Аннотация:
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.
