Title:
Computing high degree genus-0 Belyi maps with prescribed monodromy
Abstract:
The explicit calculation of three-branch-point covers (also called
Belyi maps) with prescribed monodromy is generally considered to be
quite challenging. Having a tool to deal with this topological
problem would lead to several impactful applications in different
fields of mathematics. In this talk I will establish its connections
to the Inverse Galois problem and describe a recently developed
method by Barth/W. that allows the explicit computation of high
degree genus-0 Belyi maps with prescribed monodromy groups. The main
computational results include the explicit realization of Belyi maps
with almost simple primitive monodromy groups that satisfy the well
known rational rigidity criterion (yielding polynomials with
prescribed Galois groups over Q(t)) and an explicit version of a
theorem of Magaard which lists all sporadic groups occurring as
composition factors of monodromy groups of rational functions.