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Riemann-Roch and Abel-Jacobi theory on a finite graph
Student-Run Discrete Math SeminarSpeaker: | Matt Baker, UC Berkeley |
Location: | 3106 MSB |
Start time: | Tue, May 8 2012, 11:00AM |
It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. We pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph. This is joint work with Serguei Norine.