A composition of birational maps given by Laurent polynomials need not be a Laurent polynomial. When it does, we talk about the Laurent phenomenon. A large variety of examples of the Laurent phenomenon for commuting variables comes from the theory of cluster algebras introduced by Fomin and Zelevinsky. Much less is know in the noncommutative case. I will discuss various noncommutative Laurent phenomena including examples coming from noncommutative triangulations of polygons and oriented surfaces. As a byproduct of the theory, I will outline a proof of Laurentness of a noncommutative […]

I will present several examples of group actions by birational transformations in free noncommuting variables. One of examples is related to the talk of V.Retakh on noncommutative Laurent phenomenon, while another (a noncommutative generalization of the Coble action of Coxeter groups of series E) is definitely not cluster.

The talk will be an introduction to the new theory of the refined Jones and Quantum Group invariants of torus knots based on double affine Hecke algebras. This approach provides formulas (though mainly conjectural) for Poincare polynomials of stable Khovanov-Rozansky homology, also called super-polynomials, related to the BPS states from String theory. Khovanov-Rozansky theory will be touched upon only a little