Dept. of Mathematics, Yeungnam University, Kyeongsan, Korea.

A map is a 2-cell embedding of a graph into a closed surface and a regular map is a highly symmetric map like five Platonic solids. A map is not merely a topological object. It is also a sequence of permutations, which provides a relation to group theory, and a ramified covering of the Riemann sphere, which gives a relation to Riemann surface. Furthermore, it can be realized by a complex algebraic curve called Belyi function.

In this talk, classifications of regular maps are considered from several aspect. And, some connections between maps and permutation group, Riemann surface and Belyi functions and their applications are given.

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