# Sandpile Groups (15w5119)

## Organizers

Luis Garcia Puente (Sam Houston State University)

Dino Lorenzini (University of Georgia)

Criel Merino (National Autonomous University of Mexico)

David Perkinson (Reed College)

Carlos Enrique Valencia Oleta (National Polytechnic Institute)

## Objectives

The study of the sandpile group has received much attention recently. At the

2011 Joint Mathematics Meeting, there was an MAA invited address and paper

session on the sandpile model (under the rubric "Laplacian growth"). In July

2013, the American Institute of Mathematics held a week-long workshop on the

sandpile model, "Generalizations of chip-firing and the critical group". The

sandpile group also appeared on the cover article [10] of the September

2010 issue of the *Notices* of the American Mathematics Society. Our main

objective is to build on this momentum and bring together researchers from a

wide variety of mathematical communities studying the sandpile model. The

workshop will focus on four main questions, related to the sandpile group

topics described in the previous section.

First, note that graphs and Riemann surfaces are the one-dimensional cases of

simplicial complexes and smooth algebraic varieties, respectively. The natural

question that the workshop will address is then: how can the chip-firing game be

extended to higher dimensions? Of special interest is the question of whether

there is a generalization of the Riemann-Roch theorem for graphs to simplicial

complexes in higher dimensions. The workshop will also explore natural generalizations of the Riemann-Roch theorem to abelian sandpile models, in the spirit of the recent works [1] and [14].

Second, generalizing the theory of sandpiles from graphs to matroids suggests

itself naturally as an avenue for attacking Stanley's h-vector conjecture.

Thus, the workshop will investigate the possibility of generalizing the sandpile

model to some families of matroids besides graphic matroids.

Third, the two most widely studied examples of abelian networks are the abelian

sandpile model and the rotor-router or Eulerian walkers model [18].

These are examples of unary networks, i.e., of abelian networks in which the

underlying finite automata each have alphabet of cardinality 1. How much more

general are abelian networks than unary networks? In general, we would like to

explore the complexity hierarchy of the various known examples of abelian

networks.

Fourth, we would like to identify the scaling limit for patterns in sandpile

aggregation for various lattices.

We envision that interactions among diverse scientists will help define and

shape these problems more precisely. We hope that the synergies of the

workshop participants, working in different areas of mathematics and with

different techniques and tools, will lead to cross-pollination and advances in

the understanding of the sandpile model.

## Bibliography

[1] O. Amini and M. Manjunath, Riemann–Roch for sub-lattices of the root lattice An. Electronic Journal of Combinatorics 17, no. 1 (2010): Research Paper 124, 50 pp.

[2] P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. (4), 59:381–384, 1987.

[3] B. Bond and L. Levine, Abelian networks: Foundations and Examples. Preprint, 2013.

[4] D. Dhar, Self-organized critical state of sandpile automaton models, Phys. Rev. lett., 1990.

[5] Björner and L. Lovász and P. W. Shor, Chip-firing games on graphs. European J. Combin., 12(4):283–291, 1991.

[6] B. Benson, D. Chakrabarty, and P. Tetali, G-parking functions, acyclic orientations and spanning trees. Discrete Math., 310(8):1340–1353, 2010.

[7] C. Merino, The chip-firing game. Discrete Math., 302(1-3):188–210, 2005.

[8] R. Bacher, P. de la Harpe, and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph. Bull. Soc. Math. France, 125(2):167–198, 1997.

[9] A. E. Holroyd, L. Levine, K. Mészáros, Y. Peres, J. Propp, and D. B. Wilson, Chip-firing and rotorrouting on directed graphs. In and out of equilibrium. 2, volume 60 of Progr. Probab., pages 331–364. Birkha¨user, Basel, 2008.

[10] L. Levine and J. Propp, What is . . . a sandpile? Notices Amer. Math. Soc. 57 (2010), no. 8, 976—979.

[11] N. L. Biggs, Chip-firing and the critical group of a graph. J. Algebraic Combin., 9(1):25–45, 1999.

[12] K. A. Berman, Bicycles and spanning trees, SIAM J. Alg. Meth. 7(1):1–12, 1986.

[13] D. Lorenzini, Arithmetical graphs, Math. Ann 285:481–501, 1989.

[14] D. Lorenzini, Two-variable zeta-functions on graphs and Riemann-Roch theorems. Int. Math. Res. Not. IMRN, 22:5100–5131, 2012.

[15] M. Baker and S. Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math., 215(2):766–788, 2007.

[16] D. Dhar, Theoretical studies of self-organized criticality. Phys. A, 369(1):29–70, 2006.

[17] C. Merino, The chip firing game and matroid complexes. In Discrete models: combinatorics, computation, and geometry (Paris, 2001), Discrete Math. Theor. Comput. Sci. Proc., AA, pages 245–255 (electronic). Maison Inform. Math. Discr`et. (MIMD), Paris, 2001.

[18] V. B. Priezzhev, D. Dhar, A. Dhar, and S. Krishnamurthy, Eulerian walkers as a model of self-organised criticality. Phys. Rev. Lett., 77:5079–5082, 1996.

[19] W. Pegden and C. Smart, Convergence of the abelian sandpile. Duke Math J. 162: 627–642, 2013.