Topology Seminar: Dessins d'enfants and the generalized Hurwitz numbers

A multi-graph, embedded into a closed connected oriented surface, is called  a 'dessin d'enfant' if its complement is homeomorphic to a disjoint union of cells. The (appropriately defined) category of dessins d'enfants turned out to be equivalent to the category of 'Belyi pairs' that belongs to arithmetic geometry; a first part of the talk will be devoted to the foundations of this theory.

Then the certain problems of enumeration of dessins (and their generalizations) will be introduced. The particular cases of these problems, e.g., recursions for the so-called 'Hurwitz numbers', were studied intensively during the last decades; the corresponding recent results will be mentioned. However, the general case is currently out of reach, and during the second part of the talk a certain project, based on the above category equivalence, will be presented. Hopefully, realization of this project will promote the understanding of general case.