Dessins d'enfants theory, initiated by Alexander Grothendieck in
the 1970's, establishes an equivalence between the category of certain
graphs on topological surfaces and some arithmetic-geometric category
(of Belyi pairs, i.e. (X, beta)'s, where X is a curve and beta: X ---> P^1
is a covering with 3 branch points). It turned out (Mumford-Penner-Kontsevich-...) that the decorated moduli spaces of curves
M_{g,N} (C) x R_{>0}^N admit the orbifold cell decomposition in which the
cells are parametrized by certain dessins d'enfants. In 1992 Kontsevich
applied this construction to the proof of the famous Witten conjecture; the discrete version of this theory was suggested in [2].

This discrete version relates Belyi pairs with the Deligne-Mumford
stack \overline{M_{g,N}} over Spec Z and over finite fields; the corresponding
counting problems were discussed in [1] and [3]. The talk will be devoted to the overview of the corresponding constructions, results and open problems; some examples will be presented.

References

[1] N. Amburg, E. Kreines and G. Shabat, Poincare polynomial for the moduli space \overline{M_{0,n}}(C) and the number of points in \overline{M_{0,n}}(F_q). arXiv: 1811.120976v1 [math.AG] 29 Nov 2018.
[2] P. Norbury, Counting lattice points in the moduli space of curves. Math. Res.
Lett. 17, 467-481 (2010).
[3] G.Shabat, Counting Belyi pairs over finite fields. In 2016 MATRIX Annals (eds. David R. Wood, Jan de Gier, Cheryl E. Praeger, Terence Tao). MATRIX Book Series, Volume 1, pp. 305-322, 2018.