Read This!

The MAA Online book review column

Briefly Noted

May 2006

At this point, we are adding some 30 reviews a month to MAA Reviews. This column represents an idiosyncratic selection of some of the reviews that have appeared there (fairly) recently.

We should persuade the New York Times to translate and syndicate George Szpiro’s columns from the Swiss newspaper Neue Zürcher Zeitung (NZZ) and its Sunday edition NZZ am Sonntag. That way we would not have to wait years to read his musings. He writes about mathematics for the general reader, and he has a knack for that, for giving readers “…an understanding not only of the importance, but also of the beauty and the elegance of the subject.” In The Secret Life of Numbers, the subjects include solved and unsolved problems, mathematical personalities, historical tidbits, and a potpourri of applications.

Szpiro’s background is worth a brief mention. His primary occupation is journalism, though he has done graduate work in mathematics, has an MBA from Stanford and a PhD in finance and mathematical economics from Hebrew University. Consequently, he is a specialist in virtually none of the areas he writes about and this is a great advantage. As a journalistic polymath of sorts, he conveys general concepts very clearly with well-chosen details.

Since most of the pieces in Szpiro’s book are only three or four pages long, he faces the challenge of communicating something meaningful about complex concepts to a general reader in an entertaining way, and doing so in at most a few thousand words. Generally he does so very well. For example, he discusses the Poincaré conjecture and Perelman’s work in three and a half pages and manages to tell the history, describe the problem in a comprehensible fashion and give a general idea of the proposed solution.

There were several pieces that I particularly enjoyed. There is a sly portrait called “God’s Gift to Science” about Stephen Wolfram and A New Kind of Science. (The title of the piece speaks for itself.) There are three articles about Thomas Hales’ work on the Kepler conjecture and honeycomb problem. A selection on the computer scientist Danny Hillis talks about his career as an “Imagineer” for Disney, his work on a 10,000 year clock, and his new company called Applied Minds. I also enjoyed a piece on a Swedish student’s purported solution of Hilbert’s 16th problem.

Generally, the applied pieces in the book’s last section don’t work quite as well as the other articles. There is, however, a good account of Lawrence Sirovich’s information-theoretic analysis of Supreme Court verdicts.

This is an excellent book to dip into, to give to friends or relatives who wonder what you do, or to recommend to students.

[William J. Satzer; posted to MAA Reviews 05/04/2006]

According to the description on the back of the book, Michael J. Panik's Advanced Statistics from an Elementary Point of View "captures the flavor of a course in mathematical statistics without imposing rigor for its own sake." Your reaction to that quotation will pretty well summarize your feelings towards the book, depending on whether or not you appreciate rigor "for its own sake." If all blurb-writers were this honest about their books, then there would be no need for MAA Reviews. This book covers a wide range of topics in statistics — it doesn't even define means or medians until the second chapter, but it ends with chapters about contingency tables and bivariate linear regressions. In between there are full chapters on parametric probability distributions, sampling, Chi-Square distributions, point estimation, and tests of parametric statistical hypotheses.

Panik's goal is to make the book very accessible, and in this goal he succeeds. His exposition is quite clear and I found it quite easy to follow his many examples. There are also a large number of exercises, many of which have solutions given and some of which seem quite interesting. Panik certainly does not impose much rigor, however, and many of the explanations were not as fleshed out or precise as I would have liked. While I imagine that many of my students would appreciate the lack of formal proofs throughout the book, I know that many others — and certainly most mathematicians or statisticians I know — would find this book highly deficient for this very reason. Panik preemptively addresses this issue in his introduction, pointing out that many of the theorems have their proofs developed in the exercises, but this reader was still disappointed.

Panik is an economist, and I imagine that his book would work well for a statistics course for economics majors — or for the similar courses in many of the departments on my campus that focus on actually using statistical techniques rather than on why they are true. But while it did indeed "capture the flavor" of a course I would want to see in a mathematics department, it would need quite a bit of extra meat to be substantial enough for the whole meal that I would want from a textbook.

[Darren Glass; posted to MAA Reviews 04/11/2006]

Algebraic Numbers and Algebraic Functions is an AMS/Chelsea reprint of a book first published in 1967 by Gordon and Breach, based on a series of lectures given at Princeton in 1950–51. The course was a revised version of one offered at New York University in the summer of 1950, the notes for which were published in 1951 by NYU. So what we have here is a record of how Emil Artin presented algebraic number theory and its close cognate, the theory of algebraic function fields, in the early 1950s.

Artin was, of course, a master of the subject. The analogy between number fields and function fields (in one variable) was an important theme in his work, from his thesis onward. Here, he takes a unified approach to both kinds of fields, based on the theory of valuations and of complete valued fields. Artin seems to have taken this approach to the subject already in 1933 (see Artin, "Algebraische Zahlentheorie," Hamburger Beiträge zur Geschichte der Mathematik. Mitt. Math. Ges. Hamburg 21 (2002), 159–223).

The first two parts are strictly local, and cover the theory of extensions of complete valued fields (Part I) and local class field theory (Part II). They stike me as fairly "standard," probably because Artin's approach was so influential that it was adopted almost universally.

The third chapter is more interesting. It introduces the notion of "PF-Fields," i.e., fields in which the product formula is true. These turn out, of course, to be algebraic number fields and algebraic function fields in one variable. Tensor products of fields (here called "Kronecker products") are used to treat the semi-local theory and "valuation vectors" (now called adèles) and idèles are used to move from local to global results. Part III culminates with a treatment of differentials and the Riemann-Roch theorem.

What is obviously missing here is a treatment of global class field theory. This was left to a second set of lecture notes, which became, I think, the well-known Class Field Theory, by Artin and Tate.

The exposition is (as usual with Artin) quite elegant, and the parallel treatment of number fields and function fields can be illuminating as well as efficient. But this remains a set of lecture notes, so the reader shouldn't expect much discussion and contextualization. An example is the chapter on the Riemann-Roch theorem, in which two whole sections of preparation (one of which is entitled "The First Proof") go by before the theorem is even stated. Similarly, while the notion of a differential is defined for all PF-fields, there is little discussion of how this relates to the classical notion in the function field case, nor of what it means in the number field case.

Such problems, however, are only to be expected in a book like this. They are more than compensated for by the insight that will come from working through it, particularly if the reader is prepared to sort out how this approach relates to more classical (or more modern!) ones. Thus, students of number theory and algebraic geometry can learn a lot from this book. It is a true classic in the field.

[Fernando Q. Gouvêa; posted to MAA Reviews 05/06/2006]

This book contains a description of the New Mexico High School Contest, as well as problems in preparation for it (a chapter in Number Theory and Algebra and another in Geometry and Combinatorics). The problems have been proposed by Professor Liong-Shin Hahn, who has been a faculty member of the University of New Mexico’s Department of Mathematics and Statistics responsible for composing the problems for the Contest between 1990 and 1999. His enthusiasm and hard work have raised the New Mexico Mathematics Contest to national prominence. The two chapters with problems are followed by complete and very well explained solutions to all the problems.

The book also contains five appendices including:

The book is not only a homage to Professor Hahn and his wonderful work as director of the New Mexico Mathematics Contest, but a very good source for students interested in high-school mathematics competitions, as well as teachers working with these students.

In short: this is a very good book, very well-written and useful for any high school student and any teacher, from any country. A true gem!

[Mihaela Poplicher; posted to MAA Reviews 05/25/2006]

Publication Data

The Secret Life of Numbers: 50 Easy Pieces on How Mathematicians Work and Think, by George G. Szpiro. Joseph Henry Press, 2006. Hardcover, 199 pages, $27.95. ISBN 0309096588.

Advanced Statistics from an Elementary Point of View, by Michael J. Panik. Elsevier, 2005. Hardcover, 802 pages, $99.95. ISBN 0-12-088494-1.

Algebraic Numbers and Algebraic Functions, by Emil Artin. AMS Chelsea Publishing, 2006. Hardcover, 349 pages, $49.00. ISBN 0821840754.

New Mexico Contest Problem Book, by Liong-Shin Hahn. University of New Mexico Press, 2005. Paperback, 202 pages, $29.95. ISBN 0-8263-3534-9.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Darren Glass (dglass@gettysburg.edu) is an Assistant Professor at Gettysburg College.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME.

Mihaela Poplicher is an assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu .

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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 Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.

MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu).
Last modified: Sun May 28 15:06:42 Eastern Daylight Time 2006