AIMS Cameroon: Abelian Sandpile Model

Monday (pdf, 2x2pdf). The dollar game. Divisors, linear equivalence, linear systems.
Tuesday (pdf, 2x2pdf). Review. Firings scripts. The discrete Laplacian.
Wednesday (pdf, 2x2pdf). The discrete Laplacian and vertex firing. The abelian property.
Thursday (pdf, 2x2pdf). Review. The greedy algorithm. Relation of the complete linear system to polyhedral geometry.
Friday (pdf, 2x2pdf). Implementing the greedy algorithm in Sage. Reduced divisors. Handout: greedy.pdf .

HW 1.
Template for greedy algorithm implementation: greed_template.sage. See greedy.pdf for pseudocode.

Monday (pdf, 2x2pdf). Superstable configurations. Dhar's algorithm. Handouts: qreduced.pdf, dhar.pdf, Monday_Sage.pdf.
Tuesday (pdf, 2x2pdf). Dhar's algorithm and acyclic orientations.
Wednesday (pdf, 2x2pdf). Maximal unwinnable divisors. The rank of a divisor.
Thursday (pdf, 2x2pdf). Riemann-Roch Theorem.
- Proof from class.
- The original paper: Riemann-Roch and Abel-Jacobi theory on a finite graph, by Baker and Norine.
- Source for the proof presented in class: The Riemann-Roch theorem for graphs and the rank in complete graphs, by Cori and LeBorgne.
- A version of Riemann-Roch solely in terms of orientations: Riemann-Roch Theory for Graph Orientations, by Backman.
Friday Smith normal form. (pdf, 2x2pdf).
- Smith normal form wiki page.

HW 2.
Template for Dhar's algorithm implementation: dhar.pdf. See dhar_template.sage for pseudocode.

Monday (pdf, 2x2pdf). Structure of the Picard group.
Tuesday. (pdf, 2x2pdf). Matrix-tree theorem. Recurrent elements.
- Proof of matrix-tree theorem.
- Sage recurrents experiment.
Wednesday. (pdf, 2x2pdf). The sandpile group.
Thursday. (pdf, 2x2pdf). Tutte polynomial.
Friday. (pdf, 2x2pdf). Self-organized criticality. Sandpiles on the computer.
- Sandpiles in sage.
- When does a graph have 5 as an invariant factor?