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 Kontsevich recursion. The talk is based on our joint work with Arkady Berenstein on noncommutative cluster algebras.
- Géométrie et Quantification