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Explore the rich symmetries of discrete and ultradiscrete integrable systems in this lecture from the Workshop on "Non-commutative Geometry meets Topological Recursion" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the connections between these systems and various mathematical fields, including algebraic/tropical geometry, combinatorics, and crystal base theory. Examine the 'tropical limit' process that transforms rational maps into piecewise-linear maps. Begin with an introduction to the symmetries of the discrete KdV equation and discrete Toda lattice, highlighting their ability to maintain integrability during the tropical limit. Then, investigate the box-ball system (BBS), an integrable cellular automata related to the aforementioned discrete systems through 'ultradiscretization.' Discover how the BBS serves as a bridge between classical and quantum integrable systems, showcasing the intersection of tropical geometry and crystal base theory.
