Title: Computing Belyi maps

Abstract: A Belyi map is a finite, branched cover of the complex projective line that is unramified away from 0, 1, and infinity. Belyi maps arise in many areas of mathematics, and their applications are just as numerous. They gained prominence in Grothendieck's program of dessins d'enfants, a topological/combinatorial way to study the absolute Galois group of the rational numbers.

In this talk, we survey computational methods for Belyi maps, and we exhibit a uniform, numerical method that works explicitly with power series expansions of modular forms on finite index subgroups of Fuchsian triangle groups. This is joint work with Jeroen Sijsling and with Michael Klug, Michael Musty, and Sam Schiavone.