Read This!

The MAA Online book review column

Briefly Noted

March 2002

Writing the history of mathematics in the 20th century is a challenging job. To begin with, there is so much technical detail to master that it's hard to imagine any single historian who can cover everything. In this and in the previous volume on The Development of Mathematics, 1900-1950, Jean-Paul Pier opts for the most natural solution: ask a collection of experts to write on their areas of expertise. The articles, in English and French, discuss a very wide range of subjects. Inevitably, there is a lot of variation in style, depth, and level of detail. As a result, the book does not really provide a systematic survey of mathematics in the last 50 years.

Most of the articles in the book opt for detail rather than broad sweep. So, for example, Fouvry's article on "Fifty Years of the Analytic Theory of Numbers" has as subtitle "A point of view among others: sieve methods." In other words, Fouvry will focus only on one aspect of Analytic Number Theory, even though his is the only article about that field. There are articles on transcendental numbers and on cryptography, but no broad overview of number theory. The coverage is broader in other areas, but the majority of articles have a tight focus and an "internal" approach, and many assume the reader knows much of the relevant background mathematics. As a result, most readers will probably only want to read those articles that are closer to their interests.

In addition to the articles, the book contains three interviews (with Adrien Douady, Mickhael Gromov, and Friedrich Hirzebruch), lists of Fields Medal and Nevanlinna Prize winners, a list of speakers and topics at the ICMs, a list of expository articles published in the Bulletin of the American Mathematical Society and in Uspehi Matematiceskih Nauk, and a valuable preliminary bibliography on the history of mathematics in the last fifty years. While few people will read the book from cover to cover, it is likely to prove to be very valuable resource. [Fernando Q. Gouvêa]

Teaching differential geometry to undergraduates is, I think, always a challenge. While the initial pre-requisites aren't too heavy (basically, vector calculus and linear algebra), the subject can get quite technical very fast. This creates some dilemmas for the instructor. Should one teach only the geometry of curves and surfaces in three-dimensional space? If so, should one stick to the older, classical language, or should one introduce the language of differential forms and tangent bundles? Which theorem(s) should serve as a goal of the course?

Wolfgang Kühnel's Differential Geometry: Curves — Surfaces — Manifolds illustrates one set of answers to these questions. The author describes his first four chapters as "a course in classical differential geometry", which is followed by four more chapters which make up "a course on Riemannian geomtery". For an undergraduate course, one would presumably want to use only the first half. The pre-requisites are described as "linear algebra and calculus", but it should be clear that we are talking about enough linear algebra and calculus to comfortably handle n-dimensional space. This includes some notions about analysis in Rⁿ, including the derivative as a linear transformation and the implicit function theorem.

Kühnel starts by discussing the geometry of curves in Rⁿ. He develops the general case, then considers curves in the plane, in space, and in Minkowski space in more detail. In addition, the first chapter goes into the global geometry of curves, including such results as the theorem on turning tangents, the four vertex theorem, and Fenchel's theorem on the total curvature of space curves. All this is done by page 52. There follow chapters on the local geometry of surfaces and the intrinsic geometry of surfaces to complete the section on classical differential geometry. The remaining chapters treat Riemannian manifolds, the curvature tensor, spaces of constant curvature, and Einstein spaces.

I suspect most undergraduates would find this book very hard to read. For the exceptional student who can handle this level of abstraction, however, it might be useful as a survey of and introduction to one of the more beautiful theories in mathematics. [Fernando Q. Gouvêa]

Publication Data

The Development of Mathematics, 1950-2000, ed. by Jean-Paul Pier. Birkhäuser, 2000. Hardcover, 1372pp., $169.00. ISBN 3-7643-6280-4.

Differential Geometry: Curves — Surfaces — Manifolds, by Wolfgang Kühnel. Student Mathematical Library, volume 16. American Mathematical Society, 2002. Paperback, 360pp., $49.00 ($39.00 to AMS members). ISBN 0-8218-2656-5.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of FOCUS and MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

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.

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