Explore a comprehensive lecture on stationary measures for integrable probabilistic systems delivered at the International Centre for Theoretical Sciences. Delve into the mathematical foundations of discrete integrable systems, focusing on difference equations, cluster algebras, and probabilistic models. Learn about exactly solvable systems and their conserved quantities, examining both continuous and discrete integrable systems derived from theoretical physics and mathematics. Understand the three main research aspects: integrable difference equations and their soliton solutions, cluster algebra structures in discrete integrable systems, and integrable probability including interacting particle systems and stochastic growth models. Gain insights into tools from quantum integrable systems and symmetric functions theory, while exploring connections to statistical physics, string theory, combinatorics, representation theory, geometry, and probability. The lecture is part of a broader program featuring mini-courses, problem sessions, and workshop components, designed for both junior and senior researchers interested in discrete integrable systems and related areas. Read more