Thomas Lam (UMich)

Title: From Toda lattices to box-ball systems

Abstract: The Toda lattice is one of the most important and classical
algebraic integrable systems.  I will survey relations between the
(continuous time and space) Toda lattice, the (discrete time and
continuous space) discrete Toda lattice, and the (discrete time and
discrete space) box-ball system.

The box-ball system, first studied by Takahashi and Satsuma, turns out
to be intimately related to classical combinatorial algorithms such as
RSK insertion, and jeu-de-tauqin, and also to the theory of affine
crystal graphs.  The discrete Toda lattice appears in the study of
networks on surfaces, the octahedron recurrence, the dimer model, and
in total positivity.  I will try to explain some of these connections.