Discrete Riemann surfaces

Abstract

We survey the theory of discrete Riemann surfaces. The theory takes place on a cellular decomposition of a surface, together with its Poincaré dual, equipped with a discrete conformal structure. Many theorems of the continuous theory follow through to the discrete case; we will define the discrete analogs of period matrices, Riemann’s bilinear relations, exponential of constant argument and series. We present the notion of criticality and its relationship with integrability.