# Hurwitz numbers of Riemann sphere and integrable hierarchies.

2019/05/23 Thu 10:30 - 12:00

Room 006, RIMS

The Hurwitz numbers count the topologically non-equivalent types
of finite ramified coverings of a given Riemann surface. When the base
Riemann surface is the Riemann sphere, these numbers are known to be
related to intersection numbers of the Hodge classes and the psi classes
on the moduli space of stable complex curves. On the other hand, the
same numbers can be expressed in a genuinely combinatorial form in terms
of symmetric groups. The latter expression reveals that the generating
functions of a particular class of Hurwitz numbers of the Riemann sphere
become tau functions of the KP hierarchy and its relatives. I will review
these facts for non-experts of integrable systems.