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Séminaire Bourbaki, Volume 2010/2011, Exposés 1027–1042

Nicolas Bourbaki
Publisher: 
Société Mathématique de France
Publication Date: 
2012
Number of Pages: 
496
Format: 
Paperback
Series: 
Astérisque 348
Price: 
141.00
ISBN: 
9782856293515
Category: 
Anthology
[Reviewed by
Michael Berg
, on
06/5/2013
]

I first came across a live copy of Astérisque when, a number of years ago, I needed to learn about perverse sheaves. After reading the corresponding fabulous discussion presented in what is surely one of the best-written books in the genre, namely, Sheaves on Manifolds by Masaki Kashiwara and Pierre Schapira, the time had come for even more explicitly perverse material, so to speak. I presently ordered a copy of Astérisque’s issue 100, dating to 1982, which had begun to appear more and more prominently on my radar (or computer) screen. Issue 100 is, like many Astérisque volumes, devoted in its entirety to a single article, the seminal Faisceaux pervers, by Beilinson-Bernstein-Deligne, providing the foundation of the entire subject so named.

It is however not the rule with Astérisque that only articles of such size and scope, de facto books, should appear in its pages: the journal’s charter provides that it is a venue for “excellent research monographs in French or in English, and proceedings of prestigious seminars or of outstanding international meetings … Each volume is devoted to a single topic, chosen, in principle, from the whole spectrum of mathematics.” It is consonant with this claim that issues 101 and 102 actually joined issue 100 to form a trio devoted to the proceedings of the 1981 Luminy Colloquium dealing with the analysis and topology of singular spaces.

Along not dissimilar lines, the volume under review is in fact a presentation of exposés 1027–1042 of the redoubtable Séminaire Bourbaki. It should be noted right off that the assertion that the journal, issue by issue, should be devoted to a single topic is evidently to be interpreted somewhat liberally: in this volume we find articles (exposés) on such disparate themes as fluid mechanics, Kervaire invariants, and the p-adic Langlands correspondence. Manifestly in this case the ruling (or over-ruling) principle was the bill of fare at the Séminaire. On the other hand, it seems that the broad subject claiming the largest number of articles is this volume is the happy union of algebraic geometry and number theory (taken in the broadest sense).

The exposés are obviously pitched at a high level: even the survey material is aimed at specialists in at least proximate fields. This having been said, the collection of offerings in 348 is rich; many mathematicians browsing the list will find appealing topics there (and, yes, algebraic geometry, in its broadest interpretation, dominates). But there should be no doubt that the journal, this issue in particular, is not aimed at the browsing novice: the reader should be prepared to “go deep,” as they say in football and scuba-diving. And it’s obviously a big plus if this reader is also able to handle mathematical French.

 


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

Novembre 2010

  • O. Glass -- La méthode du retour en contrôlabilité et ses applications en mécanique des fluides
  • E. Kowalski -- Crible en expansion
  • H. Miller -- Kervaire invariant one
  • W. Werner -- Analyticité discrète du modèle d'Ising

Janvier 2011

  • C. Breuil -- Correspondance de Langlands p-adique, compatibilité local-global et applications
  • A. Chambert-Loir -- Relations de dépendance et intersections exceptionnelles
  • A. Naor -- Sparse quadratic forms and their geometric applications
  • F. Santambrogio -- Inégalités isopérimétriques quantitatives via le transport optimal

Avril 2011

  • T. C. Hales -- The fundamental lemma and the Hitchin fibration
  • A. Oancea -- Invariants de Welschinger
  • T. Scanlon -- A proof of the André-Oort conjecture via mathematical logic
  • C. Voisin -- Sections rationnelles de fibrations sur les surfaces et conjecture de Serre

Juin 2011

  • C. Houdayer -- Invariant percolation and measured theory of nonamenable groups
  • D. Huybrechts -- A global Torelli theorem for hyperkähler manifolds
  • P. Mironescu -- Le déterminant jacobien
  • F. Planchon -- Existence globale et scattering pour les solutions de masse finie de l'équation de Schrödinger cubique en dimension deux
  • Table par noms d'auteurs

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