Combinatorics of Hurwitz numbers

Abstract

The goal of this dissertation, is to count branched covering of P1. The formal count was rst instigated by A. Hurwitz in his landmark paper of 1981. Hence the numbers associated to the count of branched covering are called Hurwitz numbers. The general idea is to count the number of holomorphic functions to the complex projective line P1 by xing some geometrical conditions to guarantee the nite count. It is shown in this thesis that this number is always nite and using combinatorial and representation theoretic techniques we provide some examples.