Stanislav Smirnov: Discrete complex analysis

It is well-known that discrete harmonic functions can be defined on any graph, e.g. by requiring the mean value property: the function at a vertex is the mean of the values at its neighbors. Such discrete functions share many properties of their continuous counterparts and have been very extensively studied. The theory of discrete analytic functions also has a long history, but is less developed. One starts with a planar graph, e.g. the square lattice, and then asks a function to satisfy some discretization of the Cauchy-Riemann equations. There are many possible discretizations, and some of them have deep connections to integrable systems. It turns out that much of the usual complex analysis can be transferred to the discrete setting, albeit with some difficulties. Sometimes discrete theories can be applied to the continuous case and even lead to easier proofs of several results, including the Riemann uniformization theorem. There are also exciting connections to combinatorics, probability, geometry and even computer science.

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