Editing Alexander Grothendieck (section)

== Life ==

=== Family and childhood ===
Grothendieck was born in [[Berlin]] to [[Anarchism|anarchist]] parents. His father, [[Sascha Schapiro|Alexander "Sascha" Schapiro]] (also known as Alexander Tanaroff), had [[Hasidic Judaism|Hasidic Jewish]] roots and had been imprisoned in Russia before moving to Germany in 1922, while his mother, [[Hanka Grothendieck|Johanna "Hanka" Grothendieck]], came from a [[Protestant]] German family in [[Hamburg]] and worked as a journalist.{{efn|Testimony by [[Pierre Cartier (mathematician)|Pierre Cartier]] asserts that his mother was of Jewish German descent: "what I know of his life comes from Grothendieck himself".{{sfn|Cartier|2001}}}} As teenagers, both of his parents had broken away from their early backgrounds.<ref name="siam.org">{{Cite web |url=http://www.siam.org/news/news.php?id=1405 |title=The Early Background of Genius |access-date=15 June 2011 |url-status=dead |archive-date=15 June 2011 |archive-url=https://web.archive.org/web/20110615185446/http://www.siam.org/news/news.php?id=1405}}</ref> 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.<ref name="siam.org"/> 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]].{{sfn|Jackson|2004a}}<ref>{{cite news| url=https://www.lemonde.fr/sciences/article/2019/05/06/les-archives-insaisissables-d-alexandre-grothendieck_5459049_1650684.html| title=Trésor scientifique ou vieux papiers illisibles? Les mystérieuses archives d'Alexandre Grothendieck| trans-title=Scientific treasure or unreadable old paper? The mysterious archives of Alexandre Grothendieck| language=fr| website=[[Le Monde]]| author=Philippe Douroux| date=6 May 2019}}</ref> According to [[Winfried Scharlau]], during this time, his parents took part in the [[Spanish Civil War]] as non-combatant auxiliaries.{{sfn|Scharlau|2008|p=931}}<ref>{{harvnb|Scharlau|n.d.|p=2}}: "Beide beteiligten sich am Spanischen Bürgerkrieg, nicht aktiv kämpfend, aber unterstützend."</ref> However, others state that Schapiro fought in the anarchist militia.{{sfn|Hersh|John-Steiner|2011|p=109}}

=== World War II ===
In May 1939, Grothendieck was put on a train in Hamburg for France. Shortly afterward his father was interned in [[Camp Vernet|Le Vernet]].<ref name="Aczel" /> He and his mother were then interned in various camps from 1940 to 1942 as "undesirable dangerous foreigners."<ref>Piotr Pragacz, 'Notes on the Life and Work of Alexander Grothendieck,' in Piotr Pragacz (ed.), [https://books.google.com/books?id=9l4w1rqCKVIC&pg=PR11 ''Topics in Cohomological Studies of Algebraic Varieties: Impanga Lecture Notes,''] Springer Science & Business Media, 2006 pp-xi-xxviii p.xii.</ref> The first camp was the [[Rieucros Camp]], where his mother may have contracted the tuberculosis that would eventually cause her death in 1957. While there, Grothendieck managed to attend the local school, at [[Mende, Lozère|Mende]]. Once, he managed to escape from the camp, intending to assassinate [[Adolf Hitler|Hitler]].<ref name="Aczel" /> Later, his mother Hanka was transferred to the [[Gurs internment camp]] for the remainder of [[World War II]].<ref name ="Aczel">Amir D. Aczel,[https://books.google.com/books?id=fRCH-at7wgYC&pg=PA12 ''The Artist and the Mathematician,''] Basic Books, 2009 pp.8ff.pp.8–15.</ref> Grothendieck was permitted to live separated from his mother.<ref name ="Viale">Luca Barbieri Viale, 'Alexander Grothendieck:entusiasmo e creatività,' in C. Bartocci, R. Betti, A. Guerraggio, R. Lucchetti (eds.,) [https://books.google.com/books?id=nx2J7hqZ43EC&pg=PA237 ''Vite matematiche: Protagonisti del '900, da Hilbert a Wiles,''] Springer Science & Business Media, 2007 pp.237–249 p.237.</ref>

In the village of [[Le Chambon-sur-Lignon]], he was sheltered and hidden in local boarding houses or [[Pension (lodging)|pension]]s, although he occasionally had to seek refuge in the woods during Nazi raids, surviving at times without food or water for several days.<ref name="Aczel" /><ref name="Viale" />

His father was arrested under the [[Vichy anti-Jewish legislation]], and sent to the [[Drancy internment camp]], and then handed over by the [[Vichy France|French Vichy government]] to the Germans to be sent to be murdered at the [[Auschwitz concentration camp]] in 1942.<ref name="nytob" />{{sfn|Ruelle|2007|p=[https://books.google.com/books?id=B3A1bjOkOaEC&pg=PA35 35]}}

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.<ref name="Liberation-Douroux-Obit-2014-11-13" />

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.<ref>{{Cite web |language=fr |url=http://images.math.cnrs.fr/Alexandre-Grothendieck.html#nh14 |author=Philippe Douroux |date=February 8, 2012 |title=Alexandre Grothendieck: Un voyage à la poursuite des choses évidentes |trans-title=Alexander Grothendieck: A journey in pursuit of the obvious |website=Images des mathématiques |publisher=CNRS}}</ref> 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.{{sfn|Jackson|2004a}}

Initially, Grothendieck attended [[Henri Cartan]]'s Seminar at {{lang|fr|[[École Normale Supérieure]]|italic=no}}, but he lacked the necessary background to follow the high-powered seminar. On the advice of Cartan and [[André Weil]], he moved to the [[University of Nancy]] where two leading experts were working on Grothendieck's area of interest, [[topological vector space]]s: [[Jean Dieudonné]] and [[Laurent Schwartz]]. The latter had recently won a Fields Medal. Dieudonné and Schwartz showed the new student their latest paper ''La dualité dans les espaces ({{mathcal|F}}) et ({{mathcal|LF}})''; it ended with a list of 14 open questions, relevant for [[locally convex space]]s.<ref>{{cite journal |author1=[[Jean Dieudonné]] |author2=[[Laurent Schwartz]] |title=La dualité dans les espaces ({{mathcal|F}}) et ({{mathcal|LF}}) |journal=Annales de l'Institut Fourier |year=1949 |volume=1 |pages=61–101 |issn=0373-0956 |url=https://eudml.org/doc/73677 }}</ref> Grothendieck introduced new mathematical methods that enabled him to solve all of these problems within a few months.<ref>{{cite arXiv |last1=Peixoto |first1=Tatiana |last2=Bietenholza |first2=Wolfgang |title=To the Memory of Alexander Grothendieck: a Great and Mysterious Genius of Mathematics |eprint=1605.08112 |date=2016 |class=math.HO }}</ref><ref>{{cite journal |author=Alexander Grothendieck |title=Sur la complétion du dual d'un espace localement convexe |journal=C. R. Acad. Sci. Paris |year=1950 |volume=230 |pages=605–606 |url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/AG/AG-1.pdf}}</ref><ref>{{cite journal |author=Alexander Grothendieck |title=Quelques résultats relatifs à la dualité dans les espaces {{mathcal|F}} |journal=C. R. Acad. Sci. Paris |year=1950 |volume=230 |pages=1561–1563 |url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/AG/AG-2.pdf }}</ref><ref>{{cite journal |author=Alexander Grothendieck |title=Critères généraux de compacité dans les espaces vectoriels localement convexes. Pathologie des espaces {{mathcal|LF}} |journal=C. R. Acad. Sci. Paris |year=1950 |volume=231 |pages=940–941 | url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/AG/AG-3.pdf}}</ref><ref>{{cite journal |author=Alexander Grothendieck |title=Quelques résultats sur les espaces vectoriels topologiques |journal=C. R. Acad. Sci. Paris |year=1951 |volume=233 |pages=839–841 | url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/AG/AG-4.pdf}}</ref><ref>{{cite journal |author=Alexander Grothendieck |title=Sur une notion de produit tensoriel topologique d'espaces vectoriels topologiques, et une classe remarquable d'espaces vectoriels liée à cette notion |journal=C. R. Acad. Sci. Paris |year=1951 |volume=233 |pages=1556–1558 |url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/AG/AG-5.pdf}}</ref><ref>{{cite journal |author=Alexander Grothendieck |title=Critères de compacité dans les espaces fonctionnels généraux |journal=Amer. J. Math. |year=1952 |volume=74 |issue=1 |pages=168–186 |doi=10.2307/2372076 |jstor=2372076 |url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/AG/AG-6.pdf}}</ref>

In Nancy, he wrote his dissertation under those two professors on [[functional analysis]], from 1950 to 1953.{{sfn|Cartier |Illusie |Katz |Laumon |2007|loc="Foreword"}} At this time he was a leading expert in the theory of topological vector spaces.<ref>{{cite journal |department=Book Reviews |title=Topological vector spaces, by A. Grothendieck, ... |journal=Bulletin of the American Mathematical Society |url=https://www.ams.org/journals/bull/1976-82-04/S0002-9904-1976-14076-1/S0002-9904-1976-14076-1.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/1976-82-04/S0002-9904-1976-14076-1/S0002-9904-1976-14076-1.pdf |archive-date=2022-10-09 |url-status=live |last=Horvâth |first=John |volume=82 |issue=4 |pages=515–521 |doi=10.1090/S0002-9904-1976-14076-1 |date=July 1976|doi-access=free}}</ref> In 1953 he moved to the [[University of São Paulo]] in Brazil, where he immigrated by means of a [[Nansen passport]], given that he had refused to take French nationality (as that would have entailed military service against his convictions). He stayed in São Paulo (apart from a lengthy visit in France from October 1953 to March 1954) until the end of 1954. His published work from the time spent in Brazil is still in the theory of topological vector spaces; it is there that he completed his last major work on that topic (on "metric" theory of [[Banach space]]s).

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.{{sfn|Schneps|n.d.}}{{sfn|Colmez|Serre|2004}} 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 "[[Grothendieck's Tôhoku paper|Tôhoku paper]]".<ref>{{Citation |last1=Grothendieck |first1=Alexander |author1-link=Alexander Grothendieck |title=Sur quelques points d'algèbre homologique |mr=0102537 |year=1957| journal=[[Tohoku Mathematical Journal]] |series=Second Series |issn=0040-8735 |volume=9 |issue=2 |pages=119–221 |doi=10.2748/tmj/1178244839| doi-access=free |language=fr}}</ref>

In 1957 he was invited to visit [[Harvard University]] 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.{{sfn|Hersh|John-Steiner|2011|p=113}}

Comparing Grothendieck during his Nancy years to the {{lang|fr|[[École Normale Supérieure]]|italic=no}}-trained students at that time ([[Pierre Samuel]], [[Roger Godement]], [[René Thom]], [[Jacques Dixmier]], [[Jean Cerf]], [[Yvonne Bruhat]], [[Jean-Pierre Serre]], and [[Bernard Malgrange]]), [[Leila Schneps]] said:

{{blockquote|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.<ref>{{cite book |title=Who Is Alexandre Grothendieck: Anarchy, Mathematics, Spirituality |volume=2 |chapter=Chapter 3. From student to celebrity: 1949-1952 |chapter-url=http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathematics/chap3.pdf}}</ref>}} 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 [[EPR paradox|Einstein–Podolsky–Rosen paradox]] in [[quantum physics]].<ref>{{Cite web |author=Guillaume Aubrun |date=March 17, 2020 |title=1953 : un « Résumé » aux développements illimités|trans-title=1953: a "Summary" with unlimited developments|language=fr|url=https://images.math.cnrs.fr/1953-un-Resume-aux-developpements-illimites.html|website=Images des Mathématiques|publisher=CNRS}}</ref>

=== IHÉS years ===
In 1958, Grothendieck was installed at the [[Institut des Hautes Études Scientifiques|Institut des hautes études scientifiques]] (IHÉS), a new privately funded research institute that, in effect, had been created for [[Jean Dieudonné]] and Grothendieck.{{sfn|Cartier|2004}} 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).{{sfn|Jackson|2004a}} 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.<ref name="artist">{{cite book |title=The Artist and the Mathematician |author=Amir D. Aczel |date=2009 |publisher=Basic Books}}</ref>

Officially during this time, he had as students [[Michel Demazure]] (who worked on SGA3, on [[group scheme]]s), {{interlanguage link|Monique Hakim|fr}} ([[Grothendieck's relative point of view|relative schemes]] and [[classifying topos]]), [[Luc Illusie]] (cotangent complex), [[Michel Raynaud]], [[Michèle Raynaud]], [[Jean-Louis Verdier]] (co-founder of the [[derived category]] theory), and [[Pierre Deligne]]. Collaborators on the SGA projects also included [[Michael Artin]] ([[étale cohomology]]), [[Nick Katz]] ([[monodromy theory]], and [[Lefschetz pencil]]s). [[Jean Giraud (mathematician)|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]].{{sfn|Cartier |Illusie |Katz |Laumon |2007|loc="Foreword"}} His first (pre-IHÉS) discovery in algebraic geometry was the [[Grothendieck–Hirzebruch–Riemann–Roch theorem]], a generalisation of the [[Hirzebruch–Riemann–Roch theorem]] proved algebraically; in this context he also introduced [[K-theory]]. Then, following the programme he outlined in his talk at the 1958 [[International Congress of Mathematicians]], he introduced the theory of [[scheme (mathematics)|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.{{sfn|Jackson|2004a}} 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 group]]s of schemes giving birth to the now famous [[étale fundamental group]] and he then conjectured the existence of a further generalization of it, which is now known as the [[fundamental group scheme]]. As a framework for his [[coherent duality]] theory, he also introduced [[derived category|derived categories]], which were further developed by Verdier.<ref>{{cite book |chapter=Notes on derived categories and Grothendieck duality |first=Joseph|last=Lipman|author-link=Joseph Lipman |title=Foundations of Grothendieck Duality for Diagrams of Schemes |series=Lecture Notes in Mathematics |volume=1960 |pages=1–259 |publisher=Springer-Verlag |location=New York |date=2009|mr=2490557 |doi=10.1007/978-3-540-85420-3 |chapter-url=http://www.math.purdue.edu/~lipman/Duality.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.math.purdue.edu/~lipman/Duality.pdf |archive-date=2022-10-09 |url-status=live|isbn=978-3-540-85419-7}}</ref>

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.{{sfn|Jackson|2004a}}

=== Political activism ===
Grothendieck's political views were [[Political radicalism|radical]] and [[pacifist]]ic. He strongly opposed both United States [[Vietnam War|intervention in Vietnam]] and [[Soviet Empire|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.<ref>''The Life and Work of Alexander Grothendieck'', ''[[American Mathematical Monthly]]'', vol. 113, no. 9, footnote 6.</ref> In 1966, he had declined to attend the International Congress of Mathematicians (ICM) in Moscow, where he was to receive the Fields Medal.{{sfn|Jackson|2004b}} He retired from scientific life around 1970 after he had found out that IHÉS was partly funded by the military.<ref>SGA1, Springer Lecture Notes 224, {{pp.|xii, xiii}}</ref> 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 (mathematician)|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.<ref name="IHÉS at Forty">{{cite journal |first=Allyn |last=Jackson |date=March 1999 |url=https://www.ams.org/notices/199903/ihes-changes.pdf |title=The IHÉS at Forty |journal=Notices of the AMS |volume=46 |issue=3 |pages=329–337}}</ref> 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.{{sfn|Cartier|2004}} 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]].{{efn|Ruelle<!--not Lorenz?--> 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.}} [[Biology]] interested Grothendieck much more than physics, and he organized some seminars on biological topics.<ref name="IHÉS at Forty"/>

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.{{sfn|Pragacz|2005}} Grothendieck devoted the next three years to this group and served as the main editor of its bulletin.{{sfn|Scharlau|2008}}

Although Grothendieck continued with mathematical enquiries, his standard mathematical career mostly ended when he left the IHÉS.<ref name="nytob"/> After leaving the IHÉS, Grothendieck became a [[temporary professor]] at [[Collège de France]] for two years.{{sfn|Pragacz|2005}} 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 [[Centre National de la Recherche Scientifique|CNRS]].{{sfn|Scharlau|2008}}

=== 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]]''.<ref name="esquissefr">Alexandre Grothendieck, [http://matematicas.unex.es/~navarro/res/esquissefr.pdf Esquisse d'un Programme], [http://matematicas.unex.es/~navarro/res/esquisseeng.pdf English translation]</ref> It also includes a study of [[Teichmüller space|Teichmüller theory]].

In 1983, stimulated by correspondence with [[Ronald Brown (mathematician)|Ronald Brown]] and Tim Porter at [[Bangor University]], Grothendieck wrote a 600-page manuscript entitled ''[[Pursuing Stacks]]''. It began with a letter addressed to [[Daniel Quillen]]. This letter and successive parts were distributed from Bangor (see [[#External links|External links]] below). Within these, in an informal, diary-like manner, Grothendieck explained and developed his ideas on the relationship between algebraic [[homotopy theory]] and [[algebraic geometry]] and prospects for a [[noncommutative]] theory of [[Stack (mathematics)|stack]]s. 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''.{{sfn|Jackson|2004b}} Much of this work anticipated the subsequent development during the mid-1990s of the [[A¹ homotopy theory|motivic homotopy theory]] of [[Fabien Morel]] and [[Vladimir Voevodsky]].

In 1984, Grothendieck wrote the proposal ''[[Esquisse d'un Programme]]'' ("Sketch of a Programme")<ref name="esquissefr"/> for a position at the [[Centre National de la Recherche Scientifique]] (CNRS). It describes new ideas for studying the [[moduli space]] of complex curves. Although Grothendieck never published his work in this area, the proposal inspired other mathematicians to work in the area by becoming the source of [[dessin d'enfant]] theory and [[anabelian geometry]]. Later, it was published in two-volumes and entitled [https://books.google.com/books?id=Hlf1j2XmXkcC&q=Geometric+Galois+Actions ''Geometric Galois Actions''] (Cambridge University Press, 1997).

During this period, Grothendieck also gave his consent to publishing some of his drafts for EGA on [[Theorem of Bertini|Bertini-type theorem]]s (''EGA''&nbsp;V, published in Ulam Quarterly in 1992–1993 and later made available on the [http://www.grothendieckcircle.org/ Grothendieck Circle] web site in 2004).

In the extensive autobiographical work, ''Récoltes et Semailles'' ('Harvests and Sowings', 1986), Grothendieck describes his approach to mathematics and his experiences in the mathematical community, a community that initially accepted him in an open and welcoming manner, but which he progressively perceived to be governed by competition and status. He complains about what he saw as the "burial" of his work and betrayal by his former students and colleagues after he had left the community.{{sfn|Jackson|2004a}} ''Récoltes et Semailles'' was finally published in 2022 by Gallimard{{sfn|Grothendieck|2022}} and, thanks to French science historian Alain Herreman,{{sfn|Jackson|2004b}} is also available on the Internet.<ref>{{Cite web |title=Récoltes et Semailles |url=https://jmrlivres.files.wordpress.com/2009/11/recoltes-et-semailles.pdf |last=Grothendieck |first=Alexandre |access-date=17 September 2024 |language=fr}}</ref> An English translation by [[Leila Schneps]] will be published by [[MIT Press]] in 2025.<ref>{{cite web|title=Publication Announcement| date=25 January 2023| url=https://mitpress.mit.edu/book-deals-january-2023-edition/}}</ref> A partial English translation can be found on the Internet.<ref>{{Cite web |title=Harvests and Sowings |url= https://web.ma.utexas.edu/users/slaoui/notes/recoltes_et_semailles.pdf |access-date=15 September 2024}}</ref> 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".<ref>{{cite web |title=Visiting Alexandre Grothendieck|url=https://www.fermentmagazine.org/quest88|author=Roy Lisker |access-date=25 January 2022}}</ref><ref>{{cite web|title=Chapter 23. Récoltes et Semailles |last1=Scharlau |first1=Winfried |url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Spirituality/Spirituality23.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Spirituality/Spirituality23.pdf |archive-date=2022-10-09 |url-status=live|access-date=25 January 2022}}</ref><ref>{{cite book|title=Suugakusha no kodokuna bōken : suugaku to jiko no hakken eno tabi |trans-title= The Solitary Adventures of a Mathematician: A Journey into Mathematics and Self-Discovery |author= Grothendieck, Alexander|translator= Tsuji Yuichi|edition=2nd|publisher=Gendai Sūgaku-sha|year=2015|location= Kyoto| language= ja}}</ref><ref>{{cite book|title= Sūgaku to hadaka no ōsama: Aru yume to sūgaku no maisō |trans-title=Mathematics and the Naked King: A Dream and the Burial of Mathematics |author= Grothendieck, Alexander|translator= Tsuji Yuichi|edition=2nd|publisher=Gendai Sūgaku-sha|year=2015|location= Kyoto| language= ja}}</ref><ref>{{cite book|title= Aru yume to sūgaku no maisō: In to yō no kagi |trans-title= A Dream and the Burial of Mathematics: The Key to Yin and Yang |author= Grothendieck, Alexander|translator= Tsuji Yuichi|edition=2nd|publisher=Gendai Sūgaku-sha|year=2016|location= Kyoto| language= ja}}</ref><ref>{{cite book|title= Maisō (3) aruiwa yottsu no sōsa |trans-title=Burial (3) or Four Operations |author= Grothendieck, Alexander |translator= Tsuji Yuichi |year=1998 |type=Unpublished manuscript| language= ja}}</ref> Parts of ''Récoltes et Semailles'' have been translated into Spanish,<ref>{{cite web |url=http://matematicas.unex.es/~navarro/res/ |title=Récoltes et Semailles; La Clef des Songes|language=es}}</ref> as well as into a Russian translation that was published in Moscow.<ref>{{Cite web|url=http://www.mccme.ru/free-books/grothendieck/RS.html|title=Free books: Récoltes et semailles|website=www.mccme.ru|access-date=2017-09-12}}</ref>

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."<ref name=columbia>{{cite web|url=http://www.math.columbia.edu/~lipyan/CrafoordPrize.pdf |title=Crafoord Prize letter, English translation |access-date=2005-06-17 |url-status=dead |archive-url=https://web.archive.org/web/20060106062005/http://www.math.columbia.edu/~lipyan/CrafoordPrize.pdf |archive-date=6 January 2006}}</ref>

''La Clef des Songes'',<ref>{{cite web |last1=Grothendieck |first1=Alexander |title=La Clef des Songes |url=http://cm2vivi2002.free.fr/AG-biblio/AG-clesonges.pdf |access-date=2 December 2021 |archive-url=https://ghostarchive.org/archive/20221009/http://cm2vivi2002.free.fr/AG-biblio/AG-clesonges.pdf |archive-date=2022-10-09 |url-status=live}}</ref> 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 [[Existence of God|deity]] exists.{{sfn|Scharlau|2008|p=940}} 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.{{sfn|Scharlau|2008}} The only mathematician on his list was [[Bernhard Riemann]].<ref>{{Citation |last=Scharlau |first=Winfried |title=Die Mutanten – Les Mutants – eine Meditation von Alexander Grothendieck |date=2 May 2023 |language=de |url=http://www.scharlau-online.de/DOKS/Die%20Mutanten.pdf}}</ref> 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.{{sfn|Scharlau|2008}} 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.{{sfn|Jackson|2004b}}

The ''Grothendieck Festschrift'', published in 1990, was a three-volume collection of research papers to mark his sixtieth birthday in 1988.{{sfn|Cartier |Illusie |Katz |Laumon |2007}}

More than 20,000 pages of Grothendieck's mathematical and other writings are held at the University of Montpellier and remain unpublished.<ref>[http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-maths_830399 Le trésor oublié du génie des maths] {{in lang|fr}}</ref> They have been digitized for preservation and are freely available in open access through the Institut Montpelliérain Alexander Grothendieck portal.<ref>[http://www.liberation.fr/sciences/2015/06/17/les-gribouillis-d-alexandre-grothendieck-enfin-sauvegardes_1331465 Les «gribouillis» d'Alexandre Grothendieck enfin sauvegardés] {{in lang|fr}}</ref><ref>{{Cite web|title=IMAG|trans-title=Welcome|website=Institut Montpelliérain Alexander Grothendieck|language=fr|url=https://grothendieck.umontpellier.fr/archives-grothendieck/}}</ref>

=== 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.{{sfn|Scharlau|2008}} Very few people visited him afterward.<ref name="Rivka#">{{cite magazine |last=Galchen |first=Rivka |url=https://www.newyorker.com/magazine/2022/05/16/the-mysterious-disappearance-of-a-revolutionary-mathematician |title=The Mysterious Disappearance revolutionary mathematician |magazine=The New Yorker |date=May 9, 2022}}</ref> Local villagers helped sustain him with a more varied diet after he tried to live on a staple of [[Taraxacum|dandelion]] soup.<ref>{{cite book |author=[[John Derbyshire]] |url=https://books.google.com/books?id=_hsWNzBvr3oC&pg=PA314 |title=Unknown Quantity: A Real and Imaginary History of Algebra |publisher=National Academies Press |year=2006 |page=314| isbn=9780309164801 }}</ref> 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".<ref>{{cite web |last=Leith |first=Sam |author-link=Sam Leith |title=The Einstein of maths |date=2004-03-20 |url=http://www.spectator.co.uk/features/12036/the-einstein-of-maths/ |work=The Spectator |access-date=2019-12-26 |url-status=dead |archive-date=2016-08-11 |archive-url=https://web.archive.org/web/20160811200940/http://www.spectator.co.uk/2004/03/the-einstein-of-maths/}}</ref> 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]].<ref>{{cite news |author1=Stéphane Foucart |author2=Philippe Pajot |title=Alexandre Grothendieck, le plus grand mathématicien du XXe siècle, est mort |trans-title=Alexandre Grothendieck, the greatest mathematician of the 20th century, is dead |language=fr |url=http://www.lemonde.fr/disparitions/article/2014/11/14/le-mathematicien-alexandre-grothendieck-est-mort_4523482_3382.html |newspaper=Le Monde |date=14 November 2014}}</ref>

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.<ref>{{Cite news|title=Grothendieck's letter|date=2010-02-09|work=Secret Blogging Seminar |url=https://sbseminar.wordpress.com/2010/02/09/grothendiecks-letter|access-date=2017-09-12}}</ref> He characterized a website devoted to his work as "an abomination".<ref>{{cite web |url=http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/ |title=Grothendieck Circle |access-date=2015-10-13 |archive-url=https://web.archive.org/web/20140929144122/http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/ |archive-date=29 September 2014 |url-status=live}}</ref> His dictate may have been reversed in 2010.<ref>{{cite web|url=http://www.math.u-psud.fr/~laszlo/sga4.html |title=Réédition des SGA |access-date=2013-11-12 |url-status=dead |archive-date=29 June 2016 |archive-url=https://web.archive.org/web/20160629235119/http://www.math.u-psud.fr/~laszlo/sga4.html}}</ref>

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.<ref name="Hoad" >Phil Hoad, [https://www.theguardian.com/science/article/2024/aug/31/alexander-grothendieck-huawei-ai-artificial-intelligence ‘He was in mystic delirium’: was this hermit mathematician a forgotten genius whose ideas could transform AI – or a lonely madman?], [[The Guardian]] 31 August 2024</ref> On 13 November 2014, aged 86, Grothendieck died in the hospital of [[Saint-Lizier]]<ref name="Hoad" /> or  [[Saint-Girons, Ariège]].<ref name="Liberation-Douroux-Obit-2014-11-13">{{cite web |url=http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-d-un-genie-qui-voulait-se-faire-oublier_1142614 |title=Alexandre Grothendieck, ou la mort d'un génie qui voulait se faire oublier |date=13 November 2014 |work=Libération Sciences |access-date=14 November 2014 |language=fr}}</ref><ref>{{cite web |url = http://www.telegraph.co.uk/news/obituaries/11231703/Alexander-Grothendieck-obituary.html |title = Alexander Grothendieck - obituary |archive-url=https://web.archive.org/web/20141115004729/http://www.telegraph.co.uk/news/obituaries/11231703/Alexander-Grothendieck-obituary.html |archive-date=15 November 2014 |url-status=dead}}</ref>

=== Citizenship ===
Grothendieck was born in [[Weimar Republic|Weimar Germany]]. In 1938, aged ten, he moved to France as a refugee. Records of his nationality were destroyed in the [[third Reich|fall of Nazi Germany]] in 1945 and he did not apply for [[French citizenship]] after the war. Thus, he became a [[Statelessness|stateless]] person for at least the majority of his working life and he traveled on a [[Nansen passport]].{{sfn|Douroux|2012}}{{sfn|Cartier|2004|loc=p. 10, footnote 12}}{{sfn|Kleinert|2007}}<!--Note: Douroux 2012's 1971 date is incorrect; given Cartier's other statements, 1981 (i.e. during the 1980s) is more credible (see Cartier 2001, p.&nbsp;393: "...an absurd trial in 1977 due to a 1945 regulation that made it a misdemeanor to meet with a foreigner," indicating that Grothendieck did not have French nationality in 1977.--> 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).{{sfn|Cartier|2004}}{{sfn|Kleinert|2007}}{{sfn|Cartier|2001}} 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.{{sfn|Cartier|2004}}

=== Family ===
Grothendieck was very close to his mother, to whom he dedicated his dissertation. She died in 1957 from [[tuberculosis]] that she contracted in camps for [[displaced persons]].{{sfn|Pragacz|2005}}

He had five children: a son with his [[landlady]] during his time in Nancy;{{sfn|Cartier|2004}} three children, Johanna (1959), Alexander (1961), and Mathieu (1965) with his wife Mireille Dufour;{{sfn|Scharlau|2008}}{{sfn|Hersh|John-Steiner|2011|p=113}} and one child with Justine Skalba, with whom he lived in a [[intentional community|commune]] in the early 1970s.{{sfn|Scharlau|2008}}