ABSTRACT
ABSTRACT: 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. 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, with stress made on their combinatorial rather than geometric nature.
