Combinatorial Algorithms, Tropical Dynamics and Integrable Systems
In this talk we will develop a link between a certain class of solvable cellular automata (box-ball exclusion processes) and integrable dynamical systems (related to the Toda Lattice particle system). This link between discrete space-discrete time processes and continuous space-continuous time dynamical systems is, remarkably, provided by a combinatorial algorithm due to Robinson,Schensted and Knuth (RSK) that underlies the classical Schur-Weyl approach to the representation theory of compact Lie groups. This talk will focus on this RSK link. A subsequent talk will develop connections to the Lie theory and stochastic processes related to random matrix theory.