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MAA  Review 

Teichmüller Theory and Applications to Geometry, Topology, and Dynamics  

Volume I: Teichmüller Theory

Following are excerpts from an MAA Review by David Roberts of the University of Minnesota, Morris. Posted with permission of the MAA.
MAA members and site subscribers can  log into the MAA web site to access the entire review.

There is an ambitious new publishing house on the mathematics scene, Matrix Editions, with lead author John H. Hubbard. As a motto, Matrix Editions has chosen "Serious mathematics, written with the reader in mind."  The volume under review is the first volume of a two-volume book.  It beautifully exemplifies the motto.

The serious mathematics in this volume is Teichmüller theory, a theory of Riemann surfaces blending analysis, geometry, topology, and algebra.  The book is aimed at readers who have completed at least a year of graduate school, in conformity with the advanced level of the material.  The writing keeps such readers in mind in many ways, as we'll see.

The book is dedicated in unusually strong language to Fields medalist William Thurston: "Only one dedication is possible for this book. Thanks, Bill, for teaching us all the meaning of geometry." Indeed, Thurston is behind much of the mathematics in the book, as is evident from the forthcoming second volume's title, "Four theorems by William Thurston."  But also some of his philosophical views on mathematics, put forth in a foreword of independent interest, are reflected in the book's communicative writing style.....


"Whole mind" exposition

A theme in Thurston's foreword is that the way mathematics resides in our brains is quite different from the way we typically commit it to paper.  If we all wrote mathematics in a way which actually took into account how humans understand mathematics, then the task of readers trying to make sense of the literature would be much easier. ...
In particular, Thurston is adamant that our understanding of mathematics has an emotional component.  He writes, "In mathematics, what is intriguing, puzzling, interesting, surprising, boring, tedious, exciting is crucial," since these attitudes actually shape our cognitive understanding .  Even more importantly, at least when the subject is something like Teichmüller theory, our understanding has a very large geometric component.  Good mathematical writing should include some direct appeals to our "spatial and visual senses," as well as the usual appeals to the "linguistic, symbol-handling areas" of our brains.

Thurston writes that "John Hubbard approaches mathematics with his whole mind," and indeed he does.  Definitions, theorems, proofs, and remarks are embedded in a coherent narrative.  Geometry is made visual whenever possible.

The extra narration serves to keep the reader oriented.  For example, Hubbard introduces quasiconformal maps on page 111 by explaining that it took him a long time to get used to their paradoxical nature: they are smooth enough for some of calculus to hold but too rough for other parts to hold.  Intuitive preliminaries like this one help readers interpret the rigorous mathematics which follows.

The geometrical support comes at all levels.  At lower levels, we're taught how to canoe and drive cars in the hyperbolic plane (Section 2.3).  We likewise learn that hyperbolic trousers fit us better than Euclidean trousers (Figure 3.5.1).  At higher levels, we're taught to think of Beltrami forms as ellipse fields (Figure 4.8.2) and to literally model a Riemann surface with a quadratic differential using ruled paper and tape (Figures 5.3.3-4).  Many figures, such as the cover figure, capture central notions in ways that humans naturally understand them; the corresponding text makes sense only after one has internalized the picture.

Hubbard also keeps the reader in mind in ways besides the two which figure prominently in Thurston's foreword.  He aims to be self-contained and appeal to as many readers as possible.  Accordingly, there is a two page notation summary, a thirteen page glossary, and a fourteen page index.  The glossary is particularly handy, with seventy-two definition-based entries, starting with "act freely" and ending with "upper semi-continuous."  There is also even a ninety page appendix of background advanced topics which come to the forefront only briefly in the main development.  These background advanced topics are indeed best isolated in an appendix; the topics are very wide-ranging, from Dehn twists, to holomorphic functions on Banach manifolds, to Serre duality.

It should be emphasized that keeping the reader in mind does not at all mean skirting difficult points.  In fact, Hubbard's aim is to present complete preliminaries and complete proofs, even all the way through the end of the second volume.  This self-containment at times becomes quite demanding of the reader.  For example, much of the text concerns arbitrary Riemann surfaces, not just those of finite topological type, and this necessarily complicates the presentation.


In his preface, Hubbard cites books by Ahlors (1966), Abikoff (1980), Nag (1988), Imayoshi & Taniguchi (1992), and Gardiner & Lakic (1999).  He says they are excellent and recommends them highly.  At this point, readers do indeed have many excellent introductions to Teichmüller theory. The volum
e under review is already a highly competitive newcomer to the list.  It will become even more attractive when its sequel volume appears.