A mathematician of staggering accomplishment (one reference work described him as "the mathematician whose work was to lead to a unification of geometry, number theory, topology and complex analysis"), Grothendieck's ubiquitous presence in almost all branches of pure mathematics between 1955 and 1970 revolutionised the subject, in recognition of which he was awarded the Fields Medal (the mathematics equivalent of the Nobel Prize) in 1966.
His extraordinary creativity expressed itself in the form of thousands of pages of mathematical literature, notably in the monumental Eléments de Géométrie Algˇbrique and Séminaire de Géométrie Algébrique -- although his achievement was matched only by the impossibility of explaining it to anyone without at least a degree in Pure Mathematics. (Grothendieck's most important single accomplishment, for example, was said to be "the invention of the étale and l-adic cohomology theories".)
Grothendieck's most creative period was spent at the French Institut des Hautes Etudes Scientifiques (IHES), and his time there was regarded ever after as the institute's "Golden Age" during which a whole new school of mathematics flourished under Grothendieck's charismatic leadership. He established the IHES as a world centre of algebraic geometry, with him as its driving force.
There was, however, another Grothendieck, a man who felt deeply about the world's injustices. As a young student, he had decided not to study physics (despite his love for the subject), as he saw the discipline, after Hiroshima, as hopelessly compromised. During the Vietnam War, to protest against American imperialism, he gave lectures on category theory in the forests around Hanoi while the city was being bombed.
In the 1960s he refused to participate in conferences supported by Nato, Nasa or other defence interests. In some cases conference organisers went to the length of securing alternative funding in order to secure his participation. In 1966, when he was awarded the Fields Medal, he refused to travel to Moscow for the ceremony in protest at Soviet militarism.
His career reached a crisis in 1970 when he discovered that IHES was being funded in part, and indirectly, by the French Ministry of Defence. This triggered a bitter debate between Grothendieck and the founder of IHES, Leo Motchane, who maintained a clear division between scientific matters, which were left to the professors, and financial ones, which were the director's domain.
Grothendieck poured scorn on the ease with which colleagues had accepted the situation, observing that their willingness to accept military funding had not prevented them "from professing the ideas 'of the Left' or from being indignant at colonial wars. They generally justify this by saying that this did not limit in any sense their independence or freedom of thought. They refuse to see that this collaboration gives an aura of respectability and liberalism to this apparatus of control, destruction and depreciation. This is something that shocked me."
When Grothendieck failed to secure an immediate halt to the offending subsidy, he felt he had no choice but to resign. His attempts to find a position at an alternative top-ranking university or institute failed, either because the authorities were wary of his fiery reputation or because they did not fulfil his exacting preconditions.
He found work at lesser institutions, but not in areas of advanced research; and in any case, he had become increasingly preoccupied by politics.
In the 1970s, declaring himself a "militant activist", he founded a small group called Survivre et Vivre, an anti-war, anti-imperialist, environmental movement. But partly because of its founder's increasingly eccentric behaviour, the group failed to establish a popular base.
By the 1980s, Grothendieck had become seriously psychologically unstable. Finally he secluded himself in a small hamlet in the Pyrenees. In 1991, after burning thousands of pages of manuscript in the garden of his then girlfriend, he vanished altogether.
Alexander Grothendieck was born to Jewish parents in Berlin on March 28 1928. His father, Shapiro, was a Russian-born anarchist who had taken an active part in the Revolution and, after falling out with Lenin, in various Leftist movements in Germany, where he married the equally radical Johanna ("Hanka"). In 1933 Alexander's parents moved to Paris to escape the Nazis, leaving their five-year-old son with a family in Hamburg, where he went to school. During this time his father fought in the Spanish Civil War.
In 1939 Alexander came to France, and in 1940 was interned with his mother as an "undesirable" (German, then after the German invasion, as a Jew) in the Rieucros camp near Mende. Shapiro, meanwhile, was interned in the camp of Le Vernet, from where he was deported to Auschwitz and died in 1942. Though Alexander never really knew his father, he held him in great esteem. His office at the IHES had no decoration except a portrait of his father.
Life in Vichy France was not easy. But in 1942, after the Grothendiecks had been moved to a detention camp at Gurs, Alexander was able to attend the Coll&eagrave;ge Cévénol, a school run by Protestant resisters at the village of Chambon-sur-Lignon, where he obtained his baccalauréat.
After the war Alexander and his mother moved to a small village near Montpellier, where he found part-time work on a farm while studying Mathematics at the university. In 1948 he went to Paris, carrying a letter of introduction from his former school to the mathematician Henri Cartan. Cartan advised him to go to Nancy, where he studied for a doctorate under Jean Dieudonnˇ.
Grothendieck then spent several years travelling and teaching in Brazil and America . In 1956 he returned to France and, in 1959, he and Dieudonnˇ accepted appointments as professors at the newly-established IHES in Bures-sur-Yvette, where, over the next 12 years, Grothendieck completely revolutionised the theory of algebraic geometry.
After leaving the IHES, Grothendieck tried but failed to get a post at the Coll¸ge de France in Paris. Instead, in 1973, he accepted a professorship at Montpellier University, where he mainly taught elementary subjects such as linear algebra and calculus. He became estranged from the high-level mathematical community.
During the 1980s Grothendieck wrote thousands of pages of mathematical and non-mathematical meditations, much of it mixing philosophical invective, paranoid attacks on rivals and, here and there, insights of pure genius. These included his autobiographical Récoltes et Semailles (1983-85), a paranoid 1,000-page treatise in which he set out his dissatisfactions with the mathematical world but also laid the groundwork for a new field known as anabelian geometry; La clef des songes (1986), in which he explained how the reality of dreams convinced him of God's existence; and Esquisse d'un programme (1984), a proposal for a position at the Centre National de la Recherche Scientifique, in which he described new ideas for studying the "moduli space" of complex curves. Although Grothendieck never published work in this area, the idea became the inspiration for other mathematicians and the source of the new theory of dessins d'enfants (children's drawings).
In May 1988 the Swedish Academy awarded him the Crafoord Prize, a belated attempt to repair the neglect of Alfred Nobel in not creating a prize in mathematics, which came with a cash award of $160,000. But Grothendieck astonished the mathematics world by rejecting it, and in a rambling letter to Le Monde explained his decision as motivated by disgust at the dishonesty and corruption of the scientific and political establishment .
In August 1991, Grothendieck left his home in the Pyrenees, suddenly and without warning, for an unknown location. Severing contacts with friends, family and colleagues, he refused practically all human contact. Over the next few years various rumours circulated. Some suggested that he had remained in the Pyrenees and become a Buddhist. Others maintained that he was living in the Ardèche, herding goats and entertaining radical ecological theories. According to another rumour he was working on a 50-volume manuscript addressing, among other things, the physics of free will.
One of the members of the mathematical establishment to come into contact with him was Leila Schneps who, with her future husband, Pierre Loschak, tracked him down and found him "obsessed by the devil which he sees at work everywhere in the world". In a subsequent letter to Leila Schneps, Grothendieck said he would be prepared to share his research into physics with her if she could answer one question: "What is a metre?"
In 2010 he tried to eradicate all trace of his past life, writing a letter to one of his students demanding that his entire back catalogue be removed from libraries and refusing to allow republications.
Alexander Grothendieck, who was twice married and had four children, died in hospital at Saint-Girons in south-west France.
Alexander Grothendieck, born March 28 1928, died November 13 2014
14 November 2014 © Telegraph