Riemann-Roch theory via partial graph orientations

Graph Theory Seminar
Thursday, March 6, 2014 - 12:05
1 hour (actually 50 minutes)
Skiles 005
Math, GT
This talk is a sequel to the speaker's previous lecture given in the January 31st Combinatorics Seminar, but attendance at the first talk is not assumed. We begin by carefully reviewing our generalized cycle-cocyle reversal system for partial graph orientations. A self contained description of Baker and Norin's Riemann-Roch formula for graphs is given using their original chip-firing language. We then explain how to reinterpret and reprove this theorem using partial graph orientations. In passing, the Baker-Norin rank of a partial orientation is shown to be one less than the minimum number of directed paths which need to be reversed in the generalized cycle-cocycle reversal system to produce an acyclic partial orientation. We conclude with an overview of how these results extend to the continuous setting of metric graphs (abstract tropical curves).