Ben Brubaker's 2022 Project

This project will explore certain “solvable lattice models”, sometimes referred to as “ice models.” These models are built on simple grids that originated in the study of sheets of crystallized materials, but have connections to a remarkable number of areas of mathematics. Specifically, we will focus on the word “solvable” whose meaning is many-faceted, but which is defined by a famous relationship called the Yang-Baxter equation. For the simplest types of lattice models, it is known which ones are solvable, but when we introduce “color” into our grids, only special cases are known. We will attempt to find precisely when these more general lattice models are solvable, as well as explore consequences of this powerful equation.