Grothendieck was born in Berlin to anarchist parents. His father, Alexander Schapiro, had Hasidic Jewish roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, Johanna Grothendieck, came from a Protestant family in Hamburg and worked as a journalist. Both had broken away from their early backgrounds in their teens. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and his birthname was initially recorded as "Alexander Raddatz." The marriage was dissolved in 1929 and Schapiro acknowledged his paternity, but never married Hanka.
Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to Paris to evade Nazism, followed soon thereafter by his mother. They left Grothendieck in the care of Wilhelm Heydorn, a Lutheran pastor and teacher in Hamburg. During this time, his parents took part in the Spanish Civil War, according to Winfried Scharlau, as non-combatant auxiliaries, though others state that Sascha fought in the anarchist militia.
In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterwards his father was interned in Le Vernet. He and his mother were then interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners". The first was the Rieucros Camp, where his mother contracted the tuberculosis which eventually caused her death and where Alexander managed to attend the local school, at Mende. Once Alexander managed to escape from the camp, intending to assassinate Hitler. Later, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II. Alexander was permitted to live, separated from his mother, in the village of Le Chambon-sur-Lignon, sheltered and hidden in local boarding houses or pensions, though he occasionally had to seek refuge in the woods during Nazis raids, surviving at times without food or water for several days. His father was arrested under the Vichy anti-Jewish legislation, and then handed over by the French Vichy government to the Germans to be sent to be murdered at the Auschwitz concentration camp in 1942. In Chambon, Grothendieck attended the Collège Cévenol, a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists. Many of the refugee children hidden in Chambon attended Cévenol, and it was at this school that Grothendieck apparently first became fascinated with mathematics.
After the war, the young Grothendieck studied mathematics in France, initially at the University of Montpellier where he did not initially perform well, failing such classes as astronomy. Working on his own, he rediscovered the Lebesgue measure. After three years of increasingly independent studies there, he went to continue his studies in Paris in 1948.
Initially, Grothendieck attended Henri Cartan's Seminar at École Normale Supérieure, but he lacked the necessary background to follow the high-powered seminar. On the advice of Cartan and André Weil, he moved to the University of Nancy where he wrote his dissertation under Laurent Schwartz and Jean Dieudonné on functional analysis, from 1950 to 1953. At this time he was a leading expert in the theory of topological vector spaces. From 1953 to 1955 he moved to the University of São Paulo in Brazil,
His first works on topological vector spaces in 1953 have been successfully applied to physics and computer science, culminating in a relation between Grothendieck inequality and the Einstein-Podolsky-Rosen paradox in quantum physics.
In 1958, Grothendieck was installed at the Institut des hautes études scientifiques (IHÉS), a new privately funded research institute that, in effect, had been created for Jean Dieudonné and Grothendieck. Grothendieck attracted attention by an intense and highly productive activity of seminars there (de facto working groups drafting into foundational work some of the ablest French and other mathematicians of the younger generation). Grothendieck himself practically ceased publication of papers through the conventional, learned journal route. He was, however, able to play a dominant role in mathematics for around a decade, gathering a strong school.
During this time, he had officially as students Michel Demazure (who worked on SGA3, on group schemes), Luc Illusie (cotangent complex), Michel Raynaud, Jean-Louis Verdier (cofounder of the derived category theory) and Pierre Deligne. Collaborators on the SGA projects also included Michael Artin (étale cohomology) and Nick Katz (monodromy theory and Lefschetz pencils). Jean Giraud worked out torsor theory extensions of nonabelian cohomology. Many others like David Mumford, Robin Hartshorne, Barry Mazur and C.P. Ramanujam were also involved.
Alexander Grothendieck's work during the "Golden Age" period at the IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory and complex analysis. His first discovery in algebraic geometry was the Grothendieck–Hirzebruch–Riemann–Roch theorem, a generalisation of the Hirzebruch–Riemann–Roch theorem proved algebraically; in this context he also introduced K-theory. Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time. He went on to introduce the étale cohomology theory of schemes, providing the key tools for proving the Weil conjectures, as well as crystalline cohomology and algebraic de Rham cohomology to complement it. Closely linked to these cohomology theories, he originated topos theory as a generalisation of topology. He also provided an algebraic definition of fundamental groups of schemes and more generally the main structures of a categorical Galois theory. As a framework for his coherent duality theory he also introduced derived categories, which were further developed by Verdier.
Grothendieck's political views were radical and pacifist, and he strongly opposed both United States intervention in Vietnam and Soviet military expansionism. He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam War. He retired from scientific life around 1970, having found out that IHÉS was partly funded by the military. He returned to academia a few years later as a professor at the University of Montpellier.
In 1970, Grothendieck, with two other mathematicians, Claude Chevalley and Pierre Samuel, created a political group called Survivre. The group published a bulletin and was dedicated to antimilitary and ecological issues, and also developed strong criticism of the indiscriminate use of science and technology. Grothendieck devoted the next three years to this group.
Although Grothendieck continued with mathematical enquiries his standard mathematical career, for the most part, ended when he left the IHÉS. After leaving the IHÉS Grothendieck became a temporary professor at Collège de France for two years. He then became a professor at the University of Montpellier, where he became increasingly estranged from the mathematical community.
While not publishing mathematical research in conventional ways during the 1980s, he produced several influential manuscripts with limited distribution, with both mathematical and biographical content.
- La Longue Marche à travers la théorie de Galois
- Esquisse d'un Programme
- Récoltes et semailles (Autobiographical)
In 1988 Grothendieck declined the Crafoord Prize with an open letter to the media. He wrote that established mathematicians like himself had no need for additional financial support and criticized what he saw as the declining ethics of the scientific community, characterized by outright scientific theft that, according to him, had become commonplace and tolerated.
Over 20,000 pages of Grothendieck's mathematical and other writings, held at the University of Montpellier, remain unpublished. They have been digitized for preservation and are freely available in open access through the Institut Montpelliérain Alexander Grothendieck portal.
In 1991, Grothendieck moved to a new address which he did not provide to his previous contacts in the mathematical community. Very few people visited him afterward.
In January 2010, Grothendieck wrote the letter "Déclaration d'intention de non-publication" to Luc Illusie, claiming that all materials published in his absence have been published without his permission. He asks that none of his work be reproduced in whole or in part and that copies of this work be removed from libraries.
On 13 November 2014, aged 86, Grothendieck died in the hospital of Saint-Girons, Ariège.
