The article, “Gromov-Witten Theory, Hurwitz Theory, and completed cycles,” by Okounkov and Pandharipande (*Ann. Math.*, 163 (2006), pp. 517–560), contains the following revealing sentences:

… Gromov-Witten theory and Hurwitz theory are both enumerative theories of maps …

The Hurwitz theory of a non-singular curve \(X\) concerns the enumeration of covers of \(X\) with prescribed ramification,

and

While any particular Hurwitz number may be calculated, very few explicit formulas are available.

This is very appealing, of course, and suggests precisely what the authors of the book under review alert the reader to at the outset, namely, that “Hurwitz theory is a beautiful algebro-geometric theory that studies maps of Riemann surfaces.”

They go on to say, however, that

Despite being (relatively) unsophisticated, it is typically unapproachable at the undergraduate level because it ties together several branches of mathematics that are commonly treated separately.

This is a suggestive observation, given that the subject’s origin and early development belong to a time and place when mathematics was regarded as much more of a holistic affair than it is today. (At least recently, but, as the Dylan song has it, the times they are a-changing: Gromov and Witten are certainly prime examples of the perestroika that over the last decades has brought Riemannian geometry, low-dimensional topology, and quantum field theory together in a common cause.) Regarding Hurwitz, then, and his work, I think the definitive historical source is Constance Reid’s wonderful biography, *Hilbert*. Indeed, my first encounter with Adolf Hurwitz took place in the pages of this fine account, not just of the life of its title character but of German academic and scientific life in the late 19th and early 20th centuries (Hurwitz’s dates: 1859–1919).

As Reid describes so evocatively, Hurwitz was a mentor to none other than mathematics’ Damon and Pythias, Hilbert and Minkowski, and it is noteworthy that Hurwitz’s work should possess telling similarities with Hilbert’s seminal formulation of class field theory in his Zahlbericht, what with the centrality of the notion of ramification, and that Minkowski should be so closely associated with the geometry of numbers. Hurwitz was an effective mentor.

Thus, the pedigree of the subject is impeccable and the subject is beautiful. It is a marvelous thing, then, to encounter a treatment that, the earlier *caveat* notwithstanding,

intends to present Hurwitz theory to an undergraduate audience, paying special attention to the connections between algebra, geometry and complex analysis that it brings about.

So, let’s get down to at least a modicum of nuts and bolts: say Cavalieri and Miles,

Hurwitz theory is the enumerative study of analytic functions between Riemann surfaces — complex compact manifolds of dimension one. A Hurwitz function counts the number of such functions when the appropriate set of discrete invariants is fixed.

Cutting to the chase (on p.56):

Let \(f:X\longrightarrow Y\) be a non-constant, degree \(d\), holomorphic map of compact Riemann surfaces. Denote by \(g_{X}\) (respectively \(g_{Y}\)) the genus of \(X\) (respectively \(Y\)). \ Then \[ 2g_{X}-2=d(2g_{Y}-2)+\sum_{x\in X} \nu_{X}, \] where \(\nu_{X}\) … is the differential length of \(f\) at \(x\).

Note the suggestive form of two of the terms in the game here: surely we’re going to play around with the Euler characteristic. And so it proves: in the proof, the authors state,

Given that we are comparing Euler chatacteristics of \(X\) and \(Y\), a natural strategy is to choose a suitable “good” graph on \(Y\) and “lift” it to a good graph on \(X\) which we use to compute \(\chi (X)\).

Well, this fragment is revealing on a number of counts. First, we get a glimpse of what the whole business is about. Second, we see that the geometry of Riemann surfaces meets some algebraic topology. Third, the whole business does indeed smell like algebraic geometry. (In any case it’s a very good smell.) And, fourth, it reveals a little bit about the authors’ pedagogical style.

To wit, the book is indeed well-suited to advanced undergraduates who know some serious algebra, analysis (complex analysis in particular), and are disposed to hit themes in algebraic topology and (to a limited degree) algebraic geometry. It would make a good text for a senior seminar.

The lay-out is as follows. First there’s an introduction to Riemann surfaces and manifolds, and, immediately thereafter, the focus is placed on compact Riemann surfaces and maps between them. After that it’s topology: “loops and lifts.” Then we hit some beefy stuff, namely themes in representation theory (including monodromy representations). Then it’s the pay-off: representations of certain symmetric groups, specifics on Hurwitz numbers, and stuff on the Hurwitz potential. Finally there are four sizeable appendices, written by guest authors, including considerations of what happens in positive characteristic, tropical Hurwitz theory, and even material on Hurwitz theory and string theory. Cool.

It’s a well-written book, and even though its objective is quite ambitious, it succeeds: the seniors will have to work hard, but they’ll learn something beautiful.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.