Read This! The MAA Online book review column What's Happening in the Mathematical Sciences, Volume 5, by Barry Cipra Reviewed by David P. Roberts

The American Mathematical Society suspected it had found a winner back in the early 1990's. It was willing to title a collection of ten mathematics articles by Barry Cipra not just What's Happening in the Mathematical Sciences, but rather What's Happening in the Mathematical Sciences, Volume 1.

A winner indeed it was, and we are now at Volume 5. Cipra must have begun on some sort of probation, since his name did not appear on the cover of Volumes 1 and 2. But since Volume 3, he's been acknowledged on the cover as the force behind this very successful series. Only once did the editors insert an article by another author—a certain Henri Poincaré. One can say without overstatement that the standards in these volumes are very high indeed.

What makes these volumes so well-received? Here are five reasons. Budding expositors of mathematics, take note!

First, the articles are all written in a very lively style. Cipra's personal style involves plenty of word play. In Volume 5, two of the titles are "Nothing to Sphere but Sphere Itself" and "Ising on the Cake." Another title is "A Celestial Pas de trois," and the metaphor of periodic solutions to the three body problem as dances is effectively pursued well into the article. Throughout all the articles, the witty and sophisticated writing adds an extra level of interest. All articles come with photos, figures, and sidebars. Readers can enter at least somewhat into each article effortlessly, as if they were reading a non-technical magazine.

Second, the articles give a balanced treatment of what is indeed happening in the mathematical sciences. Cipra functions as an investigative reporter and he faithfully covers his beat. As in the other volumes, Cipra treats topics across the pure-applied spectrum. In Volume 5, the applied end is represented by articles on protein folding, traffic jams, and the shape of the universe. Cross-disciplinary material also attracts Cipra's attention. In Volume 5, one of the articles is about a novel interpretation of a 4000-year-old Babylonian clay tablet and its relation to the "Pythagorean" theorem.

Third, the primary goal of reaching readers besides professional mathematicians is kept in sight throughout. Topics which fit into the story but are too technical for the readership are appropriately finessed. The article "Think and Grow Rich" on the Clay Mathematics Institute's seven prize problems devotes some five paragraphs to each problem. But what to do about the Hodge conjecture, so removed as it is from the general reader's experience? Having just introduced the concepts of manifold and higher dimensions in the discussion of the Poincaré conjecture, Cipra writes "The Hodge conjecture concerns the analysis of high-dimensional manifolds defined by systems of algebraic equations. It says, very roughly, that everything you always wanted to know about algebraically defined manifolds (but were afraid to ask) is to be found in the theory of calculus." These two sentences are a good start towards capturing the general nature of the Hodge conjecture. Cipra then goes into more detail, making these sentences clearer in a way appropriate to his readership.

Fourth, the articles are short and sharply focused. The whole book is less than a hundred pages long! The first article traces the grand epic which starts with Taniyama's conjecture in 1955 that every elliptic curve is modular, goes through the proof of Fermat's last theorem, and continues to this day in the framework of the Langlands program. How to present all this mathematics and history to a wide audience in ten large-margin photo-packed pages? Cipra keeps to a narrow path: nothing about Taniyama's tragic death, nothing about the controversial assignment of credit for the now-proved conjecture. As a reward, Cipra gets to conclude by communicating in an understandable way some stunning recent work on the Langlands program, some of the deepest current happenings in the mathematical sciences.

Fifth, there is mathematical meat in every article. The exposition is informal throughout, but professional mathematician readers will sometimes suddenly even get the feeling, "you know, I think I might be able to piece together the exact statement of that theorem." For example, readers are given a very good feel for a new theorem in which the inverse-square law for interaction in "small world networks" is strikingly distinguished from all other power laws.

As an indication that the above praise is genuine, rather than derived from some sort of Minnesota solidarity, let me offer a negative comment as well. I myself would be happier if each of Cipra's articles came with a short bibliography. It might detract somewhat from the "light" feel of the series, but it would allow readers who have been attracted to a certain topic to more easily pursue their newly kindled interest. A volume of What's Happening is similar in some ways to an issue of Scientific American, and articles in Scientific American have bibliographies.

Each of the five volumes of What's Happening has exactly ten articles. Together the fifty articles are remarkable not only for their excellence but also for the consistency of style maintained over a ten-year period. The mathematical community should thank Barry Cipra and congratulate him for a job very well done. I hope to see another five volumes so that our present mathematical generation can be viewed by future historians as enjoyably accessible via the Cipra Decameron!

Publication Data: What's Happening in the Mathematical Sciences, Volume 5 by Barry Cipra. American Mathematical Society, 2002. Softcover, 95pp, $19.00, ISBN 0821829041.

Volume 4 was reviewed in the August 1999 briefly noted column.

David Roberts is an assistant professor of mathematics at the University of Minnesota, Morris.

Posted January 6, 2003

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