Empire Maps on Surfaces

Abstract

This report is an introduction to mathematical map colouring and the problems posed by Heawood in his paper of 1890. There will be a brief discussion of the Map Colour Theorem; then we will move towards investigating empire maps in the plane and the recent contributions by Wessel. Finally we will conclude with a discussion of all known results for empire maps on higher genus surfaces and prove Heawoods Empire Conjecture in a previously unknown case.