Year 3 project – The mathematics of chip-firing

This project explores chip-firing on graphs, a deceptively simple game whose theory reveals deep connections with various areas of mathematics including combinatorics, algebra, geometry, and mathematical physics.

Background and motivation

The fundamental idea of chip-firing is simple: given a graph where each vertex holds a certain number of “chips,” a vertex can “fire” by distributing one chip to each of its neighbours. If you want to see this in practice here you can play a videogame inspired by chip-firing.

Despite its elementary rules, chip-firing gives rise to complex behaviours, making it a fascinating subject in discrete mathematics. One of the most famous incarnations of chip-firing is the abelian sandpile model, which originated in theoretical physics as a model for self-organized criticality. This model leads to remarkable limit shapes with fractal-like structure and has been widely studied. The study of recurrent configurations in sandpile models reveals deep connections with algebraic structures such as the Jacobian group of a graph, an abelian group whose order is given by the number of spanning trees of the graph, and the Laplacian matrix of the graph, which governs the dynamics of chip-firing.

Group project

In the group project, we will learn the algebraic foundations of the theory, starting from the first three chapters of the book by Corry and Perkinson [CP]. The books has some nice exercises, and further questions might arise along the way. By the end of the group project, we will have learned:

• The relationship between divisors and chip firing
• The definition of the Picard and Jacobian groups
• The definition of the Laplacian matrix and how to use it to compute Picard and Jacobian groups
• Algorithms to determine if a chip-firing game can be won and – if that is the case – a strategy for winning.

Depending on the interests of the group, we might explore some extra topics in combinatorics (such as chapter 4 of [CP]) or algebra (such as chapter 5 of [CP]).

Mode of operation & evidence of learning (group project)

You will be asked to present a section of the book each, and engage with each other’s presentation of the book’s material at our regular meetings. The work produced through solving problems, exploring examples, and investigating theoretical applications will form the content of your group presentation. Your work will be assessed on clear communication of the material in both written and oral formats.

Individual project

The individual project will build on the knowledge you have gained in the group project and will explore additional advanced topics. In algebraic and tropical geometry, chip-firing appears through the lens of divisors on graphs, which serve as a discrete analogue of divisors on algebraic curves. The Riemann-Roch theorem for graphs, developed by Baker and Norine, mirrors its classical counterpart in algebraic geometry but in a purely combinatorial setting. This perspective has led to applications in Brill-Noether theory, moduli spaces of curves, and tropical geometry, where chip-firing plays a role in understanding metric graphs and degeneration techniques in algebraic geometry.

A few examples of topics you will be able to choose from are: * Studying sandpile dynamics and recurrent configurations on different graph structures. * Investigating combinatorial games related to chip-firing and analyzing winning strategies. * Exploring the algebraic structure of the Jacobian group and its connections to spanning trees. * Examining the Riemann-Roch theorem for graphs and its implications in tropical geometry.

Mode of operation & evidence of learning (individual project)

The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Prerequisites and Co-requisites

It is essential that students have taken Algebra II and Complex Analysis II. It is suggested (but not essential) that students take one of Geometry III or Decision Theory III (depending on the students interest) during the course of the project.

Additional information

The first part of the project is at the intersection of algebra and combinatorics, but potential topics for the individual project span a larger set of mathematical knowledge (including dynamical systems and mathematical physics). As such, I have not indicated a research area for it. You are welcome to contact me for more information.

References

• [CP] - Divisors and Sandpiles: An Introduction to Chip-Firing by Scott Corry and David Perkinson Link
• [LP] - What is… a sandpile? by Lionel Levine and James Propp Link
• [K] “The mathematics of chip-firing” by Caroline Klivans Durham University Library link