Alexander Grothendieck, later Alexandre Grothendieck in French (; German: [ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk] ; French: [ɡʁɔtɛndik]; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century.
Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He received the Fields Medal in 1966 for advances in algebraic geometry, homological algebra, and K-theory. He later became professor at the University of Montpellier and, while still producing relevant mathematical work, he withdrew from the mathematical community and devoted himself to political and religious pursuits (first Buddhism and later, a more Catholic Christian vision). In 1991, he moved to the French village of Lasserre in the Pyrenees, where he lived in seclusion, still working on mathematics and his philosophical and religious thoughts until his death in 2014.
Grothendieck was born in Berlin on March 28, 1928, to anarchist parents. His father, Alexander "Sascha" Schapiro (also known as Alexander Tanaroff), had Hasidic Jewish roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, Johanna "Hanka" Grothendieck, came from a Protestant German family in Hamburg and worked as a journalist. As teenagers, both of his parents had broken away from their early backgrounds. At the time of his birth, Grothendieck's mother was married to the journalist Johannes Raddatz and initially, his birth name was recorded as "Alexander Raddatz." That marriage was dissolved in 1929 and Schapiro acknowledged his paternity, but never married Hanka Grothendieck. Grothendieck had a maternal sibling, his half sister Maidi.
Grothendieck lived with his parents in Berlin until the end of 1933, when his father moved to Paris to evade Nazism. His mother followed soon thereafter. Grothendieck was left in the care of Wilhelm Heydorn, a Lutheran pastor and teacher in Hamburg. According to Winfried Scharlau, during this time, his parents took part in the Spanish Civil War as non-combatant auxiliaries. However, others state that Schapiro fought in the anarchist militia.
In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterward 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 camp was the Rieucros Camp, where his mother may have contracted the tuberculosis that would eventually cause her death in 1957. While there, Grothendieck managed to attend the local school, at Mende. He once 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. Grothendieck was permitted to live separated from his mother.
In the village of Le Chambon-sur-Lignon, he was sheltered and hidden in local boarding houses or pensions, although he occasionally had to seek refuge in the woods during Nazi raids, surviving at times without food or water for several days.
His father was arrested under the Vichy anti-Jewish legislation, and sent to the Drancy internment camp, and then handed over by the French Vichy government to the Germans to be sent to the Auschwitz concentration camp in 1942 where he was murdered.
In Le Chambon, Grothendieck attended the Collège Cévenol (now known as the Le Collège-Lycée Cévenol International), a unique secondary school founded in 1938 by local Protestant pacifists and anti-war activists. Many of the refugee children hidden in Le Chambon attended Collège Cévenol, and it was at this school that Grothendieck apparently first became fascinated with mathematics.
In 1990, for risking their lives to rescue Jews, the entire village was recognized as "Righteous Among the Nations".
Studies and contact with research mathematics
After the war, the young Grothendieck studied mathematics in France, initially at the University of Montpellier where at first he did not 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 two leading experts were working on Grothendieck's area of interest, topological vector spaces: Jean Dieudonné and Laurent Schwartz. The latter had recently won a Fields Medal. Dieudonné and Schwartz showed the new student their latest paper La dualité dans les espaces (F) et (LF); it ended with a list of 14 open questions, relevant for locally convex spaces. Grothendieck introduced new mathematical methods that enabled him to solve all of these problems within a few months.
In Nancy, he wrote his dissertation under those two professors on functional analysis, from 1950 to 1953. At this time he was a leading expert in the theory of topological vector spaces. In 1953 he moved to the University of São Paulo in Brazil, where he immigrated by means of a Nansen passport, given that he had refused to take French nationality (as that would have entailed military service against his convictions). He stayed in São Paulo (apart from a lengthy visit in France from October 1953 to March 1954) until the end of 1954. His published work from the time spent in Brazil is still in the theory of topological vector spaces; it is there that he completed his last major work on that topic (on "metric" theory of Banach spaces).
Grothendieck moved to Lawrence, Kansas at the beginning of 1955, and there he set his old subject aside in order to work in algebraic topology and homological algebra, and increasingly in algebraic geometry. It was in Lawrence that Grothendieck developed his theory of abelian categories and the reformulation of sheaf cohomology based on them, leading to the very influential "Tôhoku paper".
In 1957 he was invited to visit Harvard University by Oscar Zariski, but the offer fell through when he refused to sign a loyalty oath, a pledge promising not to work to overthrow the United States government—a refusal which, he was warned, threatened to land him in prison. The prospect of prison did not worry him, so long as he could have access to books.
Comparing Grothendieck during his Nancy years to the École Normale Supérieure-trained students at that time (Pierre Samuel, Roger Godement, René Thom, Jacques Dixmier, Jean Cerf, Yvonne Bruhat, Jean-Pierre Serre, and Bernard Malgrange), Leila Schneps said:
He was so completely unknown to this group and to their professors, came from such a deprived and chaotic background, and was, compared to them, so ignorant at the start of his research career, that his fulgurating ascent to sudden stardom is all the more incredible; quite unique in the history of mathematics.
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.