This publication, part of the CBMS Regional Conference Series, is presented as a companion to a lecture series by Scott Wolpert given at a July 2009 conference taking place at Central Connecticut State University. We learn from the Preface that since “[t]he study of Riemann surfaces continues to be an interface for algebra, analysis, geometry, and topology … [Wolpert intends] to suggest the [according] interaction to the audience and [the] reader.” Wolpert’s goal is not to be exhaustive in his coverage, but, nonetheless, to present “a generally self-contained course for graduate students and postgraduates … [with topics running] across research areas.” Accordingly, he takes pains to present a good deal of guidance as regards the indicated research literature and, indeed, each chapter ends with a useful recommendation of further readings.
Before getting to the structure proper of the book, here is a sample of what kind of mathematics we are dealing with. Already on p. 7 we encounter the uniformization theorem: “A Riemann surface homeomorphic to the [Riemann] sphere [i.e. the complex projective line] is conformally equivalent to the sphere. A Riemann surface, not homeomorphic to the sphere, is equivalent either to a quotient C/Γ or H/Γ for a discrete torsion-free group Γ. The cases are distinguished by the Euler characteristic of the surface. Furthermore, the coverings are universal.” Does it get any more beautiful than that? Not being an insider in the area, I did a bit of internet snooping and found out that this uniformization theorem goes back to Paul Koebe and Henri Poincaré, with later proofs given by (presumably Émile) Borel and Lipman Bers: a wonderful genealogy.
The appearance of C/Γ and H/Γ in the game makes, of course, for connections to elliptic modular functions and hyperbolic structure. Indeed, as a number theorist my first exposure to Riemann surfaces in any non-trivial way came about in connection with the action of Γ = (P)SL(2,Z) on the complex upper half plane H, yielding in the form of the accompanying fundamental domain H/PSL(2,Z) a Riemann surface compactifiable by means of adjoining the point at infinity; to boot, the measure of this quotient space is y–2dxdy: we’re in hyperbolic space. Indeed, we’re also clearly at the aforementioned interface of algebra, analysis, geometry, and topology.
Well, the, what about the book under review? What does Wolpert cover? After a first chapter devoted to “Preliminaries,” he hits some very sexy stuff indeed with Ch 2 devoted to Teichmüller space. Ch 3 introduces some symplectic geometry, and Ch 4, 5, and 6 deal with “Geometry of the Augmented Tiechmüller space” in three parts. Then things begin to get geometric in the sense of Thurston, so to speak: Ch 7 is titled, “Deformations of hyperbolic metrics and the curvature tensor, and then the last three chapters deal with, respectively, “Collar expansions and exponential-distance sums, “Train tracks and the Mirzakhani volume recursion”, and the “Mirzakhani prime simple geodesic theorem.” Pretty heavy stuff.
Being the write-up of a set of lectures pitched to an audience of, at worst, fellow travelers, the material in Families of Riemann Surfaces and Weil-Petersson Geometry is quite sophisticated: Wolpert does not have the option to waste time, and he does not do so. I do find it entertaining that neither André Weil nor Hans Petersson is mentioned in the book. It just goes to show how far the subject has evolved, I suppose. And the reader should take account of that fact. The book under review is manifestly aimed at would-be specialists. And it looks to be very successful in meeting its goals.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.