We exhibit several finite monoids each of which is not contained in any variety generated by a finitely based finite semigroup but is contained in a finitely based locally finite variety. The smallest monoid with these properties we found so far has as few as 9 elements. The result solves a 20+ years old problem and has numerous implications for the finite basis problem for finite semigroups.
This is joint work with Sergey Gusev (Ekaterinburg, Russia) and Olga Sapir (Beersheba, Israel).
Quantum vertex algebras are deformations of vertex algebras introduced by Etingof and Kazhdan in 1998. They are families of vertex operators with relations deformed by a solution of the quantum Yang-Baxter equation. There is also a dual notion of quantum vertex coalgebras which deform vertex coalgebras. In this talk I will explain how to construct two distinct structures of a quantum vertex coalgebra on the Yangian associated to any simple finite-dimensional complex Lie algebra, and how to induce quantum vertex algebra structures on the dual Yangian. This talk is based on ongoing work with Alex Weekes and Curtis Wendlandt.