In 1987, physicists Bak, Tang, and Wiesenfeld created an idealized version of a sandpile in which grains of sand are piled on the vertices of a (combinatorial) graph and are subjected to certain avalanching rules. In the last three decades, there has been a broad effort in the statistical physics community to understand the dynamics of this model, which has a natural underlying algebraic structure. The sandpile monoid and the sandpile group encode the short and long-term dynamics of the system. Disguised under different names, these algebraic structures have been widely studied in diverse contexts including algebraic combinatorics, arithmetic geometry, and algebraic geometry.
In this talk we give an introduction to the sandpile model and the algebraic structures attached to it. We provide a broad overview of the theory and discuss some of the more celebrated results. We end the talk with a discussion about an ongoing project that showcases some of the unresolved fundamental questions accessible to undergraduates.