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Séminaire Bourbaki Volume 2015-2016, Exposés 1104-1119

Nicolas Bourbaki, editor
Société Mathématique de France
Publication Date: 
Number of Pages: 
Astérisque 390
[Reviewed by
Fernando Q. Gouvêa
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Many mathematical careers begin at large research universities but are lived out mostly in smaller places. One of the most noticeable differences between those two settings is one’s ability to know “what is going on” in mathematics as a whole. While in graduate school or a post-doctoral position, it’s easy: anyone who does something significant is likely to be invited to give a talk, and even if they don’t come the new results will be discussed in common rooms and offices. Not so in a small college or university: one tries to keep up with what is going on in one’s narrow field of specialization, and even that is difficult. It’s unlikely that a number theorist will hear about that big breakthrough in ergodic theory.

One of the standard ways to address that problem is to organize a seminar at a national meeting. In the US, this has happened since 2003 at the “Current Events Bulletin” session at the Joint Mathematics Meetings. In France, it happens at the Séminaire Bourbaki. Indeed, the “Current Events” session seems to have been created in conscious imitation of the older French institution, which goes back to the 1948–49 academic year.

As everyone knows, Nicolas Bourbaki is the “collective name” of a group of mathematicians. Organized in the 1930s, the group set out to write a massive textbook on all of “basic” mathematics. Somewhat later, Bourbaki began organizing his Séminaire, now held four times each academic year at the Institut Henri Poincaré in Paris. The tradition is that Bourbaki invites a mathematician (usually not the main discoverer) to discuss important new results. In addition to giving a talk, participants are expected to write an “exposé” to be distributed at the meeting; these are eventually collected in the annual Séminaire Bourbaki volume. The exposés are sequentially numbered; this volume brings them up to number 1,119.

Many Bourbaki write-ups are marvels of high-level mathematical exposition. As a graduate student, I often found that a good way to begin to learn new results in my area was to read the Bourbaki account; it often served as a guide to reading the (always difficult) original paper.

Since then, however, I have often used the annual volumes as a source of knowing “what’s new.” Of course, this is a skewed view of mathematics: these are the new results that Bourbaki found interesting; as everyone knows, old Nicolas has his favorite topics. Nevertheless, I enjoy reading the first few paragraphs of these expository articles to get a sense of what has been happening.

So what interested Bourbaki in 2015–16? As usual, there are several articles on algebraic geometry, most of them rather fancy stuff involving derived categories, perverse sheaves, and the like. This year it was complex algebraic geometry that got the most attention. Lie theory, another Bourbaki favorite, shows up as well. One nice article explains the proof of the Hilbert-Smith conjecture in dimension 3. Eigenvalues and eigenfunctions of the Laplacian operator also appear more than once in unusual contexts involving words such as “fractal” and “stochastic.” There are several more: analytic number theory, model theory, solving under-determined linear systems (very important in many applications, including compressed sensing), fluid dynamics related to “vortex filaments.”

Giving a Bourbaki talk seems to encourage authors to give “big picture” summaries of complicated topics. One author says, for example, that it has long been known that “every natural deformation problem is controlled by a differential graded Lie algebra,” a vague principle that guides the theory and has recently been made more precise. Another author talks about how “the psychological blockage of the example of Ornstein-Weiss” was recently overcome, so that the “subject [of sofic entropy] has flowered in bouquets since 2008.” I particularly like D. Gaitsgory’s comment on how readers who do not know derived algebraic geometry might approach his paper: he proposes simplifying assumptions for a first and second reading and then “For the third reading… learn the theory.”

Twelve of the sixteen articles are in French, with the remaining four in English, so the Séminaire won’t always be useful to non-Francophones. (All of my quotes are translated.) Very few readers will be able to understand all of the articles, but those who can read French and are willing to struggle through the first few pages of each article can get a good sense of a lot of interesting mathematics.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.