目次

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.