Making waves discretely by putting balls into boxes and using crystals
University of Queensland
Date: Tuesday 17 September 2019
Time: 2:00 p.m.
Place: Room 241, 2nd floor, Science III building
In August, 1834, John Scott Russell followed a wave traveling through a narrow channel and noticed that as the wave propagated, it did not change shape nor speed. This observation was then given a mathematical theory starting with Boussinesq in 1871, and is now known as the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation. In particular, the KdV equation admits solutions for such waves, which are called solitons. We first can make the time steps discrete, which was done by Hirota in 1977, and we will make the height and position of a wave discrete following Takahashi and Satsuma. Indeed, by using boxes that can hold at most one ball in a simple discrete dynamical system, they relate the size of a wave to a coupled collection of balls. In this talk, we will discuss the Takahashi-Satsuma box-ball system and how it can be described using Kashiwara's crystal bases, a combinatorial interpretation of representation theory arising from mathematical physics (specifically, quantum groups). This allows the system to be generalized and more tools from mathematical physics to be applied, which will also be described as time permits.