The Brownian Map

Abstract

won the 2021 Maryam Mirzakhani New Frontiers Prize for her work on random surfaces and the mathematics of quantum gravity in 2 dimensions. Here I will just explain one concept involved in her work: the 'Brownian map'. This is a fundamental object in mathematics, in some sense a 2-dimensional analogue of Brownian motion.Suppose you randomly choose a triangulation of the 2-sphere with n vertices. This is defined in a purely topological way, but the set of vertices becomes a metric space if we give each edge length 1. We thus obtain a "random compact metric space": a probability measure on the space of all compact metric spaces. It turns out that if you increase n while rescaling the edge lengths to make them equal n −1/4 , this probability measure converges as n → ∞. The limit is a random compact metric space called the "Brownian map".To make all this precise takes a bit of work, and we turn to that next. But first, here is a rough image to keep in mind, created by Thomas Budzinski: