Georgia Institute of Technology College of Sciences

The talk will begin with an elementary geometric discussion of Riemann-Roch theory for sub-lattices of the integer lattice orthogonal to some positive vector. A pair of necessary and sufficient conditions for such a lattice to have the Riemann-Roch property will be presented. By studying a certain chip firing game on a directed graph related to the lattice spanned by the rows of its Laplacian I will describe a combinatorial method for checking whether a directed graph has the Riemann-Roch property. The talk will conclude with a presentation of arithmetical graphs, which after the application of a simple transformation, may be viewed as a special class of directed graphs. Examples from this class demonstrate that either, both or neither of the Riemann-Roch conditions may be satisfied for a directed graph. This is joint work with Arash Asadi.