Abstracts of Talks
Charles Smart, MIT
Convergence of the Abelian sandpile model
Abstract: I will give a mostly self-contained proof of scaling limit of the least action principle of Fey, Levine, and Peres.
Wesley Pegden, New York University
The Apollonian structure of the sandpile PDE
Abstract: I will sketch the proof of the Apollonian structure of the sandpile PDE, with an eye towards possible generalizations to other lattices.
Katherine Stange, University of Colorado at Boulder
The sensual Apollonian circle packing
Abstract: Conway studied the values of a binary quadratic form on a topograph representing P1(Q). Here, Apollonian circle packings are rediscovered as Hermitian forms on a similar topograph for P1(Q(i)). As a result, one associates a full-rank sublattice of Z2 to any circle in an Apollonian circle packing and obtains a Descartes rule relating the lattices of quadruples of tangent circles. This rule also describes the relation between quadratic forms associated to the circles.
Guglielmo Paoletti, Université Paris Sud
Patterns in deterministic Abelian Sandpile Models
Abstract: The Abelian sandpile model was first introduced in 1987 as an archetype of self organized criticality. Recently there has been a growing interest in the description of pattern formation in some deterministic dynamics (Dhar et al, 2009). I will discuss here the emergence of 2-dimensional periodic patterns (patches) and 1-dimensional periodic defects (strings) in a number of protocols, the relation between these objects will be clarified as as well as the relation between their densities and their periodicity vectors. A full classification of these objects will be given. Finally I will then explain the connection between the structure of these objects and SL(2,Z) showing how this connection leads to the formation of unexpected Sierpinski like structures.
David B. Wilson, Microsoft Research
Local statistics of the abelian sandpile model
Abstract: We show how to compute local statistics of the abelian sandpile model on the square, hexagonal, and triangular lattices. The one-site marginals alone on the square lattice took 20 years to determine. We prove that on the square lattice, all local events are rational polynomials in 1/π, while on the hexagonal and triangular lattices they are rational polynomials in √3/π. The proofs use the burning bijection of Dhar and Majumdar relating sandpiles to spanning trees, and the methods of Kenyon and Wilson for computing grove partition functions.