Hurwitz Numbers: On the Edge Between Combinatorics and Geometry

Hurwitz numbers were introduced by A. Hurwitz in the end of the nineteenth century. They enumerate ramified coverings of two-dimensional surfaces. They also have many other manifestations: as connection coefficients in symmetric groups, as numbers enumerating certain classes of graphs, as Gromov-Witten invariants of complex curves. Hurwitz numbers belong to a tribe of numerical sequences that penetrate the whole body of mathematics, like multinomial coefficients. They are indexed by partitions, or, more generally, by tuples of partitions, which does not allow one to overview all of them simultaneously. Instead, we usually deal with some of their specific subsequences. The Cayley numbers NN-1 enumerating rooted trees on N marked vertices is may be the simplest such instance. The corresponding exponential generating series has been considered by Euler and he gave it the name of Lambert function. Certain series of Hurwitz numbers can be expressed by nice explicit formulas, and the corresponding generating functions provide solutions to integrable hierarchies of mathematical physics. The paper surveys recent progress in understanding Hurwitz numbers.