Combinatorial Methods for Enumerating Maps in Surfaces of Arbitrary Genus

Abstract

The problem of map enumeration is one that has been studied intensely for the past half century. Early work on this subject included the works of Tutte for various types of rooted planar maps and the works of Brown for non-planar maps. Furthermore, the works of Bender, Canfield, and Richmond as well as Bender and Gao give asymptotic results for the enumeration of various types of maps.

This subject also attracted the attention of physicists when they independently discovered that map enumeration can be applied to quantum field theory. The results of 't Hooft established the connection between matrix integration and map enumeration, which allowed algebraic techniques to be used. Other examples of this application can be found in the papers of Itzykson and Zuber.

One result of particular significance is the Harer-Zagier formula, which gives the genus series for maps with one vertex. This result has been proved many times in the literature, a selection of which includes the proofs of Goulden and Nica, Itzykson and Zuber, Jackson, Kerov, Kontsevich, Lass, Penner, and Zagier. An extension of this result to locally orientable maps on one vertex can be found in Goulden and Jackson, while another extension to two vertex maps can be found in Goulden and Slofstra.

In this thesis, we will extend the combinatorial techniques used in the papers of Goulden and Nica and Goulden and Slofstra, so that they can be applied to maps with an arbitrary number of vertices, when the graph being embedded is a tree with loops and multiple edges. This involves defining a new set of combinatorial objects that extends the ones used in Goulden and Slofstra, and develop new techniques for handling these objects. Furthermore, we will simplify some of the techniques and results in the existing literature. Finally, we seek to relate the techniques used in this thesis to techniques in other map enumeration problems, and briefly discuss the potential of applying our techniques to those problems.