Abstract: Hurwitz numbers are a classical part of the theory of Riemann surfaces, or algebraic curves. They count the number of maps between compact curves with specified discrete invariants. In the last thirty years, they have been intimately connected with natural inftinite-dimensional integrable systems and recursive methods, as well as with deep algebro-geometric invariants like Gromov-Witten theory of curves. This ways, they are computable and provide insight into technically more complicated counts that are important in algebraic geometry and string theory.
I will give a short introduction into these numbers, from an algebraic geometry perspective, and discuss recent developments regarding a spin extension, and hopefully also leakiness.