Children's Drawings and the Riemann-Hilbert Problem

Dessin d'enfants (French for children's drawings) serve as a unique standpoint of studying classical complex analysis under the lens of combinatorial constructs. A thorough development of the background of this theory is developed with an emphasis on the relationship of monodromy to Dessins, which serve as a pathway to the Riemann Hilbert problem. This paper investigates representations of Dessins by permutations, the connection of Dessins to a particular class of Riemann surfaces established by Belyi's theorem and how these combinatorial objects provide another perspective of solving the discrete Riemann-Hilbert problem.