Total Positivity, Directed Networks and Integrable Cluster Dynamics - Lecture 1

Explore a comprehensive lecture on total positivity, directed networks, and integrable cluster dynamics delivered at the International Centre for Theoretical Sciences as part of a two-week program on discrete integrable systems. Delve into the mathematical foundations of exactly solvable systems and their conserved quantities, with particular focus on discrete integrable systems emerging from theoretical physics and mathematics. Learn about integrable difference equations, cluster algebra structures, and integrable probability through detailed discussions of soliton solutions, singularity structures, and their connections to quantum groups, combinatorics, tropical geometry, and stochastic processes. Examine the relationships between geometric objects like Grassmannians and mathematical physics concepts such as Y-systems and dimer models. Gain insights into interacting particle systems, stochastic growth models, and their connections to the KPZ equation and random tilings, while exploring tools from quantum integrable systems and symmetric function theory.