Dessins d'enfants and modular curves associated to the sporadic group Co3 and Janko 2

Dessins d'enfants and their realization as Belyi maps of compact Riemann surfaces were originally discovered by Felix Klein. Their importance and relevance was finally understood by Alexander Grothendieck who rediscovered and named them in his "Esquisse d'un programme" in 1984. The most important aspect of dessins is the operation of the absolute Galois group on them. Accordingly, dessins d'enfants provide fascinating insights and fundamental links between different fields of mathematics like inverse Galois theory, Teichmüller spaces, hypermaps, algebraic number theory and mathematical physics. The related problem of the construction of Riemann surfaces with given automorphism group turns out to be rather challenging. In my talk I will discuss methods for constructing Belyi maps explicitly.