Chip-firing is a simple yet surprisingly rich model on graphs, where one puts chips at the vertices and reallocates them by chip-firing moves. The model was discovered independently by researchers in statistical physics, number theory, and combinatorics; people study chip-firing and their generalizations from many perspectives as well. Among these perspectives, the setting of chip-firing can be nicely encoded using ideals of polynomial rings, hence be approached using techniques from commutative algebra such as Gr?bner bases. This project will start with the essential basic of chip-firing and polynomial algebra, before defining and studying G-parking ideals and toppling ideals. A goal of this project is to provide an introduction to combinatorial commutative algebra using a concrete (and fun) combinatorial game.
Reference:
- The mathematics of chip-firing, Caroline Klivans https://www.dam.brown.edu/people/cklivans/Chip-Firing.pdf
- Gr?bner Bases in Commutative Algebra, Viviana Ene and Jurgen Herzog. (Mostly Ch. 1 and 2)
- Polynomial Ideals for Sandpiles and their Gr?bner Bases, Robert Cori, Dominique Rossin, and Bruno Salvy.
- Primer for the algebraic geometry of sandpiles, David Perkinson and John Wilmes.