Frequently in mathematics one interchanges objects in an algebraic expression. In recent years this operation was formulated as a non-trivial linear map called a braiding and obeying certain linear braid or the famous now `Yang–Baxter’ relations (YBE). This operation turned out to be the key to revolutions both in mathematical physics (where it was central to a new `quantum’ notion of symmetry, namely a quantum group) and in knot theory (in the construction of knot invariants). On the other hand, the variety of all solutions on vector spaces of a given dimension has remained rather elusive in any degree of generality. It was proposed by V.G. Drinfeld (1991), to consider the same equations in the category of sets, and in this setting major advances were made. It is clear that a set-theoretic solution extends to a linear one, but more important than this is that set-theoretic solutions lead to their own remarkable algebraic and combinatorics structures, only somewhat analogous to quantum group constructions. Set-theoretic solutions of the Yang– Baxter equation form a meeting-ground of mathematical physics, algebra, geometry, and combinatorics. Such a solution consists of a set X and a bijective map r : X × X → X × X which satisfies the braid relation. Associated to each set-theoretic solution $(X,r)$ are several algebraic constructions generated by X and with quadratic relations naturally induced by r: the braided monoid S(X,r), the braided group G(X,r), the quantum space kR[V] = kS(X,r) (the semigroup algebra) , braces, skew braces, etc. During the last two decades the study of set-theoretic solutions of the Yang-Baxter equation and related structures notably intensified . The aim of this symposium is to present some of the most recent results, open questions and strategies in the area.

Organizers:

  • Tatiana Gateva-Ivanova, American University in Bulgaria, Bulgaria, tatyana@aubg.edu

Invited participants:

  • Shahn Majid, Queen Mary, University of London
  • Edwin Beggs, Swansea University
  • Adolfo Ballester Bolinches, University of Valencia
  • Nigel Byott, University of Exeter
  • Francesco Catino, University of Salento
  • Ferran Cedó, UAB, Barcelona
  • Fabienne Chouraqui, University of Haifa
  • Tatiana Gateva-Ivanova, American University in Bulgaria
  • Florin Nichita, IMAR, Bucharest
  • Vladimir Roubtsov, University of Angers (to be confirmed)

List of talks:

  • to be announced