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New discretization of complex analysis and completely integrable systems

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Discrete Integrable Systems

New Discretization of Complex Analysis was developed recently by the present author and I.Dynnikov. Our discretization has nothing to do with geometric discretization of conformal maps. We consider complex analysis as a theory of linear Cauchy-Riemann Operator. It is based on the Equilateral Triangle Lattice in the Euclidean Plane (the classical one was based on the quadrilateral lattice). This approach allows us to borrow some crucial ideas from the modern theory of Completely Integrable Systems missing in the case of quadrilateral lattice. New phenomena appear in the case of Equilateral Lattice in Hyperbolic (Lobachevski) Plane: geometric objects became ’’stochastic’’ requiring to use technic of Symbolic Dynamics. New difficulties appear leading to the unsolved plroblems.

This talk is part of the Isaac Newton Institute Seminar Series series.

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