Read This!
The MAA Online book review column
Briefly Noted
Short Reviews by Fernando Q. Gouvêa
October 1999
Recently, a
friend was complaining to me that all the good books go out of print. That,
unfortunately, is often the case. Fortunately, however, every so often
something really special is reissued, and then it's a time to
celebrate. That's how I feel about Graph Theory 1736-1936, a
sourcebook that collects important articles about the subject over the 200
year period mentioned in the title. In the Preface, the authors explain
that Euler's famous 1736 article about the bridges of Königsberg
provided their starting point, and they chose to end their book with
material from 1936 because that was the date of publication of the first
full-length book about the subject (by Dénes König). In all,
there are thirty-seven extracts from original sources, some of which were
especially translated for this volume. There's a full bibliography,
biographical notes on the authors, and even pictures of some of the
mathematicians whose papers are included. If you're at all interested in
the subject, this one is a must-have.
Norman Macrae's
book on the life (and, to some extent, the work) of John von Neumann
was originally published by Pantheon Books in 1992. It has now been
reprinted by the AMS. John von Neumann was enormously influential on the
mathematics of the first half of the twentieth century. His work included
crucial contributions to set theory, quantum mechanics, and the theory of
operator algebras. He created game theory as a mathematical tool for
understanding economic behavior, and was one of the pioneers in the
creation of the computer. Towards the end of his life, he got involved in
the issue of nuclear deterrence, and ended up as a member of the Atomic
Energy Commission and a consultant to the government labs at Los Alamos and
Lawrence Livermore. His positions about nuclear war and international
politics were controversial at the time, and remain a source of discomfort
for many admirers of his mathematics. (Macrae, a journalist who at one time
was editor of The Economist, seems to share many of von Neumann's
political views.) It is good to have a biography of one of the most
important mathematicians of the twentieth century, even if it is a
biography that focuses much more on the man than on the mathematics.
Non-specialists may, I think, be forgiven for feeling confused by the title
of J. Madore's An Introduction to Noncommutative Differential Geometry
and its Physical Applications. It's not too easy to see in what sense
the differential geometry we know and love is "commutative" and even harder
to imagine what a "noncommutative" geometry might look like. The first
words of the introduction help us out with the first question. They point
out that if V is a set of points then the set of complex-valued functions
on V is a (finite-dimensional) commutative (and associative) algebra. If V
is a compact space, then we can restrict to continuous complex-valued
functions on V and we get an algebra C0(V) which is in fact a
"C*-algebra," and if V is a smooth manifold we can look at
smooth functions, and so on. It turns out that much of classical
differential geometry can be expressed in terms of such algebras, and the
idea of "noncommutative geometry" is to generalize this version of
differential geometry to the case of noncommutative algebras. Amazingly,
this turns out to yield a theory that is not only interesting
mathematically but also useful in understanding the mathematics of quantum
field theory. This book, volume 257 in the traditional "London Mathematical
Society Lecture Note Series", is intended as an accessible introduction to
the subject for non-specialists. It looks to me that the author has done a
good job of opening the way to understanding a difficult theory. This is
the second edition of a book first published in 1995, and the very fact
that a new edition has appeared so soon is an indication that the book has
been successful. Not for the faint of heart, but worth a look.
Publication Data
N. L. Biggs, E. K. Lloyd, and R. J. Wilson,
Graph Theory 1736-1936. Oxford University Press, 1999. Softcover,
239pp, $45.00. ISBN 0-19-853916-9.
Norman Macrae,
John von Neumann. American Mathematical Society, 1999. Hardcover,
406pp, $35.00. ISBN 0-8218-2064-8.
J. Madore,
An Introduction to Noncommutative Differential Geometry and its Physical
Applications, Second Edition (London Mathematical Society Lecture Note
Series, volume 257). Cambridge University Press, 1999. Softcover, 321pp,
$39.95. ISBM 0-521-65991-4.
Back Issues:
- April 1999: Reprints of "classics",
books on applied mathematics.
- May 1999: Books on math and physics,
both popular and technical.
- July 1999: Various kinds of geometry,
and classics in applied mathematics.
- August 1999: What's happening,
Dirichlet, and Cryptonomicon.
Read This! is the MAA
Online book review column. Contributions are welcome; contact the
editor if you'd like to be one of our reviewers. Books for review should be
sent to the editor: Fernando Gouvêa, Dept. of Math&CS, Colby
College, Waterville, ME 04901. Publishers, please check our
reviews information page.
Copyright ©1999 The Mathematical Association of America
MAA Online is edited by Fernando Q. Gouvêa
(fqgouvea@colby.edu).
Last modified: Tue Oct 12 09:46:56 -0500 1999