Discrete completely integrable systems and complex analysis

A lot of works were written in the last decade about discretization of Completely Integrable Systems preserving its fundamental properties. Our approach is based on the idea that the key elementary algebraic property leading to the famous KdV type complete integrability is factorization of linear operators. Indeed, this property plays also a key role in the great success of Complex Analysis. We developed a new approach to discretization of Complex Analysis based on these ideas (in collaboration with I. Dynnikov, P. Grinevich and R. Novikov).