The Riemann-Roch theorem is fundamental to algebraic geometry.
In 2006, Baker and Norine discovered an analogue of the Riemann-Roch
theorem for graphs. In fact, this theorem is not a mere analogue but has
concrete relations with its algebro-geometric counterpart. Since its
conception this topic has been explored in different directions, two
significant directions are i. Connections to topics in discrete geometry
and commutative algebra ii. As a tool to studying linear series on
algebraic curves. We will provide a glimpse of these developments. Topics
in commutative algebra such as Alexander duality and minimal free
resolutions will make an appearance. This talk is based on my
dissertation and joint work with i. Bernd Sturmfels, ii. Frank-Olaf
Schreyer and John Wilmes and iii. an ongoing work with Alex Fink.