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Mathematical Survey Lectures 1943-2004

Beno Eckmann
Springer Verlag
Publication Date: 
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[Reviewed by
John McCleary
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Mathematicians love a good survey article, where by “good” I mean clear, broad, and deep, and furthermore, revealing its author's point of view, enthusiasms, and intimacy with the subject. A good survey of an elementary topic can be read for a new way of thinking, or new life to an old idea, while a good survey of an advanced topic makes something new accessible to its readers, like being on the other side of a rich conversation with an expert. Beno Eckmann is masterful at the good survey, and it is delightful that Springer has assembled this selection of his papers.

The papers are arranged chronologically, and cover more than 60 years of work. Eckmann was a student of Heinz Hopf in the 1930s and so most of the topics treated are topological. There are several surveys of ideas that were pioneered by Eckmann and his collaborators — namely, fibre spaces, the homology of groups, Hilton-Eckmann duality, and l2-methods in topology. The historical accounts provide a window into the development of topology as a subject, a story that is not as simple as it is presented in textbooks. I particularly liked the article “Is topology a respectable field?” which depicts the 1930s as a time when topology was exotic, and even thought to be uninteresting.

Typical of the kind of topology that was developed in Zurich, Eckmann's results tie diverse parts of mathematics together through topological ideas. The papers in this collection on the interaction between topology and linear algebra — in the problem of real division algebras, on the Hurwitz-Radon equations, and the theory of social choice — bear out these connections. Eckmann is a lively commentator and a skilled expositor, not a little because he was among the first to develop these ideas.

Among the official languages of Switzerland are French and German, and Eckmann writes in a lively manner in both. A few of the essays are in these languages and they give another good reason that we should all learn foreign languages — to read Eckmann! The collection is rounded out by a couple essays on the place of mathematics in culture and its future.

As you read the essays in the book, you feel as though you are getting to know the author and his passions, experiences, and best thoughts. I heartily recommend adding it to your library.

John McCleary is Professor of Mathematics at Vassar College.