**Abstract:**

Hurwitz numbers are weighted counts of covers of a given Riemann surface with given ramification. Alternatively, they are counts of decompositions of the identity in the symmetric group with given cycle types. These numbers have relations to moduli spaces of curves, dessins d’enfants, and integrable hierarchies, among others. They also have a very rich combinatorial structure, which can be used to compute them recursively. In particular, many kinds satisfy topological recursion, a ubiquitous way to calculate invariants related to curves.

I will explain several of these relations, focusing on the interplay between geometry of Riemann surfaces and combinatorial aspects to give structural properties of Hurwitz numbers.

**Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.**

**Join Zoom Meeting: https://washington.zoom.us/j/ 91547335974Meeting ID: 915 4733 5974**