Alexander Grothendieck

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Alexander Grothendieck
University of Nancy
Known forRenewing algebraic geometry and synthesis between it and number theory and topology
List of things named after Alexander Grothendieck
Awards
(declined)
Scientific career
FieldsMathematics – functional analysis, algebraic geometry, homological algebra
Institutions
ThesisProduits tensoriels topologiques et espaces nucléaires (1953)
Doctoral advisors
Doctoral students

Alexander Grothendieck (

sheaf theory, and category theory to its foundations, while his so-called "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 K-theory.[12] He later became professor at the University 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 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

Protestant German family in Hamburg 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

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 non-combatant 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 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.[22][24]

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 be murdered at the Auschwitz concentration camp in 1942.[8][25]

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.[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

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

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

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.

quantum physics.[36]

IHÉS years

In 1958, Grothendieck was installed at the

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

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 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

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 ("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.[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 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 mandarin of the scientific world.[3] 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.[b] 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.[1]

Although Grothendieck continued with mathematical enquiries, his standard mathematical career mostly ended when he left the IHÉS.

CNRS.[1]

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.[43] It also includes a study of Teichmüller theory
.

In 1983, stimulated by correspondence with

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 mid-1990s of the motivic homotopy theory of Fabien Morel and Vladimir Voevodsky
.

In 1984, Grothendieck wrote the proposal

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 1,000-page 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 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 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.[1] 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.[1] 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 three-volume 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.[1] 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 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.[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 Saint-Girons, Ariège.[26][69]

Citizenship

Grothendieck was born in

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.[4][5][6] 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).[3][6][15] He eventually applied for French citizenship in the early 1980s, after he was well past the age that exempted him from military service.[3]

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;[3] three children, Johanna (1959), Alexander (1961), and Mathieu (1965) with his wife Mireille Dufour;[1][34] and one child with Justine Skalba, with whom he lived in a commune in the early 1970s.[1]

Mathematical work

Grothendieck's early mathematical work was in

Schwartz distributions, and the application of Lp 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

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

higher 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 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 non-closed 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.

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 D-modules. 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,

Séminaire Bourbaki, also contains important material.[17]

Grothendieck's work includes the invention of the

l-adic 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:

  1. Topological tensor products and nuclear spaces
  2. "Continuous" and "discrete"
    six operations
    ")
  3. Yoga of the Grothendieck–Riemann–Roch theorem K-theory relation with intersection theory
  4. Schemes
  5. Topoi
  6. l-adic cohomology
  7. motivic Galois group
    (Grothendieck ⊗-categories)
  8. Crystals and crystalline cohomology, yoga of "de Rham coefficients", "Hodge coefficients"...
  9. "Topological algebra": ∞-stacks, derivators; cohomological formalism of topoi as inspiration for a new homotopical algebra
  10. Tame topology
  11. Yoga of
    Galois–Teichmüller theory
  12. "Schematic" or "arithmetic" point of view for regular polyhedra and regular configurations of all kinds

Here the term yoga 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.[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

Bonn Arbeitstagung, topological K-theory was founded by Michael Atiyah and Friedrich Hirzebruch.[83]

Cohomology theories

Grothendieck's construction of new

point-set topology, has influenced the fields of set theory and mathematical logic.[79]

The

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 large-scale approach has been called a "visionary program".[84] The ℓ-adic 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 "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 in algebraic K-theory, 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

  • 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.
    OCLC 1315788
    .
  • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers.
    OCLC 886098
    .

See also

Notes

  1. ^ Testimony by Pierre Cartier asserts that his mother was of Jewish German descent: "what I know of his life comes from Grothendieck himself".[15]
  2. strange attractor in a dynamical system and, with the Dutch mathematician Floris Takens, produced a new model for turbulence
    during the 1970s.

References

  1. ^ a b c d e f g h i Scharlau 2008.
  2. ^ Cartier et al. 2007, p. 7.
  3. ^ a b c d e f Cartier 2004.
  4. ^ a b c Douroux 2012.
  5. ^ a b Cartier 2004, p. 10, footnote 12.
  6. ^ a b c Kleinert 2007.
  7. ^ a b c d e Jackson 2004b.
  8. ^ a b c Bruce Weber; Julie Rehmeyer (14 November 2014). "Alexander Grothendieck, Math Enigma, Dies at 86". The New York Times. Archived from the original on 1 January 2022.
  9. PMID 25592527
    .
  10. Independent.co.uk
    .
  11. ^ a b c 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
  12. ^ "Fields Medals 1966". mathunion.org. Archived from the original on 22 March 2019. Retrieved 5 January 2022.
  13. ^ Scharlau, Winfried. "Who is Alexander Grothendieck? Anarchy, Mathematics, Spirituality, Solitude" (PDF). Archived (PDF) from the original on 9 October 2022.
  14. ^ Ruelle 2007, p. 40.
  15. ^ a b Cartier 2001.
  16. ^ a b "The Early Background of Genius". Archived from the original on 15 June 2011. Retrieved 15 June 2011.
  17. ^ a b c d e f g h i j Jackson 2004a.
  18. ^ Philippe Douroux (6 May 2019). "Trésor scientifique ou vieux papiers illisibles? Les mystérieuses archives d'Alexandre Grothendieck" [Scientific treasure or unreadable old paper? The mysterious archives of Alexandre Grothendieck]. Le Monde (in French).
  19. ^ Scharlau 2008, p. 931.
  20. ^ Scharlau n.d., p. 2: "Beide beteiligten sich am Spanischen Bürgerkrieg, nicht aktiv kämpfend, aber unterstützend."
  21. ^ Hersh & John-Steiner 2011, p. 109.
  22. ^ a b c d Amir D. Aczel,The Artist and the Mathematician, Basic Books, 2009 pp.8ff.pp.8–15.
  23. ^ Piotr Pragacz, 'Notes on the Life and Work of Alexander Grothendieck,' in Piotr Pragacz (ed.), Topics in Cohomological Studies of Algebraic Varieties: Impanga Lecture Notes, Springer Science & Business Media, 2006 pp-xi-xxviii p.xii.
  24. ^ a b 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.
  25. ^ Ruelle 2007, p. 35.
  26. ^ a b "Alexandre Grothendieck, ou la mort d'un génie qui voulait se faire oublier". Libération Sciences (in French). 13 November 2014. Retrieved 14 November 2014.
  27. ^ Philippe Douroux (8 February 2012). "Alexandre Grothendieck: Un voyage à la poursuite des choses évidentes" [Alexander Grothendieck: A journey in pursuit of the obvious]. Images des mathématiques (in French). CNRS.
  28. arXiv:1605.08112 [math.HO
    ].
  29. ^ a b Cartier et al. 2007, "Foreword".
  30. doi:10.1090/S0002-9904-1976-14076-1. Archived
    (PDF) from the original on 9 October 2022.
  31. ^ Schneps n.d.
  32. ^ Colmez & Serre 2004.
  33. MR 0102537
  34. ^ a b Hersh & John-Steiner 2011, p. 113.
  35. ^ "Chapter 3. From student to celebrity: 1949-1952" (PDF). Who Is Alexandre Grothendieck: Anarchy, Mathematics, Spirituality. Vol. 2.
  36. ^ Guillaume Aubrun (17 March 2020). "1953 : un « Résumé » aux développements illimités" [1953: a "Summary" with unlimited developments]. Images des Mathématiques (in French). CNRS.
  37. ^ a b Amir D. Aczel (2009). The Artist and the Mathematician. Basic Books.
  38. MR 2490557. Archived
    (PDF) from the original on 9 October 2022.
  39. American Mathematical Monthly
    , vol. 113, no. 9, footnote 6.
  40. ^ SGA1, Springer Lecture Notes 224, p. xii, xiii
  41. ^ a b Jackson, Allyn (March 1999). "The IHÉS at Forty" (PDF). Notices of the AMS. 46 (3): 329–337.
  42. ^ a b c d e f Pragacz 2005.
  43. ^ a b Alexandre Grothendieck, Esquisse d'un Programme, English translation
  44. ^ Grothendieck 1986.
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Sources and further reading

External links

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