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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
twovolume 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,
symbolhandling 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 5.3.34). 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 selfcontained 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 seventytwo definitionbased entries, starting with "act
freely" and ending with "upper semicontinuous." 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 wideranging, 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 selfcontainment 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.
Conclusion
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 volume under
review is already a
highly
competitive newcomer to the list. It will become even more
attractive
when its sequel volume appears.

