Read This!

The MAA Online book review column

Briefly Noted

Short Reviews by Fernando Q. Gouvêa

This page collects brief notes on books that do not fit our review criteria but which nevertheless seem worthy of note. It will be updated periodically. Books mentioned here are not included in the index of MAA Online book reviews. Publication data on each of the books can be found at the bottom of the page. Links to previous "briefly noted" columns are also available below.

July 1999

So here are these two books... They are both about geometry (in some sense of the word), and they couldn't be more different. The first, Geometry and Interpolation of Curves and Surfaces, by Robin J. Y. McLeod and M. Louisa Baart, is an introduction to algebraic geometry as a tool for solving a problem in applied mathematics. The problem, which the authors call the problem of interpolation, is the following: given a set of points on the plane, how do we draw a "good" curve passing through them? How do we even know which curves are good? A variant of the problem begins with a "visual" description of the desired curve (e.g., an artist's sketch), and then the problem is to turn that into something with which one can do mathematics (and, eventually, something that can be rendered by a computer). The whole thing is fascinating, especially for someone (like me!) who knows algebraic geometry in its fancy schemes-and-cohomology clothing. This looks like a delightful way to introduce students to the subject.

The other geometry book is Mixed Hodge Structures and Singularities, by Valentine S. Kulikov. This is about singularities in algebraic or analytic varieties and how they can be described by the theory of mixed Hodge structures. Here, we are clearly in a different world, one in which the geometry is studied via an extraordinarily sophisticated formalism involving the Gauss-Manin connection, DeRham cohomology, and so on. The introduction specifies that "the reader is expected to have the knowledge and training usual in algebraic and analytic geometry. This includes knowledge of sheaf theory and the technique of spectral sequences."

Now don't get me wrong---Kulilkov's book looks very nice, and I suspect it'll be useful too. But the difference in attitude is striking. It is not just that this is an advanced monograph and the other is intended as a student text (though that is part of it, of course). It also reveals, I think, the stunning variety of approaches to mathematics itself that characterizes today's mathematical community. I think the diversity is great, and I hope both books find a warm welcome with their intended audiences.

In the same series as Kulikov's book is Paul Roberts' Multiplicities and Chern Classes in Local Algebra, which represents a further step in the process: this one is about ideas derived from geometry, but which are now simply a part of algebra. (The connection is the ring of germs of holomorphic (or meromorphic, or algebraic) functions near a point in an algebraic variety.) This is definitely algebra, not geometry, and it illustrates a different lesson: that there is extensive cross-fertilization between different mathematical fields.

Some time back, I mentioned that several publishers had series which reprinted older volumes that could be considered "classics". We just received two volumes in another such series, the "Classics in Applied Mathematics" series from SIAM. Both, coincidentally, are statistics books: Stochastic Processes, by Emanuel Parzen, and Selecting and Ordering Populations, by Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel. As I mentioned in that previous columns, it's a delight to see that publishers are working to keep worthy volumes in print. It's also great to see (by looking at the list of past volumes in the series) that SIAM is using the series to bring out new translations of material of historical significance, in this case Gauss's Theory of the Combination of Observations Least Subject to Errors. The books are affordable but well-made softcover editions.

Publication Data

Robin J. Y. McLeod and M. Louisa Baart, Geometry and Interpolation of Curves and Surfaces. Cambridge University Press, July 1999. Hardcover, 414 pages, $80.00. ISBN 0-521-32153-0.

Valentine S. Kulikov, Mixed Hodge Structures and Singularities. Cambridge Tracts in Mathematics 132. Cambridge University Press, May 1999. Hardcover, 186 pages, $44.95. ISBN 0-521-62060-0.

Paul C. Roberts, Multiplicities and Chern Classes in Local Algebra. Cambridge Tracts in Mathematics 133. Cambridge University Press, May 1999. Hardcover, 303 pages, $59.95. ISBN 0-521-47316-0.

Emauel Parzen, Stochastic Processes. Classics in Applied Mathematics 24. SIAM, June 1999. Softcover, 324 pages, $37.00 ($29.60 to SIAM members). ISBN 0-89871-441.

Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology. Classics in Applied Mathematics 26. SIAM, June 1999. Softcover, 569 pages, $59.50 ($47.60 to SIAM members). ISBN 0-89871-439-7

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

Copyright ©1999 The Mathematical Association of America

MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu).
Last modified: Thu Jul 15 20:52:48 -0500 1999