Riemann functions, their weights, and modeling Riemann-Roch formulas as Euler characteristics

In this thesis, we discuss modelling with (virtual) k-diagrams a class of functions from ℤⁿ to ℤ, which we call Riemann functions, that generalize the graph Riemann-Roch rank functions of Baker and Norine. The graph Riemann-Roch theorem has seen significant activity in recent years, however, as of yet, there is not a satisfactory homological interpretation of this theorem. This may be viewed as disappointing given the rich homological theory contained in the classical Riemann-Roch theorem. For a graph Riemann-Roch function, we will be able to express the associated graph Riemann-Roch formula as a (virtual) Euler characteristic of the modelling (virtual) k-diagram. From the development of this approach, we obtain a new formula for computing the graph Riemann-Roch rank of the complete graph Kn.Science, Faculty ofMathematics, Department ofGraduat