[LUM#1] In Search of Alexandre Grothendieck

[LUM Magazine] - Published on January 19, 2016 in Science and Society

Who remembers Alexandre Grothendieck? The man once considered the greatest mathematician of the^20th century passed away recently. Having emerged from obscurity, this pioneer of new worlds leaves behind uncharted territory: a body of handwritten work waiting to be discovered.

He transformed the landscape of mathematics. Alexandre Grothendieck passed away on November 13, 2014, at the age of 86. For many years, he had already distanced himself from the wider community. Having taken refuge in Lasserre, a small village in the Pyrenees, the enfant terrible of mathematics had been living there as a recluse since 1991. His last public act took place three years earlier, when he refused the Crafoord Prize, which was supposed to crown his career, along with the $270,000 that came with it.“Fruitfulness is recognized by one’s offspring, not by honors,”explained the man who had been showered with them, having notably received the prestigious Fields Medal.

Extraordinary genius

Jean Malgoire was his student. Now a lecturer at the University of Montpellier, he recalls the“absolute uncompromising nature” that defined both the man and the researcher.“He had complete faith in his ability to analyze things on his own, without being swayed by other points of view. A self-confidence he applied to every field…".

From a very early age, Grothendieck explored on his own this magic in which lies the secret harmony of the world—what we call mathematics. At the University of Montpellier, it wasn’t in class that he shone, but in the shadows: at the age of 18, without realizing it, he was reconstructing Lebesgue’s venerable theory of integration. Jean Dieudonné was flabbergasted when he met this extraordinary genius in 1949: that’s not how one works!

The man who would become his closest collaborator then entrusted him with 14 problems in functional analysis that neither he nor Laurent Schwartz had been able to solve. What followed has become legendary: in just a few months, the young mathematician untangled these 14 Gordian knots, each of which could have been the subject of a respectable doctoral thesis. Grothendieck was 22 years old.

From the Age of Miracles to the Age of Protest

Two decades of uninterrupted miracles lay before him. At the Institut des Hautes Études Scientifiques, founded around him, this giant with dazzling visions devoured his work. And he placed a new constellation in the mathematical sky: modern algebraic geometry, a new conceptual framework that has been universally adopted.

“When did he think? We didn’t know. He seemed to know everything about a new subject,” remarks mathematician Michel Demazure, his first doctoral student, still amazed half a century after their first meeting. “With this strange individual, who seemed to be of a fundamentally different nature, everything seemed to already be there…”

But those 20 years of grace were about to come to an abrupt end. For Grothendieck, in the aftermath of May 1968, with the Vietnam War raging, mathematics mattered less and less. “Should I continue my research?” he asked. He found his answer. In 1970, upon learning that the IHES received grants from the Ministry of Defense, he resigned. The master whom we had followed down new paths cut ties.

Exile Diaries

With his usual conviction, he held this view: the world is heading for ruin. He identified the threats:“the ecological imbalance created by contemporary industrial society (…), military conflicts.” He took“the first step on a new journey”—a journey that would lead to his exile in 1991. In the meantime, he founded, along with a group of scientists, one of the very first radical environmental movements: “Survive and Live.”

It was once again in the shadows that he continued his intellectual pursuits. At the University of Montpellier, his new refuge where a scientific star taught while increasingly isolating himself, and even during his exile in the Pyrenees, he filled page after page with writing. The master’s thoughts flowed freely there.“He never stopped doing math,” asserts Jean Malgoire, to whom Grothendieck entrusted, in 1991, voluminous boxes ofdocuments, still largely unexamined (see box).

Grothendieck had insisted that nothing be published during his lifetime. His notebooks from exile, so long awaited by the scientific community, may now be published. What will we discover there?“Seeds, no doubt, which it will be up to us to bring to fruition,” suggests Jean Malgoire. In the realm of mathematics, Alexandre Grothendieck may not yet be done shedding light on uncharted territories.

Keywords: LUM#1, LUM Magazine, Mathematics, Research, Science