| Math 412:
Topics in Algebra |
Fall 2008
David Perkinson
L316, ext. 7417
Office
hours
Course Description
This is a course on the abelian sandpile model of Dhar. Besides the basic
theory, it will cover some of the following topics:
- matrix tree theorem
- tilings
- corresponding ideals of points
- graph-theoretic Riemann-Roch
- complexity (constructing a Turing machine from sandpiles)
- sandpile algorithms
- self-organized criticality
- rotor-routers
Prequisite: Mathematics 212 and 331.
Sage Sandpiles:
A package for doing sandpile calculations in Sage.
Version 1.3, February 22, 2009. New: algebraic geometry methods.
Sandpile applet
Class summary
- Week 1
- Monday. Labor Day.
- Wednesday. Sand configurations. The graph Laplacian.
- Friday. Reduced Laplacian. Uniqueness of stabilization.
- Week 2
- Monday. Definition of the sandpile group. Characterization of
recurrent configurations.
- Wednesday. Each equivalence class of the cokernel of the transpose
of the reduced Laplacian is represented by a unique recurrent configuration.
- Friday. Smith normal form.
- Week 3
- Monday. Matrix-tree theorem.
- Wednesday. Burning configurations on a directed graph I.
- Friday. Burning configurations on a directed graph II.
- Week 4
- Monday. Burning configurations on a directed graph III.
- Wednesday. Bijection between spanning trees and recurrent
configurations.
- Friday Trees and matchings (Kenyon, Propp, Wilson).
- Week 5
- Monday. Formula for the number of domino tilings of an MxN
grid.
- Wednesday. Tiling formula, continued.
- Friday Tiling formula, finished.
- Week 6
- Monday. Presentations.
- Wednesday. Pachter's paper: Combinatorial Approaches and
Conjectures for 2-Divisibility Problems Concerning Domino Tilings of
Polyonimoes.
- Friday. The order of the all 2s element on a 2xn grid.
- Week 7
- Monday. Review of the all 2s argument from Friday. An algorithm
for computing the order of a recurrent element. The beginnings of equivariant
sandpile groups.
- Wednesday. The order of the all 2s element on a 2nx2n grid.
- Friday. Examples of equivariant configurations.
- Week 8
- Monday. Gröbner bases.
- Wednesday. Gröbner bases.
- Friday. Gröbner bases. Lattice ideals.
- Week 9
- Monday. Class presentations.
- Wednesday. Class presentations. The toppling ideal.
- Friday. Gröbner bases for toppling ideals for undirected
graphs.
- Week 10
- Monday. Gröbner bases for toppling ideals of undirected
graphs.
- Wednesday. Gröbner bases for toppling ideals of directed
graphs. Computing subtraction with sandpiles.
- Friday. Character theory.
- Week 11
- Monday. Orbits of representations of abelian groups I.
- Wednesday. Orbits of representations of abelian groups II.
- Friday. Jacob's new theorem.
- Week 12
- Monday. Orbits in projective space.
- Wednesday. Harmonic mappings, Riemann-Roch, Java presentation.
- Friday. Thanksgiving holiday.
- Week 13
- Monday. Divisors and the kernel of the Laplacian for directed
multigraphs.
- Wednesday Riemann-Roch for undirected graphs.
- Friday. The Tutte polynomial and the generating function for
recurrent configurations.
- Week 14
- Monday. Surjectivity of the discrete Abel-Jacobi map.
- Wednesday The sandpile group and duals of planar graphs.
- Friday. No class.
