Random walks, true trees and equilateral triangulations

I will start by reviewing the definition and basic properties of harmonic measure on planar domains, i.e., the first hitting distribution of a Brownian motion on the boundary of a domain. For a tree embedded in the plane, can both sides of every edge have equal harmonic measure? If so, we call this the “true form of the tree” or a “true tree” for short. These are related to Grothendieck's dessins d'enfants and I will explain why every planar tree has a true form, and what these trees can look like. The proofs involve quasiconformal maps and will only be sketched. I will also discuss the application of these ideas to Belyi functions and building Riemann surfaces by gluing together equilateral triangles. If time (and the audience) permits, I will briefly describe a generalization of these ideas from finite trees and polynomials to infinite trees and entire functions.