Read This!

The MAA Online book review column

Briefly Noted

October 2005

What a wonderful book! Alexandru Scorpan's The Wild World of 4-Manifolds is a lively introduction to 4-manifolds written as introductions should be: it focuses on the big picture and allows readers to delve into details when they are ready.

The book is divided into four parts. To indicate what sets 4-manifolds apart, Scorpan begins with a discussion the h-cobordism theorem and why it fails in dimension 4. The second part focuses on the main invariant of 4-manifolds, the intersection form, discusses Freeman's classification of topological 4-manifolds, and ends with exotic R⁴s, manifolds which are homeomorphic, but not diffeomorhpic, to R⁴ (in contrast to Rⁿ, n ≠ 4, which admits a unique smooth structure). Part three is devoted to complex surfaces as examples of 4-manifolds. The fourth and largest part gives a plethora of results coming from applications of gauge theory to 4-manifolds. Both the third and fourth parts end by constructing infinite families of homeomorphic but non-diffeomorphic 4-manifolds.

As background the reader needs a reasonable understanding of manifolds and differential topology, as well as some algebraic topology (Poincaré duality, characteristic classes). Scorpan provides a very quick listing of background definitions and facts.

A key to the book's success is its multiple layers. In addition to the main text there are footnotes, inserted notes (indented text with smaller type), proofs (also indented), and end-notes. The main text tells the story and is a very reasonable 200 or so pages. The remainder of the 600 pages gives supporting details which can be read or ignored at the reader's pleasure. A very detailed index makes going back easier. Should you need more detail, Scorpan provides an extensive bibliography.

You might wonder if all of these layers are distracting; the answer is no. The main text flows very well and is enjoyable reading. I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds.[Stephen Ahearn]

Christopher Bradley's fascinating book, Challenges in Geometry: for Mathematical Olympians Past and Present, would make a wonderful addition to the personal library of coaches of mathematical competitions, as well as anyone who has an interest in the intersection of geometry and number theory.

In this volume, Bradley explores the classes of triangles, circles, and other geometrical objects that are constrained to have various integer or rational properties such as side length, area, radius, etc. His typical approach is to define an interesting set of constraints, for example integer sided triangles with integer area and inscribed circle with integer radius, and then produce a parameterized system of variables that generates all (or some) of the solutions. As he indicates in his preface, these problems are not ones likely to be found in competition, but expose patterns of thinking and model techniques that competition questions commonly require.

Although not really set up as a textbook, the book does offer a number of exercises at the end of each section, with solutions at the back. I will certainly use it as a resource for problems for my mathematical problem solving course, making excerpts available to my students as appropriate.

While I enjoyed reading the book, pausing frequently to work out problems or proofs, I did find it to be fairly terse at times, requiring more effort than expected to connect the dots. When used with undergraduates (or even good high school students) it will most likely oblige the instructor or coach to provide a substantial amount of background and supplementation. As a book clearly targeted to this audience, I would have also liked more in the way of motivation and problem solving strategies, where instead the author presents solutions completely worked out with little hint as to how the solution was derived.

In any case, Challenges in Geometry offers a great treasure of interesting problems, potential avenues of exploration and research for students, and new insights into rational geometry. [David J. Stucki]

In one volume, Jack Copeland has assembled the influential and historically significant papers, letters and radio broadcasts of Alan Turing (1912-1954). These include Turing's work on logic, digital computers, code breaking, artificial intelligence, and artificial life. Each source is preceded by an explanation of both the scientific details of the piece as well as its subsequent impact.

Reprinted in its entirety is Turing's 1936 paper "On Computable Numbers with an Application to the Entscheidungsproblem," in which he introduces the notion of a universal computing machine and offers a negative solution to Hilbert's decision problem (Entscheidungsproblem). This universal machine has evolved into what today is known as a complier or interpreter in computer science. Copeland includes the text of Turing's 1947 lecture on the Automatic Computing Engine (ACE), compares this with John von Neumann's work on the Electronic Discrete Variable Automatic Computer (EDVAC), and traces Turing's influence on the EDVAC.

Of particular importance is Turing's breaking of the German naval enigma code during World War II, shortening the war in Europe by what is estimated to be at least two years. Reprinted are several recently declassified documents, including excerpts from Turing's "Treatise on the Enigma," Patrick Mahon's account of code breaking at Bletchley Park, as well as a letter from Turing to Winston Churchill to secure additional support for code breaking. A fairly detailed account of the Steckerbrett (plug-board) enigma machine is given in what Turing calls "Bombe and Spider," a chapter from "Treatise on the Enigma."

Finally there appears Alan's work on artificial life and artificial intelligence from the papers "Computing Machinery and Intelligence," "The Chemical Basis for Morphogenesis," "Chess," and "Solvable and Unsolvable Problems." The latter contains an informal discussion of the decision problem and its significance, suitable for a lay audience. The book is organized chronologically with bibliographic references given as footnotes and suggestions for further reading appearing at the end of some chapters. Copeland includes a brief biography of Turing, although for more information about Turing's life and untimely death at his own hands, see Andrew Hodges' Alan Turing: The Enigma, cited at the end of the introduction. [Jerry Lodder]

Not interested in sports? Don't worry; neither am I. Yet I find that real life examples, particularly in sports, can enliven any classroom discussion. The lower level articles in this anthology might provide enrichment for an introductory statistics course. The advanced articles might be useful in a senior honors seminar or project in statistics.

The 36 articles in this collection previously appeared in four different publications of the American Statistical Association (ASA): Chance, Journal of the American Statistical Association, The American Statistician, and the Proceedings of the Statistics in Sports Section of the American Statistical Association. They are now organized, along with eight introductory articles, in one volume with six parts, each on a different sport or theme: football, baseball, basketball, ice hockey, statistical methodologies and multiple sports, and miscellaneous sports. Each part includes a short introductory article followed by anywhere from three to nine articles. There are two additional articles which serve as an introduction to the entire book.

I particularly enjoyed the combination of Chapters 19, 21, 22, and 31. In the title of Chapter 19, Patrick D. Larkey, Richard A. Smith, and Joseph B. Kadane proclaim that "It's Okay to Believe in the 'Hot Hand'." (The "Hot Hand" theory in basketball says that sometimes a player is on a shooting streak and more likely to make a basket that his shooting percentage would indicate. Similar phenomena are alleged to occur in other sports.) Amos Tversky and Thomas Gilovich declare an opposing view in Chapter 21, "The Cold Facts about the 'Hot Hand' in Basketball." In Chapter 22, "Simpson's Paradox and the Hot Hand in Basketball," Robert L. Wardrop considers why both sides may be partially correct because of the phenomenon known as Simpson's Paradox. Robert Hooke takes a philosophical approach in Chapter 31, "Basketball, Baseball, and the Null Hypothesis," to speculate on why statisticians such as Tversky and Gilovich don't observe the "hot hand" phenomenon even though intuition and experience shout that it must exist.

Each chapter has its owns references. Chapter 2 is particularly helpful in this regard. This introductory survey article by editors Albert and Cochran give references containing additional examples and discussing how to teach a special probability and statistics class focused on sports.

This is an enjoyable and useful book. Even those who subscribe to the various ASA publications will appreciate having all the articles in one place. [Raymond N. Greenwell]

Publication Data

Challenges in Geometry: For Mathematical Olympians Past and Present, by Christopher J. Bradley. Oxford University Press, 2005. Paperback, 205 pp., $34.50. ISBN 0-19-856692-1.

The Essential Turing, ed. by B. Jack Copeland. Oxford University Press, 2004. Paperback, 622 pp., $24.95. ISBN 0-19-825080-0.

Anthology of Statistics in Sports, edited by Jim Albert, Jay Bennett, and James J. Cochran. SIAM, 2005. Softcover, 322 pp., $65.00. ISBN 0-89871-587-3.

Stephen T. Ahearn (ahearn@macalester.edu) teaches mathematics at Macalester College in St. Paul, MN. His primary research interests are in algebraic topology and computational topology/geometry but allows himself to be distracted by other interesting topics as in his article ``Tolstoy's Integration Metaphor from War and Peace.'' He also enjoys hiking, swimming, baking bread, and reading.

David J. Stucki teaches computer science and mathematics at Otterbein College, in Westerville, Ohio. His most recent interests are in the history and philosophy of mathematics, computer science education, and algorithmic number theory, although he also maintains an interest in artificial intelligence, theory of programming languages, and foundations/theory of computation. He has participated in Otterbein's Mathematical Problem Solving seminar and has helped to coach the Otterbein teams participating in the annual ECC Undergraduate Mathematics Competition.

Jerry Lodder teaches a variety of mathematics courses at New Mexico State University. His interests include topology, geometry, the history of mathematics and mathematics education. Teaching from primary historical sources, he has introduced logic in a beginning discrete mathematics course via excerpts from Alan Turing's "On Computable Numbers, with an Application to the Entscheidungsproblem." Jerry enjoys traveling to other colleges and universities to discuss the use of history in teaching mathematics.

Raymond N. Greenwell (matrng@hofstra.edu) is Professor of Mathematics at Hofstra University in Hempstead, New York. His research interests include applied mathematics and statistics, and he is coauthor of the texts Finite Mathematics and Calculus with Applications, both published by Addison Wesley.

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.
• June, 2004: The Grothendieck-Serre correspondence, mathematical biology, olympiad problem books, and units.
• July, 2004: Groups, Probability, Olympiads, and Isaac Newton.
• October, 2004: Klein, dogs, rivers, physics, geometry.
• November, 2004: Karl Pearson, chessboard problems, John Wallis, and contemporary art.
• January, 2005: Abel's legacy, reductive groups, curriculum foundations, and irresistible integrals.
• February, 2005: Differential equations, graphs, optimization, algorithms... and arithmetic on function fields of characteristic p.
• March, 2005: William Rowan Hamilton, π, Adolph Zeising, and Jesuits.
• April, 2005: differential equations, abstract algebra, and two kinds of number theory.
• May, 2005: algebra, chance, variations, and Chinese.
• Late May, 2005: motives, historical sources, Lebesgue, Cauchy.
• July, 2005: calculus, control theory, instabilities, biostatistics.
• August, 2005: difference equations, Galois theory, algebra, martingales in finance.

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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.