# Old and New Problems in Geometry of Algebraic Curves

Given a complex algebraic curve $X$ and ramification data for a branched covering $Y \to X$, how explicitly can we identify $Y$? Or given isomorphic algebraic curves $X_1$ and $X_2$ defined over number fields $k_1$ and $k_2$, when does there exist a model for $X_i$'s defined over $k_1\cap k_2$? Questions of this nature exist in abundance in classical theory of curves. The theory of dessins d'enfants (or simply dessins or children's drawings) provides a tool that leads to some answers to the above questions.

Dessin theory was initiated by Grothendieck and reflects his fascination with Belyi's remarkable theorem characterizing curves defined over a number field as those admitting of a nonconstant meromorphic function with at most three critical values. Grothendieck's principal motive was to use this theory to construct a geometric/combinatorial model space on which the absolute Galois group acts thus providing a new tool for algebraic number theory. The second goal of this lecture is to construct towers of dessins with surjective maps on the corresponding monodromy groups such the absolute Galois group acts as automorphisms of the corresponding monodromies. For certain towers of number fields the corresponding Galois groups are exactly the full automorphism group of the monodromies. This construction gives new geometric combinatorial models for the action of the absolute Galois group. It also leads to new questions and ideas in Teichmüller Theory.