An Introduction to Sandpile Groups

Presenter: Fabian Ramirez

Presenter Status: Undergraduate student

Academic Year: 20-21

Semester: Fall

Faculty Mentor: Natalie Hobson

Department: Mathematics

Funding Source/Sponsor: McNair

Screenshot URL: https://drive.google.com/uc?id=1SJdLXI2xPBTcx3WsCAs8PawgDKud1ICt

Abstract:
Given a directed connected graph G, one may place "grains of sand" on the vertices of the graph. If the number of grains of sand on a vertex exceeds the number of edges leaving that vertex, we say that the vertex is "unstable" and it "topples" sand to adjacent vertices. Following these rules, one may construct a free abelian group, called the sandpile group, associated with this game. As an example of a self-organizing criticality dynamical system, the sandpile group has significant application in mathematics, statistical physics, and theoretical biology. The goal of this poster is to give a general introduction to sandpile groups as well as describe its connection to other areas of mathematics such as fractal geometry and abstract algebra.