Read This!

The MAA Online book review column

Briefly Noted

June 2004

Had Weyl lived but a bit longer he would have seen the comet that was Alexandre Grothendieck. It was visibly coming to those who looked in the right place in 1954, arguably brighter than all the other celestial objects in the period 1958-1970, and then occasionally visible (but only by telescope) since. We earthbound mathematicians have trouble understanding Grothendieck, what with his extreme abstraction and his utter independence from examples. At least we see stars moving predictably in circles. What to make of the brief bright comet moving crosswise?

Grothendieck-Serre Correspondence helps us understand both Serre and Grothendieck better. It contains about forty letters in each direction, mostly from the period 1955-1964. There are some easy-to-understand parts about things like getting better jobs and how "szyzygytic" is not quite correctly spelled. Mostly, however, the book documents with detail the creation of very formidable mathematics. Difficulties for readers are alleviated somewhat by twenty-seven pages of endnotes written recently by Serre. The current bilingual edition — original French on the left, a truly excellent translation into English on the right — certainly will help many too.

As a whole, the book gives an insider's view of one of mathematical history's most important research associations. Ironically, Serre and Grothendieck never wrote a paper together. It is overdue and fitting that their signatures have finally appeared together and handsomely on a book cover. [David P. Roberts]

Essential Mathematical Biology is written as a comprehensive introduction to a new and rapidly growing field. In the words of the author, the book is aimed at junior or senior honors undergraduates.

The book is quite comprehensive for its size, covering topics such as population dynamics (for single and interacting species), infectious diseases, genetics, molecular and cellular biology, pattern formation, tumor modeling and others. This is quite impressive, as books in the field covering close to the same number of topics (classics are Murray's Mathematical Biology and Edelstein-Keshet's Mathematical Models in Biology) total at least twice the number of pages. Of course, brevity comes at a cost, and the cost is in this case assumed background. Standard techniques such as the use of difference and differential equations, and visualizations through cobweb diagrams are hidden in the appendix. This makes the text look slightly daunting, especially for an introductory course. One has to keep in mind that the book is written for honors students in the UK, which have a different curriculum. It is unlikely to find a sufficient number of highly mathematically skilled students in a typical undergraduate institution, to offer a course using this book as a text.

The book has a companion website, which has corrections and some additional materials related to chapters 3 (Infectious Diseases) and 4 (Population Genetics and Evolution). The author is planning to add more materials in the future. Currently the address for the website is http://www.maths.bath.ac.uk/~nfb/book/ (this is a change from the address published in the book).

Essential Mathematical Biology is concise but well written. The variety of topics and the summaries provided at the end of each topic would make this book an excellent choice for a "topics in mathematical biology" class, and a great source for undergraduate student projects. As a resource, it would also make a valuable library addition. [Ioana Mihaila]

International Mathematical Olympiads: 1986-1999 was written by Marcin E. Kuczma, one of the world's leading problem proposers. It is a continuation of the compilation of the International Mathematical Olympiad problems 1959-1977 and 1978-1985 by S. L. Greitzer and M. S. Klamkin respectively, published by the Mathematical Association of America. All the problems from IMO1986 to IMO1999 are treated here. Each problem is given with a detailed solution, and sometimes more than one. The solutions are extremely clear and well presented.

Besides the main part containing the problems and solutions, the book contains useful appendices: results by country from IMO1986 to IMO1999, a list of symbols, a glossary of basic mathematical identities and definitions, subject classification of the problems presented, and a bibliography.

The effectiveness of this book derives in large part from the passion with which M. E. Kuczma shares his mathematical enthusiasm and on the breadth and erudition of his discussions. I strongly recommend this book. [Mohammed Aassila]

Mathematical Olympiads: problems and solutions from around the world: 2000-2001 represents a continuation of the compilation of Mathematical Olympiad Problems from the 1998-1999, and 1999-2000 competitions published by the Mathematical Association of America. The authors have collected olympiad problems from the national contests of 22 different countries, together with 5 regional contests from 2000 and the national contest of 16 countries and 6 regional contests from 2001. Problems from 2000 are published with solutions, but the solutions for 2001 problems are notincluded.

Besides the main part containing the problems and solutions, the book contains the usual useful appendices: a glossary of basic mathematical identities and definitions, and an index of problems classified lexicographically by subject area, country of origin, and year.

The following is just a small sample of the problems discussed in the book:

1. Let M = {1,2,...,40}. Find the smallest positive integer n for which it is possible to partition M into n subsets such that whenever a, b and c (not necessarily distinct) are in the same subset, a is not equal to b+c.

2. In the plane are given 2000 congruent triangles of area 1, which are images of a single triangle under different translations. Each of these triangles contains the centroids of all the others. Show that the area of the union of these triangles is less than 22/9.

All in all this book is very well written, full of interesting problems and I warmly recommend it to anyone interested in mathematical competitions, or just in nice problems.[Mohammed Aassila]

When Bill Berlinghoff and I were working on Math through the Ages, we felt it was important to include a chapter on the history of units, and especially of metric units. After all, this topic is part of the school mathematics curriculum. But we found it difficult to find good references. The best we could find was John Roche's The Mathematics of Measurement: A Critical History, which is useful but too heavy, technical, and expensive for our intended audience. Alex Hebra's Measure for Measure: The Story of Imperial, Metric, and Other Units is an attempt to fill that gap.

In short chapters, Hebra takes us through a plethora of units for all sorts of quantities: length, time, angles, mass and force, temperature, luminosity, etc. He tells us about attempts to create units based on natural phenomena and about ways of formulating natural laws in terms of dimensionless quantities. One of the last chapters takes the reader through an interesting exercise: suppose an alien race, communicating over intergalactic distances, asked us how to build a dam; could we give them directions despite the fact that we have no idea what sort of units of measurement they use?

Despite the subtitle, there is very little history here. Hebra does include lots of stories, but he doesn't really attempt to trace in detail the evolution of the various systems and units of measurement. The organization by kind of unit (a chapter for length, a chapter for time,...) makes it hard to get a sense of historical flow, and the transition from isolated units to systems of units does not get the emphasis it deserves. There are occasional lapses; for example, in the chapter on measuring angles, the radian is defined but no explanation is given as to why one would want to use a unit of measurement that resulted in such strange numbers. (There is also a problem with a graphic in that section: π has come out as p.) Finally, both the discussion of whether the United States should "go metric" and the jokes based on unit conversions (e.g.,"28 to 29 grams of prevention are worth 0.454 kilograms of cure") get old fairly quickly.

Nevertheless, this is a useful book for anyone wanting to know more about units of measurement and their role in science (especially physics). Many of the examples would make for excellent assignments for students, and the many illustrations are very helpful. So: not exactly the book I wanted it to be, but I'll take it. [Fernando Q. Gouvêa]

Publication Data

Essential Mathematical Biology, by Nicholas F. Britten. Springer Undergraduate Mathematics Series. Springer-Verlag, 2003. Paperback, 335 pp., $34.95. ISBN 1-85233-536-X.

Grothendieck-Serre Correspondence: Bilingual Edition, American Mathematical Society, 2004. Hardcover, 600pp, $69.00. ISBN 082183424X.

Measure for Measure: The Story of Imperial, Metric, and Other Units, by Alex Hebra. Johns Hopkins University Press, 2003. Hardcover, 215 pp., $24.95. ISBN 0-8018-7072-0.

Ioana Mihaila (imihaila@csupomona.edu) is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis.

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

Mohammed Aassila is a mathematics professor whose research area is analysis. He is interested in mathematics competitions and is the author of two books on the subject: 300 Défis Mathématiques and Olympiades Internationales de Mathématiques.

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

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.
• August 2001: introductions to algebraic number theory and to topology, and problem books.
• November 2001: Wilbur Knorr, John H. Conway, and vector analysis.
• February 2002: biological time, celestial mechanics, and things rarely talked about.
• March 2002: history of recent mathematics and differential geometry.
• June 2002: mathematics in other cultures, Archimedes, probabilistic thinking, and Dover's Phoenix.
• August 2002: Euclid's Elements, Hurewicz's Lectures, and Olympiads 2001.
• October 2002: Algebra from A to Z, Problems in Probability, and a really old trigonometry textbook.
• December, 2002: group representation theory, Irish mathematicians, and integration.
• January, 2003: careers, logic, dueling idiots, and differential geometry.
• March, 2003: the Putnam, the Liber Abaci, and the Millennial Conference.
• April, 2003: analysis, reprints, and Rithmomachia.
• May, 2003: conversation starters, dissections, and Fibonacci numbers.
• July, 2003: mathematical circles, the Scottish Café, and the zeta function.
• September, 2003: history — an encyclopedia, historiography, and algebra.
• October, 2003: selected papers and second editions.
• December, 2003: calculus, relativity, and biology.
• February, 2004: cryptography, generating functions, Wallis, and Stirling.
• March, 2004: differential equations, fundamental groups, reprints, and spherical buildings.
• April, 2004: Ramanujan graphs, time, and Medieval Islam.
• May, 2004: History: of Greek mathematics, of mechanics, of tables, of Reverend Bayes.

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