Children's Drawings and the Riemann-Hilbert Problem

TL;DR

This paper explores how children's drawings, known as Dessins d'enfants, relate to complex analysis and the Riemann-Hilbert problem, offering a combinatorial approach to classical mathematical challenges.

Contribution

It develops the theory connecting Dessins d'enfants with monodromy, Riemann surfaces, and the Riemann-Hilbert problem, providing new insights into their interrelations.

Findings

Representation of Dessins by permutations

Connection to Belyi's theorem and Riemann surfaces

New perspective on solving the discrete Riemann-Hilbert problem

Abstract

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.