Read This!

The MAA Online book review column

Briefly Noted

August 2001

Aiming at such readers helps keep the book brief, but for the most part the briefness is obtained by focusing on what is essential and omitting those proofs that are not particularly illuminating, most notably in the section on class field theory. The willingness to make such decisions is part of the charm of the book: the author's opinions and personality are clearly present throughout, such as when Swinnerton-Dyer explains that "the zeta function knows everything about the number field; we just have to prevail on it to tell us." He adds that "some of what it has already told us can be found in Chapter 4."

For many people, algebraic number theory has remained a mysterious part of mathematics, mostly because of the heavy technical pre-requisites and the intricacy of some of the proofs. This little book would be a good place to find out what the subject is all about. An annotated bibliography at the end shows where we need to go to learn more. Overall, this is not an easy book to read, but it is one that will reward the reader's efforts. [Fernando Q. Gouvêa]

V. A. Vassiliev explains that his Introduction to Topology contains the lecture notes for a course he has taught several times at the Independent University of Moscow. It must have been quite a course. In little over 140 pages, the book goes all the way from the definition of a topological space to homology and cohomology theory, Morse theory, Poincaré duality, and more. The book emphasizes intuitive arguments whenever possible, but it also assumes a lot. For example, it assumes that the reader has a good algebra background and is willing to fill in a lot of the detail.

To make this more concrete, here is what is on pages 1 and 2: a short introduction, the definition of a topology, the discrete topology and the topology on a metric space as examples of topologies, the definition of a basis of a topology, an example of an "exotic" topology on R, the definition of closed sets and of the closure of a set, and an exercise: "Consider the basis consisting of parallelepipeds in Rⁿ with edges parallel to the coordinate axes. Is the topology it determines a different one?"

Though this is part of the AMS Student Mathematical Library, it's hard to imagine an undergraduate who can learn topology from this book. On the other hand, I can imagine an advanced undergraduate enjoying the book as a broad survey of the field. It is often useful to have an overall picture of a subject before engaging it in detail. For that, this book would be a good choice. [Fernando Q. Gouvêa]

Many years ago, the New Mathematical Library published Hungarian Problem Book I and Hungarian Problem Book II. These books contained problems from the Eötvös Competition, a national mathematics competition held in Hungary. These two books covered the period from 1894 to 1928.

Now, 38 years later, we get Hungarian Problem Book III, volume 42 in the Anneli Lax New Mathematical Library series, covering the problems from the period 1929-1943. (Since the competition was not held between 1944 and 1946, this is a natural stopping point.) The editor promises that one more volume, bringing the story up to 1963, will be ready soon.

Those who know the earlier Hungarian Problem Books will know that there are lots of good problems here, organized by topic and with detailed discussions. The book is enriched by a preface by Jósef Pelikán explaining the history of the competition and of the publication of these problems. Those who enjoy problems won't want to miss this book. [Fernando Q. Gouvêa]

The MAA recently created the MAA Problem Books series. The first book in the series was Mathematical Olympiads 1998-1999, by Titu Andreescu and Zuming Feng, recently reviewed on MAA Online. Here is the second volume, USA and International Mathematical Olympiads 2000, by the same authors.

The book collects three groups of problems. The first group are the problems from the United States of America Mathematical Olympiad (USAMO). The second are problems used in the process of selecting the US team for the International Mathematical Olympiad (IMO). The third group are the problems from IMO itself.

The book is in three sections. The first contains the problems from each of the three competitions. The second contains hints, and the third has complete solutions (often several solutions for each problem). An appendix gives the results of USAMO and IMO for the year 2000.

For anyone who is involved in preparing students for olympiad competition, this book is clearly indispensable. It'll also be of interest to people who enjoy difficult problems that require only elementary mathematics and lots of ingenuity to solve. [Fernando Q. Gouvêa]

Publication Data

A Brief Guide to Algebraic Number Theory, by H. P. F. Swinnerton-Dyer. Cambridge University Press, 2001. Softcover, 146pp, $24.95, ISBN 0-521-00423-3. Hardcover, 146pp, $69.95, ISBN 0-521-80292-X.

Introduction to Topology, by V. A. Vassiliev. Student Mathematical Library, volume 14. American Mathematical Society, 2001. Softcover, 149pp., $25.00 ($20.00 to AMS members). ISBN 0-8218-2162-8.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of FOCUS and 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.
• August 2000: acrostics, field theory, categories, and partial differential equations.
• September 2000: excursions, history, and some high-level expository writing.
• November 2000: a festival of math videos and the Fourier-analytic proof of quadratic reciprocity.
• December 2000: Analysis problems, mathematical mysteries, and industrial mathematics.
• January 2001: Euler, Kolmogorov, and Honsberger.
• March 2001: history of mathematics, in three flavors: Harriot, prime numbers, and geometrical exactness.
• May 2001: three very different "applied mathematics" books.

Go to... The main Read This! page The index of all reviews The MAA Online front page

Find out... About the Mathematical Association of America Why you should be a member of the MAA How you can help support the activities of the MAA

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 Gouvˆa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.