Jared Ongaro: Hurwitz spaces and Hurwitz numbers

Hurwitz spaces are coarse moduli spaces of branched coverings of a complex projective line. They are 'concrete' spaces but deeply related to more abstract moduli space of curves. On the other hand, Hurwitz numbers are geometric objects which enumerate a given branched coverings of a complex projective line by genus g complex curve. The question of computing Hurwitz numbers  is called the Hurwitz enumeration problem.

The Hurwitz enumeration problem is an old but generally unsolved, very active  research problem thanks to its connections to several modern areas of mathematics and physics. For instance, Hurwiz numbers arise natural in  combinatorics, as a counting tool in factorization of permutation into transpositions.

In this talk, I will give a toy account of Hurwitz spaces and Hurwitz numbers in a now a vast area of algebraic geometry.