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Explore a comprehensive lecture on discrete integrable systems focusing on the extensions of Discrete Toda Lattices and their applications to Box-Ball Systems. Delve into advanced mathematical concepts as part of a broader program on discrete integrable systems, difference equations, cluster algebras, and probabilistic models. Learn about exactly solvable systems and their conserved quantities, with particular emphasis on discrete systems derived from theoretical physics and mathematics. Examine the ultradiscretization of equations leading to cellular automata, their connections to quantum groups, combinatorics, tropical geometry, and stochastic processes. Understand the relationship between cluster algebra structures and geometric objects, as well as their applications in mathematical physics. Benefit from a structured program featuring mini-courses, problem sessions, and workshop components, designed for both junior and senior researchers interested in discrete integrable systems and related areas.
