The Mathematics of Chip-Firing

Caroline J. Klivans received a BA degree in mathematics from Cornell University and a PhD in applied mathematics from MIT. Currently, she is an Associate Professor in the Division of Applied Mathematics at Brown University. She is also an Associate Director of ICERM (Institute for Computational and Experimental Research in Mathematics). Before coming to Brown she held positions at MSRI, Cornell and the University of Chicago. Her research is in algebraic, geometric and topological combinatorics.

Introduction

A brief introduction. Origins/History.

Chip-firing on Finite Graphs

The chip-firing process. Confluence. Stabilization. Toppling time. Stabilization with a sink. Long-term stable configurations. The sandpile Markov chain.

Spanning Trees

Spanning trees. Statistics on Trees. Merino’s Theorem. Cori-Le Borgne bijection. Acyclic orientations. Parking functions. Dominoes. Avalanche polynomials.

Sandpile Groups

Toppling dynamics. Group of chip-firing equivalence. Identity. Combinatorial invariance. Sandpile groups and invariant factors. Discriminant groups. Sandpile torsors.

Pattern Formation

Compelling visualizations. Infinite graphs. The one-dimensional grid. Labeled chip-firing. Two and more dimensional grids. Other lattices. The identity element.

Avalanche Finite Systems

M-matrices. Chip-firing on M-matrices. Stability. Burning. Directed graphs. Cartan matrices as M-matrices. M-pairings.

Higher Dimensions

An illustrative example. Cell complexes. Combinatorial Laplacians. Chip-firing in higher dimensions. The sandpile group. Higher-dimensional trees. Sandpile groups. Cuts and flows. Stability.

Divisors

Divisors on curves. The Picard group and Abel-Jacobi theory. Riemann-Roch Theorems. Torelli’s Theorem. The Pic^g (G) torus. Metric graphs and tropical geometry. Arithmetic geometry. Arithmetical graphs. Riemann-Roch for lattices. Two variable zeta-functions. Enumerating arithmetical structures.

Ideals

Ideals. Toppling ideals. Tree ideals. Resolutions. Critical ideals. Riemann-Roch for monomial ideals.