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Arnold: Swimming Against the Tide

Boris A. Khesin and Serge L. Tabachnikov, editors
American Mathematical Society
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

Vladimir Igorevich Arnold was one of the great mathematicians of our time. His name is attached to a remarkable number of notions, conjectures and results: Arnold diffusion, Arnold tongues in bifurcation theory, Arnold conjectures in symplectic geometry, the Kolmogorov-Arnold-Moser theory, Arnold’s cat map, Arnold’s stability criterion in hydrodynamics… The list goes on and on. He solved a general version of Hilbert’s 13th problem. He also managed to write several hundred papers and quite a few books, including at least five university textbooks.

Arnold was known for his strong, uncompromising opinions and a very much in-your-face style (with “extreme and occasionally intentionally outrageous claims” according to Yakov Eliashberg.) He was famously opposed to overly formalized mathematics (what he called “criminal Bourbakization”) and once engaged in a mathematical duel of sorts with Serre on the subject.

This book is a portrait of Arnold from the inside and the outside. The first part consists mostly of Arnold’s own words. The third part has reminiscences and reflections from people who knew him. In the middle is a particularly good collection of photographs of Arnold, alone or with colleagues and family.

The first part begins with the transcript of an interview (previously unavailable in translation) that was originally published in the Russian magazine Kvant. This is followed by two essays that should appeal to any readers who might be interested in this rather remarkable man.

The first of these is a transcript of a lecture Arnold gave to attempt to describe the diversity of his mathematical interests. He begins with a discussion of his work on Hilbert’s 13th problem (establishing that any continuous function of many variables can be represented as the composition of a finite number of continuous functions of two variables). Arnold describes how this work (completed when he was 19) led him to questions in algebraic topology, then to singularity theory, to dynamical systems and symplectic geometry.

Of course, talk of labels and boundaries between areas of mathematics would be something of an anathema to Arnold. He saw mathematics as a single thing with wonderful and mysterious connections between individual parts. The piece on “polymathematics” that follows explores this idea in greater detail. Something of the spirit of Poincaré lived in Arnold, both in the breadth of his interests and the scope of his mathematical knowledge.

Another recommended selection is Arnold’s essay on Vladimir Rokhlin. It gives a clear picture of the challenges of mathematical life in Russia in the latter part of the twentieth century. (There is also a great story about Rokhlin, Arnold and a bear in the mountains of Armenia.)

And finally, a word about the title: the metaphorical sense is apparent enough, but there is also a very literal sense. Arnold, visiting San Francisco, decided to swim across San Francisco Bay parallel to the Golden Gate Bridge during an ebb tide. The currents there are quite unpredictable and are known to be occasionally vicious. Arnold described hitting “a wall of current” and having to turn back. Fighting the current was a way of life for him.

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Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.