The letter “G” abounds in great mathematicians: Gauss, Galois and Gödel being only the standard-bearers; we also have Germain, Grassmann, Green and Gromov, all worthy of an entry in the abecedary. I, however have chosen to write about Grothendieck, which is a challenge. It is difficult to be flippant, humorous, or tongue-in-cheek about Grothendieck: his whole life, his way of thinking, and his tremendous achievement call for humility and astonishment.
Let us start with the facts. Alexandre Grothendieck was born in 1928 in Berlin to a family of anarchists. The father, the one-armed Sasha (Alexander) Shapiro, who was born in the Pale and drifted West in the mid-twenties, apparently fought in Makhno’s army against the Reds during the Civil War after the October revolution, and later in Spain. Alexandre’s mother, Hanka Grothendieck, came from a Protestant background; she was as fiery and as politically committed as her partner. She also fought in Spain, during which time her son was deposited with a kindly clergyman. Then World War Two came. Sasha perished in Auschwitz, Hanka and her son were imprisoned as German citizens, and survived the war through kindness of strangers, in particular in Chambon-sur-Lignon, a shining memorial to human bravery and generosity of heart.
By that time it was clear that Alexandre was very mathematically talented; he went to study mathematics at Montpelier, found the teaching staff in Montpelier had nothing to teach him, went to Paris and thence to Nancy. Grothendieck wrote his Ph.D. thesis under Laurent Schwartz, and in 1966, already long a founding professor at the Institute des Hautes Etudes Scientifiques (IHES), after stints in Brazil and Kansas, won a Fields medal. In 1970, aged 42, he resigned his position, allegedly because he found out that part of the funding of IHES came from the French Defence Ministry.
Now Grothendieck embarked on a different, difficult, itinerant life, interspersed with ever shorter bouts of academic activity: some teaching at Montpelier, an application to CNRS, communication with colleagues through letters, more mathematical work, which is still being discussed and pondered over. But it is true to say that the focus of his life moved inward, and his thought became strongly deipetal. He filled in thousands of typescript pages with meditations and analyses of his dreams; meandering, partially autobiographical reflective treatises such as Harvestings and Sowings, The Key to Dreams are the result. In 2010 he disseminated a letter forbidding publication of his work in manuscript and republication of published results. He is still alive, in the Pyrenees somewhere, no-one knows where.
What about Grothendieck the man? I think it is true to say that Grothendieck’s commitment to whatever he was and is doing has always been complete. He was a workaholic, totally immersed, mathematician; the Vietnam war, survival of the human race in the face of the nuclear war threat, the place of science of society, occupied him incessantly and deeply, and I expect God-searching and trying to understand the essence of human existence, tasks he has sent himself in later life, allow no distractions. Let us add to this his kindness and compassion to all sufferers, his gratitude to the many teachers and supporters who allowed him, always a stateless person, to find employment, his collegiality, his demanding and yet deeply nourishing teaching (one of his students, Pierre Deligne, is also a Fields medalist).
It remains to say something about Grothendieck the mathematician. But there words fail me. He has contributed significantly to functional analysis, and most of all to algebraic geometry. He was a part of the Bourbaki group, and in a sense is the justification and the epitome of the Bourbaki approach: instead of solving a particular problem he always strove for the generality in which the problem falls out of the set-up. He has been a staunch champion of the categorical approach and that it now pervades all of mathematics, from its foundations to cosmology. Nuclear spaces, étale cohomology, schemes, Weil conjectures, topos theory, sheaf theory, Grothendieck groups and K-theory, … His influence on Artin, Serre, and more recently, on Drinfeld, Manin and Voevodsky, is well-documented. I wish someone would write a “Grothendieck for Dummies” book; I suspect such a book is impossible.
Why did he leave his position at IHES? Many reasons have been put forward: tiredness; understanding that at 42, after a dozen of years of stellar achievement, he was past his prime; purely ideological reasons; conflicts with colleagues (such as René Thom): all these have been suggested, and perhaps each of these has a grain of truth in it. But I think more is true: Grothendieck felt that the time for the next step of his evolution as a human being had come.
In a manuscript entitled Notes for the Key to Dreams, Grothendieck has inserted a self-contained narrative called The Mutants. This word in French has a more mysterious connotation than in English, of something faintly alien. And indeed, in common with Ludwig Wittgenstein, Simone Weil, and Glenn Gould, Grothendieck has an uncanny power of penetration into a sphere of human activity that seems inhuman in its seriousness, and in the depth and resolution of its vision.
W. Scharlau, Who is Alexander [sic] Grothendieck? Notices of the AMS, 44(8) (2008), 930-941.