Complete Regular Dessins

Graduate Student of the City University of New York (CUNY) Graduate Center and former Math Club President presents a special lecture on his work on Complete Regular Dessins! Below is the abstract for his talk. Consider a regular tetrahedron. As a topological surface, it is just a sphere. We can view it as a sphere with an embedded graph, namely the complete graph on four vertices. Moreover, the regular tetrahedron possesses a large number of symmetries. In a certain sense, the regular tetrahedron is "as symmetric as possible". All of this is abstracted by the notion of a complete regular map, which is an embedding of a complete graph into a surface that is "as symmetric as possible". In 1985, Lynne D James and Gareth A Jones, building on the 1971 work of Norman Biggs, classified all complete regular maps. In this talk, I will view complete regular maps from an algebro-geometric viewpoint. Namely, I will introduce the concept of a complete regular dessin and produce some explicit examples using the Weierstrass p function, ideas from Galois theory, and number theory.