Read This!

The MAA Online book review column

Briefly Noted

August 2000

The Carus Mathematical Monographs are one of the MAA's oldest and most respected series of books. The goal of the series is to "make accessible at a nominal cost a series of expository presentations of the best thoughts and keenest researches in pure and applied mathematics." Beginning with G. A. Bliss on the Calculus of Variations and continuing until today, the Carus series has been a reliable starting point for students seeking delve a little more deeply into some area of mathematics. Throughout these many years, the series has also had a consistent look. These small-sized hardcovers bound in black cloth have been a constant in many mathematics libraries. (The color of the binding has actually varied in recent years, but I must admit it's the black ones that have stuck in my memory.) So it's worthy of note that the MAA has decided to republish some of the books in the series in paperback format, starting off with Charles Hadlock's Field Theory and Its Classical Problems. Hadlock's book uses the famous Greek construction problems and the problem of solution by radicals as a motivation to develop the basics of Galois theory. This is a very nice book, and it comes to us in spiffy new clothes. If you don't have a copy yet, here's your chance.

Category Theory seems to have fallen somewhat out of fashion after its heyday some years ago, but it's still useful and important in many contexts. On such context is K-theory, and this is the reason for Berrick and Keating's Categories and Modules With K-theory in View. Their book, which seems to be part of a series leading up to a book on K-theory itself, introduces category theory in general but keeps constantly in view the example of the category of modules over a ring. This helps make the book more accessible than many other books on this subject. It also means that the book does more than what is sometimes called "abstract nonsense." It includes, for example, a chapter on localization and one on local-global methods. This seems quite a nice book, and makes me want to take another look at their earlier book, Introduction to Rings and Modules.

OK, true confession first: I know nothing about the subject treated in Introduction to Maximum Principles and Symmetry in Elliptic Problems. But how could I resist? The introduction, speaking of the course at the University of Bath that led to this book, says, after describing a pamphlet that specified the goal of the course:

Naturally, the pamphlet did not state how this goal was to be reached in twenty lectures to students who could not be assumed to have any experience whatever of partial differential equations. Nor were detailed suggestions issued to me when, in the autumn of 1988, I joined the University of Bath and was ordered to give these lectures...

The author goes on to say that "the word Introduction in the title of the book is no gloss." The book tries to make good on the promise to lead the student quickly into the theory with minimal prerequisites by including a lot of background material in five appendices. The writing is lively and has a touch of humor. While it's certainly not easy, it strives to be friendly and to help the reader along. All in all, this seems a worthwhile introduction to some interesting material.

Publication Data

Categories and Modules: With K-theory in View, by A. J. Berrick and M. E. Keating. Cambridge University Press, 2000. Hardcover, 361pp., $54.95. ISBN 0-521-63276-5.

Introduction to Maximum Principles and Symmetry in Elliptic Problems, by L. E. Fraenkel. Cambridge University Press, 1999. Hardcover, 340pp., $74.95. ISBN 0-521-46195-2.

Fernando Gouvêa (fqgouvea@colby.edu) is the editor of MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

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.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.

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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.