Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 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.
Life
Family and childhood
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.: "Beide beteiligten sich am Spanischen Bürgerkrieg, nicht aktiv kämpfend, aber unterstützend." However, others state that Schapiro fought in the anarchist militia.
World War II
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."Piotr Pragacz, 'Notes on the Life and Work of Alexander Grothendieck,' in Piotr Pragacz (ed.), Topics in Cohomological Studies of Algebraic Varieties: Impanga Lecture Notes, Springer Science & Business Media, 2006 pp-xi-xxviii p.xii. 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.Amir D. Aczel,The Artist and the Mathematician, Basic Books, 2009 pp.8ff.pp.8–15. Grothendieck was permitted to live separated from his mother.Luca Barbieri Viale, 'Alexander Grothendieck:entusiasmo e creatività,' in C. Bartocci, R. Betti, A. Guerraggio, R. Lucchetti (eds.,) Vite matematiche: Protagonisti del '900, da Hilbert a Wiles, Springer Science & Business Media, 2007 pp.237–249 p.237.
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 , 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 () et (); 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 -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:
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.
IHÉS years
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 practically ceased publication of papers through the conventional, learned journal route. However, he was able to play a dominant role in mathematics for approximately a decade, gathering a strong school.
Officially during this time, he had as students Michel Demazure (who worked on SGA3, on group schemes), (relative schemes and classifying topos), Luc Illusie (cotangent complex), Michel Raynaud, Michèle Raynaud, Jean-Louis Verdier (co-founder of the derived category theory), and Pierre Deligne. Collaborators on the SGA projects also included Michael Artin (étale cohomology), Nick Katz (monodromy theory, and Lefschetz pencils). Jean Giraud worked out torsor theory extensions of nonabelian cohomology there as well. Many others such as David Mumford, Robin Hartshorne, Barry Mazur and C.P. Ramanujam were also involved.
"Golden Age" Grothendieck's work during what is described as the "Golden Age" period at the IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory, and complex analysis. His first (pre-IHÉS) 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 (EGA) 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. Its immediate antecedent was Serre's purely algebraic definition of unramified covers, generalising Riemann's many-sheeted covering surfaces. Closely linked to these cohomology theories, he originated topos theory as a generalisation of topology (relevant also in categorical logic). He also provided, by means of a categorical Galois theory, an algebraic definition of fundamental groups of schemes giving birth to the now famous étale fundamental group and he then conjectured the existence of a further generalization of it, which is now known as the fundamental group scheme. As a framework for his coherent duality theory, he also introduced derived categories, which were further developed by Verdier.
The results of his work on these and other topics were published in the EGA and in less polished form in the notes of the Séminaire de géométrie algébrique (SGA) that he directed at the IHÉS. SGA 4 introduced Grothendieck universes as the set-theoretic device for handling large categories.
Political activism