Alexander Grothendieck

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. 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." 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.

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), Monique Hakim (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

Grothendieck's political views were radical and pacifistic. He strongly opposed both United States intervention in Vietnam and Soviet military expansionism. To protest against the Vietnam War, he gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed. In 1966, he had declined to attend the International Congress of Mathematicians (ICM) in Moscow, where he was to receive the Fields Medal. He retired from scientific life around 1970 after he had 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.

While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure from the IHÉS, those who knew him say that the causes of the rupture ran more deeply. Pierre Cartier, a visiteur de longue durée ("long-term guest") at the IHÉS, wrote a piece about Grothendieck for a special volume published on the occasion of the IHÉS's fortieth anniversary. In that publication, Cartier notes that as the son of an antimilitary anarchist and one who grew up among the disenfranchised, Grothendieck always had a deep compassion for the poor and the downtrodden. As Cartier puts it, Grothendieck came to find Bures-sur-Yvette as "une cage dorée" ("a gilded cage"). While Grothendieck was at the IHÉS, opposition to the Vietnam War was heating up, and Cartier suggests that this also reinforced Grothendieck's distaste at having become a bureaucrat of the scientific world. In addition, after several years at the IHÉS, Grothendieck seemed to cast about for new intellectual interests. By the late 1960s, he had started to become interested in scientific areas outside mathematics. David Ruelle, a physicist who joined the IHÉS faculty in 1964, said that Grothendieck came to talk to him a few times about physics. Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics.

In 1970, Grothendieck, with two other mathematicians, Claude Chevalley and Pierre Samuel, created a political group entitled Survivre—the name later changed to Survivre et vivre. The group published a bulletin and was dedicated to antimilitary and ecological issues. It also developed strong criticism of the indiscriminate use of science and technology. Grothendieck devoted the next three years to this group and served as the main editor of its bulletin. The 27th of January 1972, he was invited to the CERN to give the conference « Will we continue scientific research ? ».

Although Grothendieck continued with mathematical enquiries, his standard mathematical career mostly 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. He formally retired in 1988, a few years after having accepted a research position at the CNRS.

Manuscripts written in the 1980s

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.

Produced during 1980 and 1981, La Longue Marche à travers la théorie de Galois (The Long March Through Galois Theory) is a 1600-page handwritten manuscript containing many of the ideas that led to the Esquisse d'un programme. It also includes a study of Teichmüller theory.

In 1983, stimulated by correspondence with Ronald Brown and Tim Porter at Bangor University, Grothendieck wrote a 600-page manuscript entitled Pursuing Stacks. It began with a letter addressed to Daniel Quillen. This letter and successive parts were distributed from Bangor (see External links below). Within these, in an informal, diary-like manner, Grothendieck explained and developed his ideas on the relationship between algebraic homotopy theory and algebraic geometry and prospects for a noncommutative theory of stacks. The manuscript, which is being edited for publication by G. Maltsiniotis, later led to another of his monumental works, Les Dérivateurs. Written in 1991, this latter opus of approximately 2000 pages, further developed the homotopical ideas begun in Pursuing Stacks. Much of this work anticipated the subsequent development during the mid-1990s of the motivic homotopy theory of Fabien Morel and Vladimir Voevodsky.

In 1984, Grothendieck wrote the proposal Esquisse d'un Programme ("Sketch of a Programme") for a position at the Centre National de la Recherche Scientifique (CNRS). It describes new ideas for studying the moduli space of complex curves. Although Grothendieck never published his work in this area, the proposal inspired other mathematicians to work in the area by becoming the source of dessin d'enfant theory and anabelian geometry. Later, it was published in two-volumes and entitled Geometric Galois Actions (Cambridge University Press, 1997).

During this period, Grothendieck also gave his consent to publishing some of his drafts for EGA on Bertini-type theorems (EGA V, published in Ulam Quarterly in 1992–1993 and later made available on the Grothendieck Circle web site in 2004).

In the extensive autobiographical work, Récoltes et Semailles ('Harvests and Sowings', 1986), Grothendieck describes his approach to mathematics and his experiences in the mathematical community, a community that initially accepted him in an open and welcoming manner, but which he progressively perceived to be governed by competition and status. He complains about what he saw as the "burial" of his work and betrayal by his former students and colleagues after he had left the community. Récoltes et Semailles was finally published in 2022 by Gallimard and, thanks to French science historian Alain Herreman, is also available on the Internet. An English translation by Leila Schneps was scheduled to be published by MIT Press in 2025, but as of May 2026 it is not yet available. A partial English translation can be found on the Internet. A Japanese translation of the whole book in four volumes was completed by Tsuji Yuichi (1938–2002), a friend of Grothendieck from the Survivre period. The first three volumes (corresponding to Parts 0 to III of the book) were published between 1989 and 1993, while the fourth volume (Part IV) was completed and, although unpublished, copies of it as a typed manuscript are circulated. Grothendieck helped with the translation and wrote a preface for it, in which he called Tsuji his "first true collaborator". Parts of Récoltes et Semailles have been translated into Spanish, as well as into a Russian translation that was published in Moscow.

In 1988, Grothendieck declined the Crafoord Prize with an open letter to the media. He wrote that he and other established mathematicians had no need for additional financial support and criticized what he saw as the declining ethics of the scientific community that was characterized by outright scientific theft that he believed had become commonplace and tolerated. The letter also expressed his belief that totally unforeseen events before the end of the century would lead to an unprecedented collapse of civilization. Grothendieck added however that his views were "in no way meant as a criticism of the Royal Academy's aims in the administration of its funds" and he added, "I regret the inconvenience that my refusal to accept the Crafoord prize may have caused you and the Royal Academy."

La Clef des Songes, a 315-page manuscript written in 1987, is Grothendieck's account of how his consideration of the source of dreams led him to conclude that a God exists. As part of the notes to this manuscript, Grothendieck described the life and the work of 18 "mutants", people whom he admired as visionaries far ahead of their time and heralding a new age. The only mathematician on his list was Bernhard Riemann. Influenced by the Catholic mystic Marthe Robin who was claimed to have survived on the Holy Eucharist alone, Grothendieck almost starved himself to death in 1988. His growing preoccupation with spiritual matters was also evident in a letter entitled Lettre de la Bonne Nouvelle sent to 250 friends in January 1990. In it, he described his encounters with a deity and announced that a "New Age" would commence on 14 October 1996.

The Grothendieck Festschrift, published in 1990, was a three-volume collection of research papers to mark his sixtieth birthday in 1988.

More than 20,000 pages of Grothendieck's mathematical and other writings are held at the University of Montpellier and remain unpublished. They have been digitized for preservation and are freely available in open access through the Institut Montpelliérain Alexander Grothendieck portal.

Retirement into reclusion and death

In 1991, Grothendieck moved to a new address that he did not share with his previous contacts in the mathematical community. Very few people visited him afterward. Local villagers helped sustain him with a more varied diet after he tried to live on a staple of dandelion soup. At some point, Leila Schneps and Pierre Lochak located him, then carried on a brief correspondence. Thus they became among "the last members of the mathematical establishment to come into contact with him". After his death, it was revealed that he lived alone in a house in Lasserre, Ariège, a small village at the foot of the Pyrenees.

In January 2010, Grothendieck wrote the letter entitled "Déclaration d'intention de non-publication" to Luc Illusie, claiming that all materials published in his absence had been published without his permission. He asked that none of his work be reproduced in whole or in part and that copies of this work be removed from libraries. He characterized a website devoted to his work as "an abomination". His dictate may have been reversed in 2010.

In September 2014, almost totally deaf and blind, he asked a neighbour to buy him a revolver so he could kill himself. His neighbour refused to do so. On 13 November, 2014, at the age of 86, Grothendieck died in the hospital of Saint-Lizier or Saint-Girons, Ariège.

Citizenship

Grothendieck was born in Weimar Germany. In 1938, aged ten, he moved to France as a refugee. Records of his nationality were destroyed in the fall of Nazi Germany in 1945 and he did not apply for French citizenship after the war. Thus, he became a stateless person for at least the majority of his working life and he traveled on a Nansen passport. Part of his reluctance to hold French nationality is attributed to not wishing to serve in the French military, particularly due to the Algerian War (1954–62). He eventually applied for French citizenship in the early 1980s, after he was well past the age that would have required him to do military service.

Family

Grothendieck was very close to his mother, to whom he dedicated his dissertation. She died in 1957 from tuberculosis that she contracted in camps for displaced persons.

He had five children: a son with his landlady during his time in Nancy; three children, Johanna (1959), Alexander (1961), and Mathieu (1965) with his wife Mireille Dufour; and one child with Justine Skalba, with whom he lived in a commune in the early 1970s.