HARMONIC MEASURE: ALGORITHMS AND APPLICATIONS

Abstract

This is a brief survey of results related to planar harmonic measure, roughly from Makarov’s results of the 1980’s to recent applications involving 4-manifolds, dessins d’enfants and transcendental dynamics. It is non-chronological and rather selective, but I hope that it still illustrates various areas in analysis, topology and algebra that are influenced by harmonic measure, the computational questions that arise, the many open problems that remain, and how these questions bridge the gaps between pure/applied and discrete/continuous mathematics.