at the

American Institute of Mathematics, San Jose, California

organized by

Lionel Levine, Jeremy Martin, David Perkinson, and James Propp

- Chip-firing in higher dimensions.
Building on the work of Duval, Klivans, and Martin, we would like to develop a theory of chip-firing for general simplicial or CW-complexes. Is there a generalization of the Baker-Norine theorem to higher dimensions---perhaps a "combinatorial Hirzebruch-Riemann-Roch theorem"? Are there appropriate generalizations of the recurrent elements of the abelian sandpile model? What are the implications of a higher-dimensional theory for combinatorics?

- Abelian networks.
Abelian networks, proposed by Dhar and developed by Bond and Levine, are systems of communicating finite automata satisfying a certain local commutativity condition. As a model of computation, they implement asynchronous algorithms on graphs. The two most widely studied examples of abelian networks are the abelian sandpile model and the rotor-router or Eulerian walkers model. How much more general are abelian networks than these? Is there a computational hierarchy within the class of abelian networks? Is the halting problem for abelian networks decidable in polynomial time?

- Pattern formation.
How can one rigorously identify and classify the rich patterns that arise in identity elements of critical groups? Can the proof of existence of the sandpile scaling limit by Pegden and Smart be adapted to prove properties of the limit? Ostojic has given a heuristic, involving the conformal map $z\mapsto 1/z^2$, for the locations and features of certain sandpile patterns. Can these heuristics be converted into precise conjectures, and what tools would be required to prove these conjectures?

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Fourientations and the Tutte Polynomial

by Spencer Backman and Sam Hopkins, *Res. Math. Sci. 4 (2017), 4:18 * MR3696165

Chip-firing and energy minimization on M-matrices

by Johnny Guzmán and Caroline Klivans, *J. Combin. Theory Ser. A 132 (2015), 14-31 * MR3311336

Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle

by Lionel Levine, *Comm. Math. Phys. 335 (2015), no. 2, 1003-1017 * MR3316648

Abelian networks: foundations and examples

by Benjamin Bond and Lionel Levine, *SIAM J. Discrete Math. 30 (2016), no. 2, 856-874 * MR3493110

Rotor-routing and spanning trees on planar graphs

by Melody Chan, Thomas Church and Joshua A. Grochow, *Int. Math. Res. Not. IMRN 2015, no. 11, 3225-3244 * MR3373049

Critical Groups of Graphs with Dihedral Actions

by Darren Glass and Criel Merino, *European J. Combin. 39 (2014), 95-112 * MR3168517

The Bernardi process and torsor structures on spanning trees

by Matthew Baker and Yao Wang

Sandpiles, spanning trees, and plane duality

by Melody Chan, Darren Glass, Matthew Macauley, David Perkinson, Caryn Werner and Qiaoyu Yang, *SIAM J. Discrete Math. 29 (2015), no. 1, 461-471 * MR3319526

G-parking functions and tree inversions

by David Perkinson, Qiaoyu Yang and Kuai Yu, *Combinatorica 37 (2017), no. 2, 269–282 * MR3638345