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Séminaire Bourbaki, Volume 2013/2014, Exposés 1074-1088

Nicolas Bourbaki
Société Mathématique de France
Publication Date: 
Number of Pages: 
Astérisque 367-368
[Reviewed by
Michael Berg
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Everyone in the mathematical world, and perhaps even beyond its borders, e.g. certain lands populated by discerning physicists, knows Bourbaki and his impact, indeed his ubiquity. He has been in evidence for many decades by now, his first sightings dating back to France in the 1930s, and his enthusiastic proponents including Weil, Ehresmann, L. Schwartz, Dieudonné, Serre, Grothendieck, Chevalley, and even the occasional foreigner like Eilenberg. With striking continuity, Bourbaki’s seminars have been in place throughout, modulo cataclysms like World War II, and in their corresponding written form now add up to over a thousand exposés, on a huge variety of different subjects in mathematics. The book under review is the 2013/2014 volume, containing exposés 1074 to 1088. We read in the frontispiece:

This 66th volume of the Bourbaki Seminar contains the texts of the fifteen survey lectures done during the year 2013/2014: four … on topology and differential geometry, four … about PDEs, one … on the structure of approximate groups, one … about functional analysis, one … on the algebraic geometry of K3 surfaces, one … about the gaps between prime[s], one … on probability … and two concerning the foundations of mathematics and formal proofs…

A wonderful selection, a broad cross section of modern mathematics, and indeed something for nearly every one, or at least for folks who are reasonably comfortable with French in addition to what is now mathematics’ lingua franca, English (or as they apparently claim at Oberwohlfach, pidgin English).

What characterizes these essays or expositions (generally speaking, true mathematics survey articles, replete with examples, theorems and proofs) is that they have been prepared by specialists dealing with contemporary themes close to their hearts. As such, they are fabulous sources from which to learn about the various topics on the docket. An example of this eminently desirable state of affairs is that the recent dramatic work by Zhang and Maynard now heading fast toward a proof of the Goldbach conjecture is discussed on pp. 327–366 by the prominent number theorist, Emmanuel Kowalski. Presented in English, this article is fabulous: yes, it aims at number theorists and fellow travelers, and it is not for the uninitiated, as none of the articles in the book are, of course, but it is a wonderful way to learn about these exciting very recent developments in Zahlentheorie at a high level and with great effectiveness.

And this is the rule throughout this volume of Astérisque, as also for the entire series of the Séminaire Bourbaki. It’s worth the effort to learn French just to be able to read all these articles, not just the ones in English. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.