MSRI-UP 2016: Sandpile Groups
The research topic of the 2016 MSRI-UP program is Sandpile Groups,
a topic at the intersection of group theory, combinatorics, linear
algebra and algebraic geometry. The research program will be led by
Prof. Luis Garcia-Puente of Sam Houston State University. Students who have had a linear algebra course and a course in which they have had to write proofs are eligible to apply.
In thermodynamics, a critical point is the end point of a phase equilibrium curve. The most prominent example is the liquid-vapor critical point, the end point of the pressure-temperature curve at which the distinction between liquid and gas can no longer be made. In order to drive this system to its critical point it is necessary to tune certain parameters, namely pressure and temperature. In nature, one can also observe different types of dynamical systems that have a critical point as an attractor. The macroscopic behavior of these systems displays the spatial and/or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to precise values. Such a system is said to display self-organized criticality. This concept is thought to be present in a large variety of physical systems like earthquakes, forest fires and even some fluctuations in the stock market. Self-organized criticality is considered to be one of the mechanisms by which complexity arises in nature and has been extensively studied in the statistical physics literature during the last three decades.
In 1987, in their seminal paper, Bak, Tang and Wiesenfeld conceived a cellular automaton model as a paradigm of self-organized criticality. In this model, the system evolves in discrete time such that at each time step a sand grain is dropped onto a random grid cell of a rectangular grid. When a cell amasses four grains of sand, it becomes unstable. It relaxes by toppling whereby four sand grains leave the site, and each of the four neighboring sites gets one grain. If the unstable cell is on the boundary of the grid then, depending on its actual position, either one or two sand grains fall off the edge and disappear. As the sand percolates over the grid in this fashion, adjacent cells may accumulate four grains of sand and become unstable causing an avalanche. This settling process continues until all cells are stable. Then another cell is picked randomly, the height of the sand on that grid cell is increased by one, and the process is repeated.
In 1990, Dhar generalized the Bak, Tang and Wiesenfeld model replacing the rectangular grid with an arbitrary undirected or directed graph with a global sink. In this model, the sand grains are placed at the vertices of the graph. The toppling threshold depends on the degree (outdegree) of each vertex, and the existence of a global sink ensures that any avalanche terminates after a finite amount of topplings. The long-term behavior of the abelian sandpile model on a graph is encoded by the critical (or recurrent) configurations. These critical configurations have connections to parking functions, to the Tutte polynomial, and to the lattices of flows and cuts of a graph. Among other properties, the critical configurations of the sandpile model have the structure of a finite abelian group. This group has been discovered in several different contexts and received many names: the sandpile group for graphs and digraphs, the critical group, the group of bicycles, the group of components, and the jacobian of the graph. The sandpile group will be the main object of study during the 2016 MSRI-UP program.
Learn more about sandpile groups from Dr. Garcia-Puente in this Numberphile video filmed during MSRI-UP 2016: