# Mathematics Colloquia and Seminars

### A real-based theory for the Riemann-Roch problem on the dimensions of linear systems on edge-weighted graphs

Special Events

 Speaker: Rick Miranda, Colorado State University Location: 2112 MSB Start time: Thu, Mar 15 2012, 4:10PM

Let G be a connected graph with positive real weights attached to each edge.

We define a divisor on G to be an element of the real vector space with basis the vertices of G.

Following Baker and Norine, we define linear equivalence, and the notion of a linear system of divisors.

We give a definition of the 'dimension' of the linear system (which is a non-negative real number) and we show that a Riemann-Roch theorem is true for this theory.

The theory is valid for divisors defined with values in any subring of the real numbers, in particular the integers also. As a corollary we obtain the Baker-Norine results.

A generalization to combinatorial structures that do not arise from graphs becomes available as well.

This is joint work with Rodney James.