Discrete systems and complex analysis

Our goal is to find realization of modern mathematical ideas (such as topology, symplectic geometry, algebra and algebraic geometry) in the problems of theoretical and mathematical physics. Three examples will be discussed in this talk: quantum scattering on the graphs with tails and symplectic geometry; completely integrable systems on a trivalent tree; discretization of Complex Analysis on the equilateral triangle lattice and completely integrable systems. Some of our works in these areas are joint with I. Krichever and I. Dynnikov.