Undergraduate Seminar: "Chip-firing games and arithmetical structures on graphs"

Abstract: Given a finite connected graph, one can assign chip totals to each vertex and play a game by transferring chips between vertices along each edge. This chip-firing game (or sandpile game) reveals surprising connections between the purely combinatorial setting and algebraic geometry, and generalizing this game leads to the notion of an arithmetical structure on a graph. In this talk, we explore these constructions their connections to geometry and arithmetic, highlighting open questions of interest. In particular, we will consider the problem of determining the number of arithmetic structures on a given graph, presenting a recent result (joint with Tomer Reiter) on an upper bound for this number.

Host: Adeli Hutton