Read This!

The MAA Online book review column

Briefly Noted

December 2002

This book is an introduction to groups and representation theory aimed at physicists. It starts with the definition of a group, and already on that page there are things that I find strange. A group is a set of elements (or operators) with an operation that satisfies four conditions: closure, associativity, existence of an identity element, existence of inverses. Fine. Then: "Since in general ab is not equal to ba the order of multiplication is important. An Abelian group is one whose elements commute with one another, that is [a, b] = ab - ba = 0." So right off the bat there seems to be present another operation, called "-". In other words, these groups really are subgroups of an algebra of operators from the beginning.

And so it goes: the notation is different, the examples are different, some of the words are different. There are lots of formulas, and most symbols seem to have at least two indices on them. (I think I saw one with four subscripts and two superscripts.) Despite its strangeness to someone who comes from "another tradition," in the end this seems all to the good. These physicists have made this mathematics their own and developed their own way of understanding it. I suspect that we can learn something by reading their take on it all.[Fernando Q. Gouvêa]

I missed this neat little book when it first came out; it definitely deserves a belated notice here.

Creators of Mathematics: The Irish Connection is the result of a project undertaken by the Royal Irish Academy. It collects short biographies of various "mathematicians with Irish connections." For the most part, this means either that they were born in Ireland or that they lived and worked in Ireland for an extended time. The authors of the essays are also "mathematicians with Irish connections."

Several of the people profiled are very well known: Thomas Harriot, William Rowan Hamilton, George Boole, George Gabriel Stokes, William Thomson (Lord Kelvin), William Sealy Gosset ("Student"). These profiles are interesting but the material in them can usually be found in other sources. From the point of view of historians, the other profiles, of less well known mathematicians, may be more valuable. There is some unevenness, however. For example, some of the biographies include short bibliographies, while others include no references at all. Each profile includes a photograph or portrait of its subject.

As someone who was born in Brazil, a country also not particularly famous for its mathematics, I was interested to note that many of the more famous names in the book were born in Ireland but lived most of their lives in England or Scotland. This has happened to many Brazilian mathematicians too. (Of course, Brazil has not yet produced mathematicians as eminent as some of the figures in this book. Or at least we don't know it yet.)

Overall, this is an interesting and pleasant book. Since it is written for a wide audience, these short profiles are probably accessible to college (maybe even high school) students. Mathematicians with an Irish connection may well want to have a copy, and I definitely recommend it for undergraduate libraries. [Fernando Q. Gouvêa]

Martin Väth's Integration Theory: A Second Course does an admirable job at living up to its title. The basic approach is to start from a very abstract notion of integral and then slowly add more and more conditions so that in the end one gets the Lebesgue integral. So in the first part of the book we start with abstract measure spaces and measurable functions. Then we add a topological structure and learn about Radon measures. Then we add a group structure and discuss the Haar measure. Apply that to the real numbers and we've got Lebesgue measure.

The second half of the book feels more like a topics course. The chapters discuss the L^p spaces, convolutions, connections with logic and set theory, special properties of Lebesgue measure, and other related ideas. All in all, it would make for a good "second course" in integration theory.

Except for a few minor (but irritating) copyediting problems, the book is nicely produced. It's not clear to me that the cover has anything to do with the subject matter, but it looks pretty nice. The writing is not too friendly, but it's friendly enough that it keeps me reading.

I like the author's general approach. Not having used the book with students, I don't know how well it would work in class. In any case, it's worth a look; I think most mathematicians will find it interesting. [Fernando Q. Gouvêa]

Publication Data

Group Representation Theory for Physicists, by Jin-Quan Chen, Jialun Ping, and Fan Wong. World Scientific, 2002. Hardcover, 574pp., $86.00. ISBN 981-238-065-5.

Creators of Mathematics: The Irish Connection, ed. by Ken Houston. University College of Dublin Press, 2001. Distributed in the United States by Dufour Editions. Paperback, 150pp., $19.95. ISBN 1-900621-49-5.

Integration Theory: A Second Course, by Martin Väth. World Scientific, 2002. Hardcover, 277pp., $48.00. ISBN 981-238-115-5.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the author of several books, including, most recently, Math through the Ages, written in collaboration with William Berlinghoff.

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.

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