Graphs on Surfaces and Their Applications

The nature of this fascinating book is well captured by the story of its origin. Several Russian mathematicians were talking about the topics they had chosen to use with the high school students in their Math Circles. The idea in these is always to consider some fairly elementary and accessible mathematics, but to choose topics that can easily lead into much deeper waters, usually connected to the leader’s own research. The group of mathematicians leading the math circles in question had a wide range of interests: some were combinatorialists, others did mathematical physics, algebraic number theory, or singularity theory. To everyone’s surprise, it turned out that all of the circles were working on similar topics, centered on the study of graphs drawn on surfaces.

Two of the mathematicians present at that conversation are the authors of this book. Fascinated by the fact that the same objects were of interest in so many different areas of mathematics, they set out to understand why. “Luckily,” they say, they underestimated “the width and the depth of the sea we intended to cross, otherwise we would not have dared to start.” It turned out that graphs on surfaces were everywhere and led to very deep mathematics. This book, which is part of the Springer Encyclopaedia of Mathematical Sciences series, is the result of their explorations.

The first chapter of the book lays the foundation, discussing “maps” (i.e., embedded graphs) on the sphere and relating them to “constellations” (n-tuples of permutations that specify an embedded graph up to isomorphism) and to ramified coverings of the sphere. A whole range of fanciful names are introduced: maps, constellations, passports, cacti, etc. The ideas are mostly quite elementary and accessible, and one can see how they might work well in a math circle for (bright and interested) high school students.

Each of the following chapters takes those ideas and develops them in the context of different areas of mathematics. Grothendieck’s theory of “dessins d’enfants” is related to algebraic geometry and number theory. The matrix integrals method comes from theoretical physics and has to do with probability distributions on spaces of matrices. Moduli spaces of complex curves and the relationship between meromorphic functions and embedded graphs are part of geometric function theory; chord diagrams and bialgebras are related to knot theory and its generalizations. As one might expect, none of these chapters offers a complete treatment of their subject, but they are surprisingly readable and often quite interesting. Though many details are omitted, readers who have a good grasp of the material in the first chapter and who are willing to suspend disbelief as necessary can probably get quite a bit out of them.

Don Zagier contributes a delightful appendix on the representation theory of finite groups, in which all of the elementary theory is developed in less than ten pages. It left me dreaming of a seminar for advanced undergraduates in which one would hand out those pages and then spend a semester unpacking them. I wonder if it would work…

The book is charmingly written. The authors often address the reader directly:

Possibly you have never studied group representation theory, and don’t know what an irreducible character is. Never mind: the goal of the examples given below is to show that you may get some useful information even without a profound knowledge of the subject.

Or, when they launch into a discussion of mathematical physics:

We had a chance to give the explanations presented below at a number of seminars in the presence of physicists. Usually our statements did not produce any serious protests. Hoping that the politeness of our colleagues-physicists was not the unique reason for that…

As that sentence also shows, a few Slavicisms have made it into the text, but they are never annoying. Every so often there is a hint of truly strange things:

An even more complicated class of models considers the surfaces not as abstract entities but as geometric objects embedded (or immersed) in a certain space. See in this respect [18], where embeddings of surfaces into a space of dimension –2 are considered! (Indeed, the level of abstraction in modern physics is often far beyond that of mathematics.)

(Yes, that is spaces of dimension negative two.)

The overall effect is to draw the reader in and to open up a wide vista of little-known mathematics that is full of surprising ideas, unexpected connections, and just-plain-wonderful theorems. This is a tour worth taking.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He lives in a space whose dimension is definitely positive.