Seminário dos Alunos - Dessins d'enfant (i.e. graphs on surfaces), and their counting

Dessin d’enfant (child’s drawing, map) is a graph embedded into a two-dimensional surface. The term was introduced by A. Grothendieck in 1984 in his famous "Esquisse d’un Programme". The theory of dessins d’enfants in a beautiful way connects 19th century complex analysis with 20th century arithmetic geometry, and very simple and intuitive objects with very abstract concepts. Besides overview of the topic I will present some results in one specific area, which is enumeration of graphs of a certain type.