# True trees

## Chris Bishop, Stony Brook University

## March 29, 2016

A finite planar tree has many topologically equivalent drawings in the
plane; is there a most natural way to draw it? One possible choice is
called the "true form" of the tree. It arises from algebraic geometry
and is closely related to Grothendieck's dessins d'enfants. I will
describe the true form of a tree (a true tree) in different terms,
using harmonic measure, Brownian motion, and conformal maps, and then
prove that every planar tree has a true form by using the measurable
Riemann mapping theorem. I will then discuss the possible shapes of
true trees, e.g., can any compact connected set can be approximated by
true trees? If time permits, I will mention the analogous problem for
infinite planar trees and some applications to holomorphic dynamics.

This talk is more analysis than discrete geometry, but I will try to
explain or avoid the technicalities.