Read This!

The MAA Online book review column

Briefly Noted

January 2005

The book on The Legacy of Niels Henrik Abel is a result of the conference, but it is not your typical proceedings volume. Contributions to the book were solicited before the conference, and there is only partial overlap between the book's content and the talks given at the conference.

Several of the articles in the book are historical and/or biographical. Some, such as Arild Stubhaug's short account of the life of Abel, are mostly on the historical end of the spectrum. Others, such as Steven Kleiman's "What is Abel's Theorem Anyway?" are a blend of mathematics and history, inspecting four theorems that are often called "Abel's Theorem", looking at what Abel actually did, and looking at what came after. Several papers are straight mathematics connected, closely or distantly, to Abel's work. The result is a massive book with a lot of interesting material in it. About half of this material should be accessible to a wide audience; the other half will be of interest mostly to specialists.

With the book comes a CD-ROM containing still more material related to Abel and to the conference. In particular, one can find information on the Centennial Conference held in 1902 (including talks by Sylow and Picard), many pictures, reproductions of stamps, statues, and coins featuring Abel. There is information about the Abel Prize and about honorary doctorates granted at the celebration. There is even an ad for a special edition of Abel's collected works.

It's a delightful package, particularly for those of us who are interested in the history of mathematics and/or in number theory, algebraic geometry, and complex function theory. Just the images on the CD would make me want to have a copy. Given the size of this book and the extras on the CD, this isn't even very expensive; no library should be without a copy.[Fernando Q. Gouvêa]

Representation Theory of Finite Reductive Groups is the first book in a new series from Cambridge University Press called New Mathematical Monographs. The series is dedicated to publishing "books containing an in-depth discussion of a substantial area of mathematics." The statement of purpose also makes a very serious promise: "As well as being detailed, [the books in this series] will be readable and contain the motivational material necessary for those entering a field." The latter task is not at all easy to pull off in a monograph at a high level. One hopes that Cambridge will be able to stick to that promise.

In this first book, Marc Cabanes and Michel Enguehard introduce us to the study of the representations of a particular class of finite groups. These groups, which can be described as the groups you get by taking the points over a finite field of a reductive algebraic groups, include (in a sense) most of the finite simple groups. The approach is very high-powered: by page 55 we are reading about derived categories and duality functors. The book comes with appendices recalling the theories of derived categories, varieties and schemes, and étale cohomology.

The style is, as one might expect, quite dense. The first sentence of Chapter I, for example, is "The main functors in representation theory of finite groups are the restriction to subgroups and its adjoint, called induction." That sends, I think, the correct signal about the authors' approach: we're doing serious work with serious pre-requisites. Whether they keep the promise of including "motivational material" depends on the reader's background. This is not the place to start learning about group representations or even about reductive groups, I think. On the other hand, the book does try to lead the reader into the material, and in particular it tries to show why the heavy theoretical apparatus is necessary and helpful. For experts, then, but not a bad start to the series. [Fernando Q. Gouvêa]

If you still haven't heard of the Curriculum Foundations project, then you haven't been paying attention. Organized by Calculus Reform And the First Two Years (CRAFTY), a subcommittee of the MAA's Committee on the Undergraduate Mathematics Program, the project asked professors in other disciplines to tell us what mathematics they would like us to be teaching their students.

The project organized a series of workshops. The idea was that each workshop would be "a dialogue between representatives of the discipline under consideration, with mathematicians present merely to listen to the discussions and to provide information on current curriculum trends in mathematics." About 30 people were invited to each of these workshops, and each workshop produced a written report. These reports are collected in Curriculum Foundations Project: Voices of the Partner Disciplines. The disciplines covered are biology, business and management, chemistry, computer science, chemical engineering, civil engineering, electrical engineering, mechanical engineering, health-related biosciences, interdisciplinary core mathematics, mathematics, physics, statistics, teacher preparation, and four different kinds of technical mathematics (biological/environmental, electronics/communications, information technology, mechanical and manufacturing). An introduction by Susan Ganter and William Barker surveys the overall picture.

There is no question that this is an interesting resource that should be available to mathematics departments and pondered seriously when making curricular decisions. Common themes do emerge, including the repeated appeal for us to stress understanding rather than doing; they are ably summarized in the introductory chapter. But it is the quirks of particular disciplines that interest me more. This may not be the lightest reading material, but it should be read and pondered by anyone who teaches undergraduates. [Fernando Q. Gouvêa]

Most high school students think that all of mathematics is building up towards the final goal of calculus, and after that crescendo is reached all of mathematics is just doing calculus over and over again. I'm sure we have all had a student who has asked us what research in mathematics is and then expressed surprise when we tell them it is not just computing harder and harder integrals. These students would have their ideas (or fears?) validated if they picked up the new book Irresistible Integrals, by George Boros and Victor H. Moll. In this book, the authors write about many areas of mathematics which come up during the evaluation of integrals.

The authors claim — and for the most part correctly — to assume only the topics covered in a calculus course, and start by discussing examples arising from binomial coefficients and partial fractions. They move through topics such as Stirling's formula, Bernoulli numbers, Euler's constant, and the Riemann Zeta function, along with other ideas which arise in trigonometric functions, exponential functions, and logarithmic integration. One of the more interesting chapters discusses the integral of the function e^-x2, which the authors call the "normal integral". There is also a nice appendix describing the Wilf-Zeilberger method, which can be used to evaluate finite sums. It is not surprising that evaluating difficult integrals often involves difficult, and interesting, mathematics, and the authors succeed in presenting the material not just as a series of tricks but as an interesting and cohesive area of mathematics.

However, while I found many of the topics in the book to be quite interesting, I found it overly cumbersome to read. The authors have no qualms about using Mathematica extensively in the text of the book, and, while in the introduction they claim that the reliance on it will be minimal, I found that even to a casual user like myself it was often difficult to follow what they were doing. More importantly, the text jumped between exposition and examples and exercises and experiments (of the Mathematica sort) rapidly and in such a way that I often found myself getting confused about what the authors had shown and what they wanted the readers to show and where they were going. The book demands extremely active reading, preferably with a computer at your side. This is not necessarily a bad thing in a book, but a casual reader is not likely to devote the required energy to the topic, at least not without significantly more motivation by the authors. And that is my main complaint with the book — the integrals may be irresistible to the authors, but they did not succeed in conveying that level of excitement to me. [Darren Glass]

Publication Data

Representation Theory of Finite Reductive Groups, by Marc Cabanes and Michel Enguehard. New Mathematical Monographs 1. Cambrdige University Press, 2004. Hardcover, xviii+436 pp., $100.00. ISBN 0-521-82517-2

Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, by George Boros and Victor H. Moll. Cambridge University Press, 2004. Paperback, 320 pp., $29.99. ISBN 0-521-79636-9.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College, editor of FOCUS and FOCUS Online, and co-author of Math through the Ages.

Darren Glass is a VIGRE Assistant Professor of Mathematics at Columbia University. His research interests include number theory, algebraic geometry, and cryptography. He can be reached at glass@math.columbia.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.