Alexander Grothendieck prefers to use the spelling Alexandre for his first name: we learn this from a footnote on p. 269 of the book under review, at the start of a chapter titled “A country of which nothing is known but the name: Grothendieck and ‘motives’,” presented by Pierre Cartier. Cartier has long been associated with the famous Institut des Hautes Études Scientifiques, which served to house and support Grothendieck during the 1960s, when he was at his most creative. It was here that Grothendieck maintained his legendary work schedule (up to twelve hours daily, with no days off) and that he produced the titanic notes to the Séminaire de Géométrie Algébrique du Bois Marie, a.k.a. SGA — and much else besides, including a parade of revolutionary articles in algebraic geometry and homological algebra. It is estimated that Grothendieck’s output along these lines amounts to over 10,000 pages.
The present book is a compendium or a collage of articles having to do with the different facets of Grothendieck both as a hugely important and influential scholar and as an ultimately enigmatic individual with a remarkable history, including a past filled with childhood tragedy and strife, and a present full of mystery. There are articles by mathematicians, experts all, on the major themes of Grothendieck as an unsurpassed algebraic geometer, preceded by an initial discussion, by Diestel, of Grothendieck’s early (and also revolutionary) work in the theory of Banach spaces, courtesy of an impetus provided by Laurent Schwartz. So Karoubi writes about G. and K-theory, Raynaud about G. and schemes, Murre about G. and the fundamental group, and Illusie about G. and étale cohomology. Kleiman writes about the Picard scheme, Mumford contributes an article about “[his own] introduction to schemes and functors,” and Simpson talks about the method of descent. Along possibly more personal lines there are the contributions by Hartshorne (“An Apprenticeship”), Schneps (on the famous Grothendieck-Serre correspondence, published by Springer-Verlag), Oort on possible mathematical influences on Grothendieck, and Manin’s reflection on his own experiences with Grothendieck largely in the context of work on motives. The article by Cartier mentioned above also contains a good deal of, for lack of a better word, psychoanalysis of Grothendieck, which brings up the recent and present history of our protagonist.
Already in the 1970s Grothendieck began to distance himself from the mathematical community, becoming engaged in political and peace movements, and in due course he resigned his post at IHES and eventually became something of a recluse. Most recently his home appears to have been, and to be, an isolated location in the Pyrenees. He has written mathematical letters to, e.g., Gerd Faltings, and he is responsible for the new and burgeoning subject of dessins d’enfants as per his 1984 Esquisse d’un Programme. Schneps and Lochak have edited a set of two books on this subject. But of late there has been no physical sign of Grothendieck. He is now 86 years old.
The book under review is irresistible to anyone who has even a mild interest in and acquaintance with algebraic geometry — well, I guess more than this is required, lest the technical discussions completely fail to resonate — and who is fascinated by Grothendieck’s remarkable life, including a trajectory from a fatherless childhood as a refugee from the Nazis sheltered in southern France, to early adulthood as an outsider who nonetheless absorbed mathematics through his pores, to a stellar career as one of the unquestioned grand masters of algebraic geometry, doing things in magical and radical ways in a style all his own, and then to the mystery that he now poses. The authors represented in this “Mathematical Portrait” are uniquely positioned to comment not only on Grothendieck’s mathematics but on the man himself, his personality, his influence (and his influence on them), and his uniqueness. Yes indeed, I think the book is altogether irresistible.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.