerns the impact of the combinatorics behind the celebrated Yang-Baxter equation
(YBE) in algebra and beyond. Rather than working directly with solutions, we focus on skew braces, a
closely-related novel algebraic structure that has been the subject of intensive research over the last
few years. The theory of skew braces is a fertile meeting ground for group theory, ring theory,
geometry, and combinatorics. As such, it has the potential to produce developments in many areas.
The project's main aim is to develop ground-breaking methods that could apply not only to the
theory of skew braces and the YBE but also in other areas and then, in the long term, apply them to
solve several open questions in mathematics. Roughly speaking, the goals of this project are as
follows: a) Develop the representation theory of skew braces and study possible applications, for
example, to the theory of Hopf-Galois structures; b) develop the cohomology theory of skew braces
and the theory of extensions and Schur covers; c) study combinatorial properties of solutions to the
YBE; and d) explore the impact of the theory of skew braces in algebraic logic by applying skew
brace methods to the theory of L-algebras