Biography:Alexander Grothendieck
Alexander Grothendieck  

Alexander Grothendieck in Montreal, 1970  
Born  Berlin, Prussia, Germany  28 March 1928
Died  13 November 2014 (aged 86) SaintLizier, Ariège, France 
Nationality 

Alma mater 

Known for  Renewing algebraic geometry and synthesis between it and number theory and topology List of things named after Alexander Grothendieck 
Awards 

Scientific career  
Fields  Mathematics – functional analysis, algebraic geometry, homological algebra 
Institutions 

Thesis  Produits tensoriels topologiques et espaces nucléaires (1953) 
Doctoral advisors  
Doctoral students 
Alexander Grothendieck (/ˈɡroʊtəndiːk/; German pronunciation: [ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk] (listen); French: [ɡʁɔtɛndik]; 28 March 1928 – 13 November 2014) was a French mathematician who became the leading figure in the creation of modern algebraic geometry.^{[7]}^{[8]} 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 socalled "relative" perspective led to revolutionary advances in many areas of pure mathematics.^{[7]}^{[9]} He is considered by many to be the greatest mathematician of the twentieth century.^{[10]}^{[11]}
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 Ktheory.^{[12]} He later became professor at the University of Montpellier^{[5]} 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 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.^{[14]}
Life
Family and childhood
Grothendieck was born in Berlin 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.^{[loweralpha 1]} 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 evade Nazism. His mother followed soon thereafter. Grothendieck was left in the care of Wilhelm Heydorn, a Lutheran pastor and teacher in Hamburg.^{[17]}^{[18]} According to Winfried Scharlau, during this time, his parents took part in the Spanish Civil War as noncombatant auxiliaries.^{[19]}^{[20]} However, others state that Schapiro fought in the anarchist militia.^{[21]}
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.^{[22]} He and his mother were then interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners."^{[23]} The first camp was the Rieucros Camp, where his mother contracted the tuberculosis that would eventually cause her death in 1957. While there, Grothendieck managed to attend the local school, at Mendel. Once, he managed to escape from the camp, intending to assassinate Hitler.^{[22]} Later, his mother Hanka was transferred to the Gurs internment camp for the remainder of World War II.^{[22]} Grothendieck was permitted to live separated from his mother.^{[24]}
In the village of Le ChambonsurLignon, 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.^{[22]}^{[24]}
His father was arrested under the Vichy antiJewish legislation, and sent to the Drancy internment camp, 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.^{[8]}^{[25]}
In Le Chambon, Grothendieck attended the Collège Cévenol (now known as the Le CollègeLycée Cévenol International), a unique secondary school founded in 1938 by local Protestant pacifists and antiwar 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".
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.^{[27]} 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.^{[17]}
Initially, Grothendieck attended Henri Cartan's Seminar at École Normale Supérieure, but he lacked the necessary background to follow the highpowered 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. He showed his new student his latest paper; 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.^{[28]}
In Nancy, he wrote his dissertation under those two professors on functional analysis, from 1950 to 1953.^{[29]} At this time he was a leading expert in the theory of topological vector spaces.^{[30]} 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  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.^{[31]}^{[32]} 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".^{[33]}
In 1957 he was invited to visit Harvard by Oscar 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.^{[34]}
Comparing Grothendieck during his Nancy years to the École Normale Supérieuretrained students at that time (Pierre Samuel, Roger Godement, René Thom, Jacques Dixmier, Jean Cerf, Yvonne Bruhat, JeanPierre 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.^{[35]}
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 EinsteinPodolskyRosen paradox in quantum physics.^{[36]}
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.^{[1]} 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.^{[37]}
Officially during this time, he had as students Michel Demazure (who worked on SGA3, on group schemes), Luc Illusie (cotangent complex), Michel Raynaud, JeanLouis Verdier (cofounder 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"
Alexander 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.^{[29]} His first (preIHÉS) 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 Ktheory. 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.^{[17]} 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 (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 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.^{[38]}
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.^{[17]}
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.^{[39]} 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.^{[40]} 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 ("longterm guest") at the IHÉS, wrote a piece about Grothendieck for a special volume published on the occasion of the IHÉS's fortieth anniversary.^{[41]} 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 BuressurYvette 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 mandarin of the scientific world.^{[1]} 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.^{[loweralpha 2]} Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics.^{[41]}
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.^{[42]} Grothendieck devoted the next three years to this group and served as the main editor of its bulletin.^{[5]}
Although Grothendieck continued with mathematical enquiries, his standard mathematical career mostly ended when he left the IHÉS.^{[8]} After leaving the IHÉS, Grothendieck became a temporary professor at Collège de France for two years.^{[42]} 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.^{[5]}
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 1600page handwritten manuscript containing many of the ideas that led to the Esquisse d'un programme.^{[43]} 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 600page 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, diarylike 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.^{[7]} Much of this work anticipated the subsequent development during the mid1990s 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")^{[43]} 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 twovolumes 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 Bertinitype theorems (EGA V, published in Ulam Quarterly in 1992–1993 and later made available on the Grothendieck Circle web site in 2004).
In the 1,000page autobiographical manuscript, Récoltes et semailles (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]} The Récoltes et semailles work is now available on the internet in the French original,^{[44]} and an English translation is underway. 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".^{[45]}^{[46]}^{[47]}^{[48]}^{[49]}^{[50]} Parts of Récoltes et semailles have been translated into Spanish,^{[51]} as well as into a Russian translation that was published in Moscow.^{[52]} The French original was finally published in two volumes in January 2022, with additional texts by people of various professions who discuss certain aspects of the book.^{[53]}
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."^{[54]}
La Clef des Songes,^{[55]} a 315page manuscript written in 1987, is Grothendieck's account of how his consideration of the source of dreams led him to conclude that a deity exists.^{[56]} 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.^{[5]} The only mathematician on his list was Bernhard Riemann.^{[57]} 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.^{[5]} 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.^{[7]}
The Grothendieck Festschrift, published in 1990, was a threevolume collection of research papers to mark his sixtieth birthday in 1988.^{[58]}
More than 20,000 pages of Grothendieck's mathematical and other writings are held at the University of Montpellier and remain unpublished.^{[59]} They have been digitized for preservation and are freely available in open access through the Institut Montpelliérain Alexander Grothendieck portal.^{[60]}^{[61]}
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.^{[5]} Very few people visited him afterward.^{[62]} Local villagers helped sustain him with a more varied diet after he tried to live on a staple of dandelion soup.^{[63]} 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".^{[64]} 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.^{[65]}
In January 2010, Grothendieck wrote the letter entitled "Déclaration d'intention de nonpublication" 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.^{[66]} He characterized a website devoted to his work as "an abomination".^{[67]} His dictate may have been reversed in 2010.^{[68]}
On 13 November 2014, aged 86, Grothendieck died in the hospital of SaintGirons, Ariège.^{[26]}^{[69]}
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.^{[2]}^{[3]}^{[4]} 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).^{[1]}^{[4]}^{[15]} He eventually applied for French citizenship in the early 1980s, after he was well past the age that exempted him from military service.^{[1]}
Family
Grothendieck was very close to his mother to whom he dedicated his dissertation. She died in 1957 from the tuberculosis that she contracted in camps for displaced persons.^{[42]}
He had five children: a son with his landlady during his time in Nancy;^{[1]} three children, Johanna (1959), Alexander (1961), and Mathieu (1965) with his wife Mireille Dufour;^{[5]}^{[34]} and one child with Justine Skalba, with whom he lived in a commune in the early 1970s.^{[5]}
Mathematical work
Grothendieck's early mathematical work was in functional analysis. Between 1949 and 1953 he worked on his doctoral thesis in this subject at Nancy, supervised by Jean Dieudonné and Laurent Schwartz. His key contributions include topological tensor products of topological vector spaces, the theory of nuclear spaces as foundational for Schwartz distributions, and the application of L^{p} spaces in studying linear maps between topological vector spaces. In a few years, he had become a leading authority on this area of functional analysis—to the extent that Dieudonné compares his impact in this field to that of Banach.^{[70]}
It is, however, in algebraic geometry and related fields where Grothendieck did his most important and influential work. From approximately 1955 he started to work on sheaf theory and homological algebra, producing the influential "Tôhoku paper" (Sur quelques points d'algèbre homologique, published in the Tohoku Mathematical Journal in 1957) where he introduced abelian categories and applied their theory to show that sheaf cohomology may be defined as certain derived functors in this context.^{[17]}
Homological methods and sheaf theory had already been introduced in algebraic geometry by JeanPierre Serre^{[71]} and others, after sheaves had been defined by Jean 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 his relative point of view (pairs of varieties related by a morphism), allowing a broad generalization of many classical theorems.^{[42]} The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finitedimensional; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent; this reduces to Serre's theorem over a onepoint space.
In 1956, he applied the same thinking to the Riemann–Roch theorem, which recently had been generalized to any dimension by Hirzebruch. The Grothendieck–Riemann–Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957.^{[42]} It appeared in print in a paper written by Armand 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 in Claude Chevalley's seminar. He outlined his programme in his talk at the 1958 International 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 nonclosed generic points, which led to the theory of schemes. Grothendieck also pioneered the systematic use of nilpotents. As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings. His theory 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 use birational geometry, techniques from number theory, Galois theory, commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.^{[17]}^{[72]}^{[73]}
Grothendieck is noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation.^{[37]} Relatively little of his work after 1960 was published by the conventional route of the learned 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 of Dmodules. Although lauded as "the Einstein of mathematics", his work also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems.^{[74]}^{[75]}
EGA, SGA, FGA
The bulk of Grothendieck's published work is collected in the monumental, yet incomplete, Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA). The collection, Fondements de la Géometrie Algébrique (FGA), which gathers together talks given in the Séminaire Bourbaki, also contains important material.^{[17]}
Grothendieck's work includes the invention of the étale and ladic cohomology theories, which explain an observation made by André Weil that argued for a connection between the topological characteristics of a variety and its diophantine (number theoretic) properties.^{[42]} For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex 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 the Weil conjectures, the last of which was settled by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics.^{[17]}
Major mathematical contributions
In Grothendieck's retrospective Récoltes et Semailles, he identified twelve of his contributions that he believed qualified as "great ideas".^{[76]} In chronological order, they are:
 Topological tensor products and nuclear spaces
 "Continuous" and "discrete" duality (derived categories, "six operations")
 Yoga of the Grothendieck–Riemann–Roch theorem Ktheory relation with intersection theory
 Schemes
 Topoi
 Étale cohomology and ladic cohomology
 Motives and the motivic Galois group (Grothendieck ⊗categories)
 Crystals and crystalline cohomology, yoga of "de Rham coefficients", "Hodge coefficients"...
 "Topological algebra": ∞stacks, derivators; cohomological formalism of topoi as inspiration for a new homotopical algebra
 Tame topology
 Yoga of anabelian algebraic geometry, Galois–Teichmüller theory
 "Schematic" or "arithmetic" point of view for regular polyhedra and regular configurations of all kinds
Here the term yoga denotes a kind of "metatheory" that may be used heuristically; Michel Raynaud writes the other terms "Ariadne's thread" and "philosophy" as effective equivalents.^{[77]}
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.^{[78]}
Influence
Grothendieck is considered by many to be the greatest mathematician of the twentieth century.^{[11]} In an obituary David Mumford and John 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,^{[79]} but influenced logic, in the field of categorical logic.^{[80]}
Geometry
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 as algebraic curves and surfaces, through the study of the algebraic equations for those objects. Properties of algebraic equations are in turn studied using the techniques of ring 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 of schemes that informally can be thought of as topological spaces on which a commutative 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.^{[81]}
His generalization of the classical Riemann–Roch theorem related topological properties of complex algebraic 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 of algebraic and topological Ktheory, which explores the topological properties of objects by associating them with rings.^{[82]} After direct contact with Grothendieck's ideas at the Bonn Arbeitstagung, topological Ktheory was founded by Michael Atiyah and Friedrich Hirzebruch.^{[83]}
Cohomology theories
Grothendieck's construction of new cohomology theories, which use algebraic techniques to study topological objects, has influenced the development of algebraic number theory, algebraic topology, and representation theory. As part of this project, his creation of topos theory, a categorytheoretic generalization of pointset topology, has influenced the fields of set theory and mathematical logic.^{[79]}
The Weil conjectures were formulated in the later 1940s as a set of mathematical problems in arithmetic geometry. They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher dimension. Grothendieck's discovery of the ℓadic étale cohomology, the first example of a Weil cohomology theory, opened the way for a proof of the Weil conjectures, ultimately completed in the 1970s by his student Pierre Deligne.^{[82]} Grothendieck's largescale approach has been called a "visionary program".^{[84]} The ℓadic cohomology then became a fundamental tool for number theorists, with applications to the Langlands program.^{[85]}
Grothendieck's conjectural theory of motives was intended to be the "ℓadic" 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 in algebraic Ktheory, motivic homotopy theory, and motivic integration.^{[86]} This theory, Daniel Quillen's work, and Grothendieck's theory of Chern classes, are considered the background to the theory of algebraic cobordism, another algebraic analogue of topological ideas.^{[87]}
Category theory
Grothendieck's emphasis on the role of universal properties across varied mathematical structures brought category 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.^{[88]} His notion of abelian category is now the basic object of study in homological algebra.^{[89]} The emergence of a separate mathematical discipline of category theory has been attributed to Grothendieck's influence, although unintentional.^{[90]}
In popular culture
The novel Colonel Lágrimas (Colonel Tears in English, available by Restless Books) by Puerto Rican–Costa Rican writer Carlos Fonseca is a semibiographic novel about Grothendieck.^{[91]}
The band Stone Hill All Stars have a song named after Alexander Grothendieck.^{[92]}
In the novel When We Cease to Understand the World, Benjamin Labatut dedicates one chapter to the story of Grothendieck.^{[93]}
In the novel The Passenger and its sequel Stella Maris by Cormac McCarthy, one of the main characters is a student of Grothendieck. ^{[94]}
Publications
 Template:Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires
 Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. New York: Gordon and Breach Science Publishers. ISBN 9780677300207. OCLC 886098. https://archive.org/details/topologicalvecto0000grot.
See also
 ∞groupoid
 λring
 AB5 category
 Abelian category
 Accessible category
 Algebraic geometry
 Algebraic stack
 Approximation property – Mathematical concept
 Barsotti–Tate group
 Chern class
 Crystal (mathematics)
 Crystalline cohomology
 Deltafunctor
 Derivator
 Derived category – Homological construction
 Descent (mathematics)
 Dévissage
 DFspace
 Dunford–Pettis property
 Effaceable functor
 Excellent ring
 Fibred category – Concept in category theory
 Formally smooth map
 Fundamental groupoid
 Fundamental group scheme
 Gorenstein ring – Local ring in commutative algebra
 Grothendieck's Tôhoku paper – On the foundations of homological algebra
 Ktheory
 Hilbert scheme
 Homotopy hypothesis
 Infinitesimal cohomology
 List of things named after Alexander Grothendieck
 Local cohomology – Concept in algebraic geometry
 Nakai conjecture
 Motive (algebraic geometry) – Structure for unifying cohomology theories
 Nuclear operator
 Nuclear space
 Parafactorial local ring
 Projective tensor product
 Pursuing Stacks – Seminal math text
 Quasifinite morphism
 Quot scheme
 Ramanujam–Samuel theorem – Conditions for a divisor of a local ring to be principal
 Scheme (mathematics)
 Section conjecture
 Semistable abelian variety
 Sheaf cohomology
 Stack (mathematics)
 Standard conjectures on algebraic cycles
 Sketch of a program
 Tannakian formalism
 Theorem of absolute purity
 Theorem on formal functions
 Ultrabornological space
 Weil conjectures
 Vector bundles on algebraic curves
 Zariski's main theorem
Notes
 ↑ Testimony by Pierre Cartier asserts that his mother was of Jewish German descent: "what I know of his life comes from Grothendieck himself".^{[15]}
 ↑ Ruelle invented the concept of a strange attractor in a dynamical system and, with the Dutch mathematician Floris Takens, produced a new model for turbulence during the 1970s.
References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} Cartier 2004.
 ↑ ^{2.0} ^{2.1} ^{2.2} Douroux 2012.
 ↑ ^{3.0} ^{3.1} Cartier 2004, p. 10, footnote 12.
 ↑ ^{4.0} ^{4.1} ^{4.2} Kleinert 2007.
 ↑ ^{5.0} ^{5.1} ^{5.2} ^{5.3} ^{5.4} ^{5.5} ^{5.6} ^{5.7} ^{5.8} Scharlau 2008.
 ↑ Cartier et al. 2007, p. 7.
 ↑ ^{7.0} ^{7.1} ^{7.2} ^{7.3} ^{7.4} Jackson 2004b.
 ↑ ^{8.0} ^{8.1} ^{8.2} Bruce Weber; Julie Rehmeyer (14 November 2014). "Alexander Grothendieck, Math Enigma, Dies at 86". The New York Times. https://www.nytimes.com/2014/11/16/world/europe/alexandergrothendieckmathenigmadiesat86.html.
 ↑ Mumford, David; Tate, John (2015). "Alexander Grothendieck (1928–2014) Mathematician who rebuilt algebraic geometry". Nature 517 (7534): 272. doi:10.1038/517272a. ISSN 00280836. PMID 25592527. Bibcode: 2015Natur.517..272M.
 ↑ "Guardian obituary". https://www.independent.co.uk/news/world/europe/greatestmathematicianofthe20thcenturyandthemostimportantscientificmindyouveneverheardofalexandergrothendieckdiesaged869862306.html.
 ↑ ^{11.0} ^{11.1} ^{11.2} Alexander Grothendieck obituary by David Mumford and John Tate David Mumford at Brown and Harvard Universities: Archive for Reprints: Can one explain schemes to biologists, 14 December 2014
 ↑ "Fields Medals 1966". https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1966/index.html.
 ↑ Scharlau, Winfried. "Who is Alexander Grothendieck? Anarchy, Mathematics, Spirituality, Solitude". https://webusers.imjprg.fr/~leila.schneps/grothendieckcircle/Spirituality/SpiritualityForeword.pdf.
 ↑ Ruelle 2007, p. 40.
 ↑ ^{15.0} ^{15.1} Cartier 2001.
 ↑ ^{16.0} ^{16.1} "The Early Background of Genius". http://www.siam.org/news/news.php?id=1405.
 ↑ ^{17.0} ^{17.1} ^{17.2} ^{17.3} ^{17.4} ^{17.5} ^{17.6} ^{17.7} ^{17.8} ^{17.9} Jackson 2004a.
 ↑ Philippe Douroux (6 May 2019). "Trésor scientifique ou vieux papiers illisibles? Les mystérieuses archives d'Alexandre Grothendieck" (in fr). https://www.lemonde.fr/sciences/article/2019/05/06/lesarchivesinsaisissablesdalexandregrothendieck_5459049_1650684.html.
 ↑ Scharlau 2008, p. 931.
 ↑ Scharlau n.d., p. 2: "Beide beteiligten sich am Spanischen Bürgerkrieg, nicht aktiv kämpfend, aber unterstützend."
 ↑ Hersh & JohnSteiner 2011, p. 109.
 ↑ ^{22.0} ^{22.1} ^{22.2} ^{22.3} Amir D. Aczel,The Artist and the Mathematician, Basic Books, 2009 pp.8ff.pp.8–15.
 ↑ 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 ppxixxviii p.xii.
 ↑ ^{24.0} ^{24.1} 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.
 ↑ Ruelle 2007, p. 35.
 ↑ ^{26.0} ^{26.1} "Alexandre Grothendieck, ou la mort d'un génie qui voulait se faire oublier" (in fr). Libération Sciences. 13 November 2014. http://www.liberation.fr/sciences/2014/11/13/alexandregrothendieckoulamortdungeniequivoulaitsefaireoublier_1142614.
 ↑ Philippe Douroux (February 8, 2012). "Alexandre Grothendieck: Un voyage à la poursuite des choses évidentes" (in fr). CNRS. http://images.math.cnrs.fr/AlexandreGrothendieck.html#nh14.
 ↑ Peixoto, Tatiana; Bietenholza, Wolfgang (2016). "To the Memory of Alexander Grothendieck: a Great and Mysterious Genius of Mathematics". arXiv:1605.08112 [math.HO].
 ↑ ^{29.0} ^{29.1} Cartier et al. 2007, "Foreword".
 ↑ Horvâth, John (July 1976). "Topological vector spaces, by A. Grothendieck, ...". Bulletin of the American Mathematical Society 82 (4): 515–521. doi:10.1090/S000299041976140761. https://www.ams.org/journals/bull/19768204/S000299041976140761/S000299041976140761.pdf.
 ↑ Schneps n.d.
 ↑ Colmez & Serre 2004.
 ↑ Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique" (in fr), Tohoku Mathematical Journal, Second Series 9 (2): 119–221, doi:10.2748/tmj/1178244839, ISSN 00408735
 ↑ ^{34.0} ^{34.1} Hersh & JohnSteiner 2011, p. 113.
 ↑ "Chapter 3. From student to celebrity: 19491952". Who Is Alexandre Grothendieck: Anarchy, Mathematics, Spirituality. 2. http://webusers.imjprg.fr/~leila.schneps/grothendieckcircle/Mathematics/chap3.pdf.
 ↑ Guillaume Aubrun (March 17, 2020). "1953 : un « Résumé » aux développements illimités" (in fr). CNRS. https://images.math.cnrs.fr/1953unResumeauxdeveloppementsillimites.html.
 ↑ ^{37.0} ^{37.1} Amir D. Aczel (2009). The Artist and the Mathematician. Basic Books.
 ↑ Lipman, Joseph (2009). "Notes on derived categories and Grothendieck duality". Foundations of Grothendieck Duality for Diagrams of Schemes. Lecture Notes in Mathematics. 1960. New York: SpringerVerlag. pp. 1–259. doi:10.1007/9783540854203. ISBN 9783540854197. http://www.math.purdue.edu/~lipman/Duality.pdf.
 ↑ The Life and Work of Alexander Grothendieck, American Mathematical Monthly, vol. 113, no. 9, footnote 6.
 ↑ SGA1, Springer Lecture Notes 224, p. xii, xiii
 ↑ ^{41.0} ^{41.1} Jackson, Allyn (March 1999). "The IHÉS at Forty". Notices of the AMS 46 (3): 329–337. https://www.ams.org/notices/199903/iheschanges.pdf.
 ↑ ^{42.0} ^{42.1} ^{42.2} ^{42.3} ^{42.4} ^{42.5} Pragacz 2005.
 ↑ ^{43.0} ^{43.1} Alexandre Grothendieck, Esquisse d'un Programme, English translation
 ↑ Grothendieck 1986.
 ↑ Roy Lisker. "Visiting Alexandre Grothendieck". https://www.fermentmagazine.org/quest88.
 ↑ Scharlau, Winfried. "Chapter 23. Récoltes et Semailles". https://webusers.imjprg.fr/~leila.schneps/grothendieckcircle/Spirituality/Spirituality23.pdf.
 ↑ Grothendieck, Alexander (2015) (in ja). Suugakusha no kodokuna bōken : suugaku to jiko no hakken eno tabi (2nd ed.). Kyoto: Gendai Sūgakusha.
 ↑ Grothendieck, Alexander (2015) (in ja). Sūgaku to hadaka no ōsama: Aru yume to sūgaku no maisō (2nd ed.). Kyoto: Gendai Sūgakusha.
 ↑ Grothendieck, Alexander (2016) (in ja). Aru yume to sūgaku no maisō: In to yō no kagi (2nd ed.). Kyoto: Gendai Sūgakusha.
 ↑ Grothendieck, Alexander (1998) (in ja). Maisō (3) aruiwa yottsu no sōsa (Unpublished manuscript).
 ↑ "Récoltes et Semailles; La Clef des Songes" (in es). http://matematicas.unex.es/~navarro/res/.
 ↑ "Free books: Récoltes et semailles". http://www.mccme.ru/freebooks/grothendieck/RS.html.
 ↑ "Parution de « Récoltes et semailles » d'Alexandre Grothendieck" (in French). IHES. 13 January 2022. https://www.ihes.fr/parutionrecoltessemaillesgrothendieck/.
 ↑ "Crafoord Prize letter, English translation". http://www.math.columbia.edu/~lipyan/CrafoordPrize.pdf.
 ↑ Grothendieck, Alexander. "La Clef des Songes". http://cm2vivi2002.free.fr/AGbiblio/AGclesonges.pdf.
 ↑ Scharlau 2008, p. 940.
 ↑ Scharlau, Winfried (in de), Die Mutanten – Les Mutants – eine Meditation von Alexander Grothendieck, http://www.scharlauonline.de/DOKS/Die%20Mutanten.pdf
 ↑ Cartier et al. 2007.
 ↑ Le trésor oublié du génie des maths (in French)
 ↑ Les «gribouillis» d'Alexandre Grothendieck enfin sauvegardés (in French)
 ↑ "IMAG" (in fr). https://grothendieck.umontpellier.fr/archivesgrothendieck/.
 ↑ Galchen, Rivka (May 9, 2022). "The Mysterious Disappearance revolutionary mathematician". The New Yorker. https://www.newyorker.com/magazine/2022/05/16/themysteriousdisappearanceofarevolutionarymathematician.
 ↑ John Derbyshire (2006). Unknown Quantity: A Real and Imaginary History of Algebra. National Academies Press. p. 314. ISBN 9780309164801. https://books.google.com/books?id=_hsWNzBvr3oC&pg=PA314.
 ↑ Leith, Sam (20040320). "The Einstein of maths". The Spectator. http://www.spectator.co.uk/features/12036/theeinsteinofmaths/.
 ↑ Stéphane Foucart; Philippe Pajot (14 November 2014). "Alexandre Grothendieck, le plus grand mathématicien du XXe siècle, est mort" (in fr). Le Monde. http://www.lemonde.fr/disparitions/article/2014/11/14/lemathematicienalexandregrothendieckestmort_4523482_3382.html.
 ↑ "Grothendieck's letter". Secret Blogging Seminar. 20100209. https://sbseminar.wordpress.com/2010/02/09/grothendiecksletter.
 ↑ "Grothendieck Circle". http://webusers.imjprg.fr/~leila.schneps/grothendieckcircle/.
 ↑ "Réédition des SGA". http://www.math.upsud.fr/~laszlo/sga4.html.
 ↑ "Alexander Grothendieck  obituary". http://www.telegraph.co.uk/news/obituaries/11231703/AlexanderGrothendieckobituary.html.
 ↑ Dieudonné 2007.
 ↑ Serre 1955.
 ↑ Deligne 1998.
 ↑ McLarty, Colin. "The Rising Sea: Grothendieck on simplicity and generality I". https://webusers.imjprg.fr/~leila.schneps/grothendieckcircle/Mathbiographies/mclarty1.pdf.
 ↑ Peck, Morgen (31 January 2007). "Equality of Mathematicians". http://scienceline.org/2007/01/math_controversy_peck/. "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."
 ↑ Grothendieck 1986, p. 21.
 ↑ Michel Raynaud (October 2003). "Correspondance GrothendieckSerre". Notices of the AMS 50 (9): 1086. https://www.ams.org/notices/200309/revraynaud.pdf.
 ↑ Grothendieck 1986, p. 22.
 ↑ ^{79.0} ^{79.1} Saunders Mac Lane; Ieke Moerdijk (1992). Sheaves in Geometry and Logic: a First Introduction to Topos Theory. SpringerVerlag New York Inc.. ISBN 0387977104.
 ↑ Dov M. Gabbay; Akihiro Kanamori; John Woods, Jr. (2012). Sets and Extensions in the Twentieth Century. Elsevier. p. 733. ISBN 9780444516213. https://books.google.com/books?id=ZF_QckMFyoC&pg=PA733.
 ↑ Miles Reid (15 December 1988). Undergraduate Algebraic Geometry. Cambridge University Press. p. 115. ISBN 9780521356626. https://archive.org/details/undergraduatealg0000reid.
 ↑ ^{82.0} ^{82.1} Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: SpringerVerlag, ISBN 9780387902449
 ↑ Michael Atiyah (3 April 2014). Michael Atiyah Collected Works: Volume 7: 2002–2013. Oxford University Press. pp. 383–. ISBN 9780199689262. https://books.google.com/books?id=Rm6VAwAAQBAJ&pg=PA383.
 ↑ M. Ram Murty; V. Kumar Murty (6 October 2012). The Mathematical Legacy of Srinivasa Ramanujan. Springer Science & Business Media. pp. 156–. ISBN 9788132207696. https://books.google.com/books?id=r4tGFgO_nc8C&pg=PA156.
 ↑ R. P. Langlands, Modular forms and ladic representations, Lecture Notes in Math. 349. (1973), 361—500
 ↑ J.S. Milne (1980). Étale cohomology. Princeton University Press.
 ↑ Marc Levine; Fabien Morel (23 February 2007). Algebraic Cobordism. Springer Science & Business Media. p. viii. ISBN 9783540368243. https://books.google.com/books?id=obVAAAAAQBAJ&pg=PR8.
 ↑ Marquis, JeanPierre (2015). Zalta, Edward N.. ed. The Stanford Encyclopedia of Philosophy (Winter 2015 ed.). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/win2015/entries/categorytheory/.
 ↑ S. Gelfand; Yuri Manin (1988). Methods of homological algebra. Springer.
 ↑ Ralph Krömer (25 June 2007). Tool and Object: A History and Philosophy of Category Theory. Springer Science & Business Media. pp. 158–. ISBN 9783764375249. https://books.google.com/books?id=41bHxtHxjUAC&pg=PA158.
 ↑ "Colonel Lágrimas" (in enUS). http://www.restlessbooks.com/bookstore/colonellagrimas.
 ↑ "Alexander Grothendieck" (in enUS). https://youtube.com/watch?v=rfAuCOT5z6U.
 ↑ Labatut, Benjamín (2020). When we cease to understand the world. New York, NY. ISBN 9781681375663.
 ↑ "CORMAC MCCARTHY HAS NEVER BEEN BETTER". https://www.theatlantic.com/magazine/archive/2023/01/cormacmccarthythepassengerstellamarisbookreview/672234/.
Sources and further reading
 Grothendieck, Alexander (1986) (in fr). Récoltes et semailles: réflexions et témoignage sur un passé de mathématicien. Paris: Gallimard. ISBN 9782072889806. http://lipn.univparis13.fr/~duchamp/Books&more/Grothendieck/RS/pdf/RetS.pdf.
 Michael Artin; Allyn Jackson; David Mumford; John Tate; ((Coordinating Editors)) (March 2016). "Alexandre Grothendieck 1928–2014, Part 1". Notices of the American Mathematical Society 63 (3): 242–255. doi:10.1090/noti1336. https://www.ams.org/publications/journals/notices/201603/rnotip242.pdf.
 Michael Artin; Allyn Jackson; David Mumford; John Tate; ((Coordinating Editors)) (April 2016). "Alexandre Grothendieck 1928–2014, Part 2". Notices of the American Mathematical Society 63 (4): 401–2413. doi:10.1090/noti1361. https://www.ams.org/publications/journals/notices/201604/rnotip401.pdf.
 Cartier, Pierre (1998), "La folle journée, de Grothendieck à Connes et Kontsevich – Évolution des notions d'espace et de symétrie", Les relations entre les mathématiques et la physique théorique – Festschrift for the 40th anniversary of the IHÉS, S88, Institut des Hautes Études Scientifiques, pp. 11–19, http://www.numdam.org/item?id=PMIHES_1998__S88__23_0
 Cartier, Pierre (2001). "A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry". Bulletin of the American Mathematical Society 38 (4): 389–408. doi:10.1090/S0273097901009132. https://www.ams.org/bull/20013804/S0273097901009132/S0273097901009132.pdf. English translation of Cartier (1998).
 Cartier, Pierre (2004), "Un pays dont on ne connaîtrait que le nom (Grothendieck et les " motifs ")", in Cartier, Pierre; Charraud, Nathalie (in fr), Réel en mathématiquespsychanalyse et mathématiques, Editions Agalma, http://lipn.univparis13.fr/~duchamp/Books&more/Grothendieck/CartierHerreman2.pdf
 The Grothendieck Festschrift, Volume I: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck. Birkhäuser. 2007. ISBN 9780817645663. https://books.google.com/books?id=_BnSoQSKnNUC.
 Script error: No such module "Harvc".
 Colmez, Pierre; JeanPierre, Serre, eds (2004). GrothendieckSerre Correspondence: Bilingual Edition. AMS and the Société Mathématique de France. p. 600. ISBN 9781470469399.
 Deligne, Pierre (1998), "Quelques idées maîtresses de l'œuvre de A. Grothendieck", Matériaux pour l'histoire des mathématiques au XXe siècle – Actes du colloque à la mémoire de Jean Dieudonné (Nice 1996), Société Mathématique de France, pp. 11–19, http://smf.emath.fr/Publications/SeminairesCongres/1998/3/pdf/smf_semcong_3_1119.pdf
 Douroux, Philippe (20120208). "Alexandre Grothendieck" (in fr). http://images.math.cnrs.fr/AlexandreGrothendieck.html.
 Hersh, Reuben; JohnSteiner, Vera (2011). Loving and Hating Mathematics: Challenging the Myths of Mathematical Life. Princeton University Press. ISBN 9781400836116. https://books.google.com/books?id=gvsHAnAuIp4C&pg=PA109.
 Jackson, Allyn (2004a), "Comme Appelé du Néant – As If Summoned from the Void: The Life of Alexandre Grothendieck I", Notices of the American Mathematical Society 51 (4): 1038–1056, https://www.ams.org/notices/200409/feagrothendieckpart1.pdf
 Jackson, Allyn (2004b), "Comme Appelé du Néant – As If Summoned from the Void: The Life of Alexandre Grothendieck II", Notices of the American Mathematical Society 51 (10): 1196–1212, https://www.ams.org/notices/200410/feagrothendieckpart2.pdf
 Pragacz, Piotr (2005), "Notes on the life and work of Alexander Grothendieck", in Pragracz, Piotr, Topics in Cohomological Studies of Algebraic Varieties: Impanga Lecture Notes, Birkhäuser, http://www.landsburg.com/grothendieck/pragasz.pdf
 Rehmeyer, Julie (9 May 2008), "Sensitivity to the Harmony of Things", Science News, http://www.sciencenews.org/view/generic/id/31898/title/Sensitivity_to_the_harmony_of_things, retrieved 5 August 2008
 Ribenboim, Paulo. "Excerpt from The Grothendieck I Knew: Telling, Not Hiding, Not Judging By Paulo Ribenboim". Notices of the American Mathematical Society (August 2019): 1069–1077. doi:10.1090/noti1909.
 Ruelle, David (2007). The Mathematician's Brain. Princeton University Press. ISBN 9780691129822. https://books.google.com/books?id=B3A1bjOkOaEC.
 Scharlau, Winfried (2007), Wer ist Alexander Grothendieck?: Anarchie, Mathematik, Spiritualität, http://www.scharlauonline.de/ag_1.html First part of planned fourvolume biography.
 Scharlau, Winfried (2011). Who is Alexander Grothendieck? Part 1: Anarchy. Books on Demand. ISBN 9783842340923. OCLC 801767784. English version.
 Scharlau, Winfried (n.d.). "Wer ist Alexander Grothendieck?" (in de). pp. 1–22. http://www.scharlauonline.de/DOKS/Wer%20ist%20AG.pdf.
 Kleinert, Werner (2007). "Wer ist Alexander Grothendieck? Anarchie, Mathematik, Spiritualität. Eine Biographie. Teil 1: Anarchie." (in en). zbMATH Open. A review of the German edition
 Scharlau, Winfried (September 2008). "Who is Alexander Grothendieck ?". Notices of the American Mathematical Society (Providence, RI: American Mathematical Society) 55 (8): 930–941. ISSN 10889477. OCLC 34550461. https://www.ams.org/notices/200808/tx080800930p.pdf. Retrieved 1 September 2011.
 Scharlau, Winfried (2010). "Who is Alexander Grothendieck? Anarchy, Mathematics, Spirituality, Solitude. Part 3: Spirituality" (in en). https://webusers.imjprg.fr/~leila.schneps/grothendieckcircle/Spirituality.html. Third part of planned fourvolume biography; crowdfinanced translation into English.
 Schneps, Leila (n.d.). "Who is Alexander Grothendieck? Anarchy, Mathematics, Spirituality, Solitude. Part 2: Mathematics" (in en). https://webusers.imjprg.fr/~leila.schneps/grothendieckcircle/Mathematics/. First 4 chapters from the incomplete second part of planned fourvolume biography.
 Schneps, Leila, ed. (2014), Alexandre Grothendieck: A Mathematical Portrait, Somerville Massachusetts: International Press of Boston, Inc., ISBN 9781571462824
 Serre, JeanPierre (1955). "Faisceaux algébriques cohérents". Annals of Mathematics 61 (2): 197–278. doi:10.2307/1969915. https://www.collegedefrance.fr/media/jeanpierreserre/UPL5435398796951750634_Serre_FAC.pdf.
External links
 https://www.grothendieckcircle.org is a website devoted to the life and works of Alexandre Grothendieck.
 O'Connor, John J.; Robertson, Edmund F., "Alexander Grothendieck", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/Grothendieck.html.
 Alexander Grothendieck at the Mathematics Genealogy Project
 Séminaire Grothendieck is a peripatetic seminar on Grothendieck view not just on mathematics
 Grothendieck Circle, collection of mathematical and biographical information, photos, links to his writings
 The origins of 'Pursuing Stacks': This is an account of how 'Pursuing Stacks' was written in response to a correspondence in English with Ronnie Brown and Tim Porter at Bangor, which continued until 1991. See also Alexander Grothendieck: some recollections.
 Récoltes et Semailles
 "Récoltes et Semailles" et "La Clef des Songes", French originals and Spanish translations
 English summary of "La Clef des Songes"
 Video of a lecture with photos from Grothendieck's life, given by Winfried Scharlau at IHES in 2009
 Can one explain schemes to biologists —biographical sketch of Grothendieck by David Mumford & John Tate
 Archives Grothendieck
 "Who Is Alexander Grothendieck?, Winfried Scharlau, Notices of the AMS 55(8), 2008.
 "Alexander Grothendieck: A Country Known Only by Name, Pierre Cartier, Notices of the AMS 62(4), 2015.
 Alexandre Grothendieck 1928–2014, Part 1, Notices of the AMS 63(3), 2016.
 A. Grothendieck by Mateo Carmona
 Lesarchivesinsaisissablesdalexandregrothendieck
 Kutateladze S.S. Rebellious Genius: In Memory of Alexander Grothendieck
 AlexandreGrothendieckunemathematiqueencathedralegothique
 Lesarchivesinsaisissablesdalexandregrothendieck
Original source: https://en.wikipedia.org/wiki/Alexander Grothendieck.
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