Final week of classes
- Monday. Levine's theorem.
- Wednesday. Progress report.
- Friday. Threshold density of \(K_n\). Summary of new results from the semester.
Homework due Monday
- None.
Previous classes
Week 1
- Monday. Introduction. Divisor theory \(\leftrightarrow\)
sandpile model. Demos.
Motivating papers for our course:- Riemann-Roch and Abel-Jacobi Theory on a Finite Graph, by Matthew Baker and Serguei Norine.
- Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle, by Lionel Levine.
- Wednesday. Basic vocabulary: divisors, degree, linear equivalence, Picard group, Jacobian group, complete linear system of a divisor.
- Friday. Smith normal form.
Homework due Friday
- Read through section 2.2.
- Turn in problems 1.2, 1.3, and 1.4 from this handout.
Week 2
- Monday. Determinantal divisors. Configurations and the reduced Laplacian.
- Wednesday. Greedy algorithm; \(q\)-reduced divisors.
- Friday. Existence and uniqueness of \(q\)-reduced divisors.
Homework due Friday
- Read Chapter 3.
- Turn in problems: 2.2, 2.4, 2.7, 2.9, 3.1. Challenge: 2.3. (handout)
Week 3
- Monday. Go over HW. Introduction to Sage.
- Wednesday. Superstable configurations. Dhar's algorithm.
- Friday. Acyclic orientations.
Homework due Monday
- Sage homework: see sagehw1 on the SMC.
- Turn in problems: 2.2, 3.2, 3.3, and 3.4 from this handout.
Week 4
- Monday. Proof of bijection between acyclic orientations with unique source and maximal superstables.
- Wednesday. Class project. To prepare, please read section 3.4.1 on parking functions and Problem 3.5, on circular parking functions.
- Friday. Acyclic orientations.
Homework due Monday
- Problem 3.5, Exercise 4.13, Problems 4.1 and 4.2, Exercises 5.2, 5.3, 5.4, and 5.5 from the this handout.
Week 5
- Monday. Example of computing the rank of a divisor on \(K_n\). Sage demonstration showing the geometry underlying the computation of a complete linear system.
- Wednesday. Proof of the Riemann-Roch theorem.
- Friday. Harmonic mappings.
Homework due Monday
- Turn in problems: 5.2, 5.3, 6.1, 6.2, 6.3, and 6.4 from this handout.
Week 6
- Monday. Harmonic mappings. Riemann-Hurwitz.
- Wednesday. Matrix-tree theorem.
- Friday. Matrix-theorem, version 2. (Notes to come.)
Homework due Monday
Week 7
Homework due Monday
- Do problems 14.2, 14.4, 14.5, and 14.7 in this handout. (Replace the word "recurrent" with "superstable" and the words "sandpile group" with "Jacobian group" wherever they occur.)
Week 8
Homework due Monday
- No HW this week.
Week 9
- Monday. Go over last week's HW.
- Wednesday. Alive divisors.
- Friday. Stationary density, threshold density, statement of Levine's theorem.
Homework due Monday
- Turn in a progress report for your project (in \(\TeX\)).
- Calculate the stationary density of \(K_3\).
- Calculate the threshold density of \(D=\vec{0}\) on \(K_3\).
- Calculate the stationary density of \(K_4\).
Week 10
- Monday. Merino's theorem.
- Wednesday. Stationary density and the Tutte polynomial. Stationary density of \(K_n\).
- Friday. Comparing the closed and open sandpile Markov chains. Burst sizes.
Homework due Monday
- Turn in a progress report for your project (in \(\TeX\)).
- Turn in the problems from this handout.
Week 11
- Monday. Levine's theorem.
- Wednesday. Threshold density, \(\zeta_{\tau}(\vec{0},K_4)\). Consequences of Levine's theorem.
- Friday. Project report on symmetric recurrents.
Homework due Monday
- Turn in a progress report for your project (in \(\TeX\)).
- Turn in the problems from this handout.
Week 12
- Monday. Levine's theorem. A Markov renewal theorem.
- Wednesday. Progress report \(K_n\) asymptotics.
- Friday. Progress report on \(P_n + K_m\) and \(C_n + K_m\).
Homework due Monday
- No HW this week.