Szego Seminar: "The Dollar Game and the Riemann-Roch Theorem for Graphs"

Abstract: In this talk, we will introduce the dollar game, which motivates the study of discrete Laplacians and the theory of divisors on graphs. The dollar game, also known as chip-firing, is played on a finite, undirected, connected graph and consists of lending and borrowing moves. In a lending move, a selected vertex disperses one chip along each edge to neighboring vertices, and a borrowing move at a vertex takes one chip in from each of its neighboring vertices. A graph's configuration is the number of chips at each vertex of the graph, which gives rise to the definition of a divisor in this setting: a linear combination of the vertices based on the graph's configuration. Various linearly equivalent divisors can be obtained through sequences of lending and borrowing moves at the vertices, which will be discussed in this talk.

The theory of divisors on graphs is parallel to the theory of divisors on Riemann surfaces, and we will introduce a graph-theoretic version of the classical Riemann-Roch Theorem provided by Baker and Norine in 2007. This talk will also provide examples of applications of this discrete version of the Riemann-Roch Theorem. In addition to being useful in combinatorics, chip-firing games have diverse applications in algebraic geometry, linear algebra, and statistical physics. 

Host: Nathan Wagner