Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at theInstitut 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 theFields Medal in 1966 for advances inalgebraic geometry,homological algebra, andK-theory.[12] He later became professor at theUniversity of Montpellier[1] 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).[13] In 1991, he moved to the French village ofLasserre in thePyrenees, where he lived in seclusion, still working on mathematics and his philosophical and religious thoughts until his death in 2014.[14]
Sacha Schapiro, his fatherHanka Grothendieck, his mother
Grothendieck was born in Berlin toanarchist parents. His father,Alexander "Sascha" Schapiro (also known as Alexander Tanaroff), hadHasidic Jewish roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother,Johanna "Hanka" Grothendieck, came from aProtestant German family inHamburg and worked as a journalist.[a] As teenagers, both of his parents had broken away from their early backgrounds.[16] 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.[16] 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 evadeNazism. His mother followed soon thereafter. Grothendieck was left in the care of Wilhelm Heydorn, aLutheranpastor and teacher inHamburg.[17][18] According toWinfried Scharlau, during this time, his parents took part in theSpanish Civil War as non-combatant auxiliaries.[19][20] However, others state that Schapiro fought in the anarchist militia.[21]
TheStolperstein of Alexander Grothendieck, n°165 Brunnenstrasse,Berlin-Mitte, where he lived with his parents.
In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterward his father was interned inLe Vernet.[22] He and his mother were then interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners."[23] The first camp was theRieucros 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, atMende. He once managed to escape from the camp, intending to assassinateHitler.[22] Later, his mother Hanka was transferred to theGurs internment camp for the remainder of World War II.[22] Grothendieck was permitted to live separated from his mother.[24]
In the village ofLe Chambon-sur-Lignon, he was sheltered and hidden in local boarding houses orpensions, although he occasionally had to seek refuge in the woods during Nazi raids, surviving at times without food or water for several days.[22][24]
In Le Chambon, Grothendieck attended the Collège Cévenol (now known as theLe 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.[26]
In 1990, for risking their lives to rescue Jews, the entire village was recognized as "Righteous Among the Nations".
After the war, the young Grothendieck studied mathematics in France, initially at theUniversity of Montpellier where at first he did not perform well, failing such classes as astronomy.[27] Working on his own, he rediscovered theLebesgue measure. After three years of increasingly independent studies there, he went to continue his studies in Paris in 1948.[17]
Initially, Grothendieck attendedHenri Cartan's Seminar atÉcole Normale Supérieure, but he lacked the necessary background to follow the high-powered seminar. On the advice of Cartan andAndré Weil, he moved to theUniversity of Nancy where two leading experts were working on Grothendieck's area of interest,topological vector spaces:Jean Dieudonné andLaurent Schwartz. The latter had recently won a Fields Medal. Dieudonné and Schwartz showed the new student their latest paperLa dualité dans les espaces (F) et (LF); it ended with a list of 14 open questions, relevant forlocally convex spaces.[28] Grothendieck introduced new mathematical methods that enabled him to solve all of these problems within a few months.[29][30][31][32][33][34][35]
In Nancy, he wrote his dissertation under those two professors onfunctional analysis, from 1950 to 1953.[36] At this time he was a leading expert in the theory of topological vector spaces.[37] In 1953 he moved to theUniversity of São Paulo in Brazil, where he immigrated by means of aNansen 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 ofBanach spaces).
In 1957 he was invited to visitHarvard University byOscar Zariski, but the offer fell through when he refused to sign 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.[41]
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.[42]
In 1958, Grothendieck was installed at theInstitut des hautes études scientifiques (IHÉS), a new privately funded research institute that, in effect, had been created forJean Dieudonné and Grothendieck.[6] 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).[17] 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.[44]
The results of his work on these and other topics were published in theEGA and in less polished form in the notes of theSéminaire de géométrie algébrique (SGA) that he directed at the IHÉS.[17]
Grothendieck's political views wereradical andpacifistic. He strongly opposed both United Statesintervention in Vietnam andSoviet military expansionism. To protest against theVietnam War, he gave lectures oncategory theory in the forests surroundingHanoi while the city was being bombed.[46] In 1966, he had declined to attend the International Congress of Mathematicians (ICM) in Moscow, where he was to receive the Fields Medal.[7] He retired from scientific life around 1970 after he had found out that IHÉS was partly funded by the military.[47] He returned to academia a few years later as a professor at theUniversity 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, avisiteur 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.[48] 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 findBures-sur-Yvette as "une cage dorée" ("a gilded cage"). While Grothendieck was at the IHÉS, opposition to theVietnam War was heating up, and Cartier suggests that this also reinforced Grothendieck's distaste at having become a bureaucrat of the scientific world.[6] 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 aboutphysics.[b]Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics.[48]
In 1970, Grothendieck, with two other mathematicians,Claude Chevalley andPierre Samuel, created a political group entitledSurvivre—the name later changed toSurvivre 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.[49] Grothendieck devoted the next three years to this group and served as the main editor of its bulletin.[1]
Although Grothendieck continued with mathematical enquiries, his standard mathematical career mostly ended when he left the IHÉS.[8] After leaving the IHÉS, Grothendieck became atemporary professor atCollège de France for two years.[49] 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 theCNRS.[1]
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 theEsquisse d'un programme.[50] It also includes a study ofTeichmüller theory.
In 1983, stimulated by correspondence withRonald Brown and Tim Porter atBangor University, Grothendieck wrote a 600-page manuscript entitledPursuing Stacks. It began with a letter addressed toDaniel Quillen. This letter and successive parts were distributed from Bangor (seeExternal links below). Within these, in an informal, diary-like manner, Grothendieck explained and developed his ideas on the relationship between algebraichomotopy theory andalgebraic geometry and prospects for anoncommutative theory ofstacks. 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 inPursuing Stacks.[7] Much of this work anticipated the subsequent development during the mid-1990s of themotivic homotopy theory ofFabien Morel andVladimir Voevodsky.
During this period, Grothendieck also gave his consent to publishing some of his drafts for EGA onBertini-type theorems (EGA V, published in Ulam Quarterly in 1992–1993 and later made available on theGrothendieck 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.[17]Récoltes et Semailles was finally published in 2022 by Gallimard[51] and, thanks to French science historian Alain Herreman,[7] is also available on the Internet.[52] An English translation byLeila Schneps will be published byMIT Press in 2025.[53] A partial English translation can be found on the Internet.[54] A Japanese translation of the whole book in four volumes was completed by Tsuji Yuichi (1938–2002), a friend of Grothendieck from theSurvivre 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".[55][56][57][58][59][60] Parts ofRécoltes et Semailles have been translated into Spanish,[61] as well as into a Russian translation that was published in Moscow.[62]
In 1988, Grothendieck declined theCrafoord 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."[63]
La Clef des Songes,[64] 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.[65] 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.[1] The only mathematician on his list wasBernhard Riemann.[66] Influenced by the Catholic mysticMarthe Robin who was claimed to have survived on the Holy Eucharist alone, Grothendieck almost starved himself to death in 1988.[1] His growing preoccupation with spiritual matters was also evident in a letter entitledLettre 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.[7]
TheGrothendieck Festschrift, published in 1990, was a three-volume collection of research papers to mark his sixtieth birthday in 1988.[67]
More than 20,000 pages of Grothendieck's mathematical and other writings are held at the University of Montpellier and remain unpublished.[68] They have been digitized for preservation and are freely available in open access through the Institut Montpelliérain Alexander Grothendieck portal.[69][70]
In 1991, Grothendieck moved to a new address that he did not share with his previous contacts in the mathematical community.[1] Very few people visited him afterward.[71] Local villagers helped sustain him with a more varied diet after he tried to live on a staple ofdandelion soup.[72] 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".[73] After his death, it was revealed that he lived alone in a house inLasserre, Ariège, a small village at the foot of thePyrenees.[74]
In January 2010, Grothendieck wrote the letter entitled "Déclaration d'intention de non-publication" toLuc 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.[75] He characterized a website devoted to his work as "an abomination".[76] His dictate may have been reversed in 2010.[77]
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.[78] On 13 November 2014, aged 86, Grothendieck died in the hospital ofSaint-Lizier[78] orSaint-Girons, Ariège.[26][79]
Grothendieck was born inWeimar Germany. In 1938, aged ten, he moved to France as a refugee. Records of his nationality were destroyed in thefall of Nazi Germany in 1945 and he did not apply forFrench citizenship after the war. Thus, he became astateless person for at least the majority of his working life and he traveled on aNansen passport.[3][4][5] Part of his reluctance to hold French nationality is attributed to not wishing to serve in the French military, particularly due to theAlgerian War (1954–62).[6][5][15] 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.[6]
Grothendieck was very close to his mother, to whom he dedicated his dissertation. She died in 1957 fromtuberculosis that she contracted in camps fordisplaced persons.[49]
He had five children: a son with hislandlady during his time in Nancy;[6] three children, Johanna (1959), Alexander (1961), and Mathieu (1965) with his wife Mireille Dufour;[1][41] and one child with Justine Skalba, with whom he lived in acommune in the early 1970s.[1]
Homological methods and sheaf theory had already been introduced in algebraic geometry byJean-Pierre Serre[81] and others, after sheaves had been defined byJean Leray. Grothendieck took them to a higher level of abstraction and turned them into a key organising principle of his theory. He shifted attention from the study of individual varieties to hisrelative point of view (pairs of varieties related by amorphism), allowing a broad generalization of many classical theorems.[49] The first major application was the relative version of Serre's theorem showing that the cohomology of acoherent sheaf on a complete variety is finite-dimensional; Grothendieck's theorem shows that thehigher direct images of coherent sheaves under a proper map are coherent; this reduces to Serre's theorem over a one-point space.
In 1956, he applied the same thinking to theRiemann–Roch theorem, which recently had been generalized to any dimension byHirzebruch. TheGrothendieck–Riemann–Roch theorem was announced by Grothendieck at the initialMathematische Arbeitstagung inBonn, in 1957.[49] It appeared in print in a paper written byArmand Borel with Serre. This result was his first work in algebraic geometry. Grothendieck went on to plan and execute a programme for rebuilding the foundations of algebraic geometry, which at the time were in a state of flux and under discussion inClaude Chevalley's seminar. He outlined his programme in his talk at the 1958International Congress of Mathematicians.
His foundational work on algebraic geometry is at a higher level of abstraction than all prior versions. He adapted the use of non-closedgeneric points, which led to the theory ofschemes. Grothendieck also pioneered the systematic use ofnilpotents. As 'functions' these can take only the value 0, but they carryinfinitesimal information, in purely algebraic settings. Histheory of schemes has become established as the best universal foundation for this field, because of its expressiveness as well as its technical depth. In that setting one can usebirational geometry, techniques fromnumber theory,Galois theory,commutative algebra, and close analogues of the methods ofalgebraic topology, all in an integrated way.[17][82][83]
Grothendieck is noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation.[44] Relatively little of his work after 1960 was published by the conventional route of thelearned journal, circulating initially in duplicated volumes of seminar notes; his influence was to a considerable extent personal. His influence spilled over into many other branches of mathematics, for example the contemporary theory ofD-modules. Although lauded as "the Einstein of mathematics", his work also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems.[84][85]
Grothendieck's work includes the invention of theétale andl-adic cohomology theories, which explain an observation made byAndré Weil that argued for a connection between the topological characteristics of a variety and its diophantine (number theoretic) properties.[49] For example, the number of solutions of an equation over afinite field reflects the topological nature of its solutions over thecomplex numbers. Weil had realized that to prove such a connection, one needed a new cohomology theory, but neither he nor any other expert saw how to accomplish this until such a theory was expressed by Grothendieck.
This program culminated in the proofs of theWeil conjectures, the last of which was settled by Grothendieck's studentPierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics.[17]
In Grothendieck's retrospectiveRécoltes et Semailles, he identified twelve of his contributions that he believed qualified as "great ideas".[86] In chronological order, they are:
"Schematic" or "arithmetic" point of view forregular polyhedra and regular configurations of all kinds
Here the termyoga denotes a kind of "meta-theory" that may be used heuristically;Michel Raynaud writes the other terms "Ariadne's thread" and "philosophy" as effective equivalents.[87]
Grothendieck wrote that, of these themes, the largest in scope was topoi, as they synthesized algebraic geometry, topology, and arithmetic. The theme that had been most extensively developed was schemes, which were the framework "par excellence" for eight of the other themes (all but 1, 5, and 12). Grothendieck wrote that the first and last themes, topological tensor products and regular configurations, were of more modest size than the others. Topological tensor products had played the role of a tool rather than of a source of inspiration for further developments; but he expected that regular configurations could not be exhausted within the lifetime of a mathematician who devoted oneself to it. He believed that the deepest themes were motives, anabelian geometry, and Galois–Teichmüller theory.[88]
Grothendieck is considered by many to be the greatest mathematician of the twentieth century.[11] In an obituaryDavid Mumford andJohn Tate wrote:
Although mathematics became more and more abstract and general throughout the 20th century, it was Alexander Grothendieck who was the greatest master of this trend. His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed themselves—and then, like a magician, show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed.[11]
By the 1970s, Grothendieck's work was seen as influential, not only in algebraic geometry and the allied fields of sheaf theory and homological algebra,[89] but influenced logic, in the field of categorical logic.[90]
According to mathematicianRavi Vakil, "Whole fields of mathematics speak the language that he set up. We live in this big structure that he built. We take it for granted—the architect is gone". In the same article,Colin McLarty said, "Lots of people today live in Grothendieck's house, unaware that it's Grothendieck's house."[71]
Grothendieck approached algebraic geometry by clarifying the foundations of the field, and by developing mathematical tools intended to prove a number of notable conjectures. Algebraic geometry has traditionally meant the understanding of geometric objects, such asalgebraic curves and surfaces, through the study of the algebraic equations for those objects. Properties of algebraic equations are in turn studied using the techniques ofring theory. In this approach, the properties of a geometric object are related to the properties of an associated ring. The space (e.g., real, complex, or projective) in which the object is defined, is extrinsic to the object, while the ring is intrinsic.
Grothendieck laid a new foundation for algebraic geometry by making intrinsic spaces ("spectra") and associated rings the primary objects of study. To that end, he developed the theory ofschemes that informally can be thought of astopological spaces on which acommutative ring is associated to every open subset of the space. Schemes have become the basic objects of study for practitioners of modern algebraic geometry. Their use as a foundation allowed geometry to absorb technical advances from other fields.[91]
Hisgeneralization of the classicalRiemann–Roch theorem related topological properties of complexalgebraic curves to their algebraic structure and now bears his name, being called "the Grothendieck–Hirzebruch–Riemann–Roch theorem". The tools he developed to prove this theorem started the study ofalgebraic andtopological K-theory, which explores the topological properties of objects by associating them with rings.[92] After direct contact with Grothendieck's ideas at theBonn Arbeitstagung, topological K-theory was founded byMichael Atiyah andFriedrich Hirzebruch.[93]
TheWeil conjectures were formulated in the later 1940s as a set of mathematical problems inarithmetic geometry. They describe properties of analytic invariants, calledlocal zeta functions, of the number of points on an algebraic curve or variety of higher dimension. Grothendieck's discovery of theℓ-adic étale cohomology, the first example of aWeil cohomology theory, opened the way for a proof of the Weil conjectures, ultimately completed in the 1970s by his studentPierre Deligne.[92] Grothendieck's large-scale approach has been called a "visionary program".[94] The ℓ-adic cohomology then became a fundamental tool for number theorists, with applications to theLanglands program.[95]
Grothendieck's conjectural theory ofmotives was intended to be the "ℓ-adic" theory but without the choice of "ℓ", a prime number. It did not provide the intended route to the Weil conjectures, but has been behind modern developments inalgebraic K-theory,motivic homotopy theory, andmotivic integration.[96] This theory,Daniel Quillen's work, and Grothendieck's theory ofChern classes, are considered the background to the theory ofalgebraic cobordism, another algebraic analogue of topological ideas.[97]
Grothendieck's emphasis on the role ofuniversal properties across varied mathematical structures broughtcategory theory into the mainstream as an organizing principle for mathematics in general. Among its uses, category theory creates a common language for describing similar structures and techniques seen in many different mathematical systems.[98] His notion ofabelian category is now the basic object of study inhomological algebra.[99] The emergence of a separate mathematical discipline of category theory has been attributed to Grothendieck's influence, although unintentional.[100]
Colonel Lágrimas (Colonel Tears in English), a novel by Puerto Rican–Costa Rican writer Carlos Fonseca is about Grothendieck.[101]
TheBenjamín Labatut bookWhen We Cease to Understand the World dedicates one chapter to the work and life of Grothendieck, introducing his story by reference to the Japanese mathematicianShinichi Mochizuki. The book is a lightly fictionalized account of the world of scientific inquiry and was a finalist for theNational Book Award.[102]
Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces].Memoirs of the American Mathematical Society Series (in French).16. Providence: American Mathematical Society.MR0075539.OCLC9308061.
^abLuca 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.
^Peixoto, Tatiana; Bietenholza, Wolfgang (2016). "To the Memory of Alexander Grothendieck: a Great and Mysterious Genius of Mathematics".arXiv:1605.08112 [math.HO].
^Grothendieck, Alexander (2015).Suugakusha no kodokuna bōken : suugaku to jiko no hakken eno tabi [The Solitary Adventures of a Mathematician: A Journey into Mathematics and Self-Discovery] (in Japanese). Translated by Tsuji Yuichi (2nd ed.). Kyoto: Gendai Sūgaku-sha.
^Grothendieck, Alexander (2015).Sūgaku to hadaka no ōsama: Aru yume to sūgaku no maisō [Mathematics and the Naked King: A Dream and the Burial of Mathematics] (in Japanese). Translated by Tsuji Yuichi (2nd ed.). Kyoto: Gendai Sūgaku-sha.
^Grothendieck, Alexander (2016).Aru yume to sūgaku no maisō: In to yō no kagi [A Dream and the Burial of Mathematics: The Key to Yin and Yang] (in Japanese). Translated by Tsuji Yuichi (2nd ed.). Kyoto: Gendai Sūgaku-sha.
^Grothendieck, Alexander (1998).Maisō (3) aruiwa yottsu no sōsa [Burial (3) or Four Operations] (Unpublished manuscript) (in Japanese). Translated by Tsuji Yuichi.
^Peck, Morgen (31 January 2007)."Equality of Mathematicians".ScienceLine.Alexandre Grothendieck is arguably the most important mathematician of the 20th century...
^Leith 2004: "[A] mathematician of staggering accomplishment... a legendary figure in the mathematical world."
^Marquis, Jean-Pierre (2015). "Category Theory". In Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy (Winter 2015 ed.). Metaphysics Research Lab, Stanford University.
^S. Gelfand; Yuri Manin (1988).Methods of homological algebra. Springer.
Colmez, Pierre; Jean-Pierre, Serre, eds. (2004).Grothendieck-Serre Correspondence: Bilingual Edition. AMS and the Société Mathématique de France. p. 600.ISBN978-1-4704-6939-9.
Grothendieck, Alexander (2022).Récoltes et Semailles : réflexions et témoignage sur un passé de mathématicien (in French). Paris, France:Gallimard.ISBN978-2073004833.
Kleinert, Werner (2007). "Wer ist Alexander Grothendieck? Anarchie, Mathematik, Spiritualität. Eine Biographie. Teil 1: Anarchie" [Who is Alexander Grothendieck? Anarchy, mathematics, spirituality. A biography. Part 1: Anarchy.].zbMATH Open.Zbl1129.01018. A review of the German edition
Schneps, Leila, ed. (2014),Alexandre Grothendieck: A Mathematical Portrait, Somerville Massachusetts: International Press of Boston, Inc.,ISBN978-1-57146-282-4
Centre for Grothendieckian Studies (CSG) is a research centre of the Grothendieck Institute, with a dedicated mission to honour the memory of Alexander Grothendieck.
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