Hurwitz numbers and integrable hierarchies

Hurwitz numbers enumerate non-isomorphic coverings of a Riemann sphere by a genus g Riemann surface, such that all except one critical values are simple.

On the other hand Hurwitz numbers have purely combinatorial nature, due to the equivalent definition in terms of symmetry group Sn. By a famous theorem of Ekedahl, Lando, Shapiro and Vainshtein, Hurwitz numbers are closely related to the intersection theory on moduli spaces of algebraic curves.

In this talk we will describe a connection between Hurwitz numbers and integrable hierarchies: we will show that the generating series for Hurwitz numbers is an element of a certain family of combinatorial τ-functions of the KP-hierarchy.

Speaker

Dr Boris Bychkov obtained his PhD in 2016 from the Faculty of Mathematics of the Higher School of Economics, Moscow. Since then he has worked at the same Faculty as a Researcher at the International Laboratory of Representation Theory and Mathematical Physics. In 2018 he also became a part-time Researcher at the Centre of Integrable Systems of the P.G. Demidov Yaroslavl State University. His main scientific interests are Hurwitz numbers, moduli spaces of algebraic curves, integrable hierarchies and topological recursion.

This talk will be hosted by Dr Georgi Grahovski.