View source for Biography:Alexander Grothendieck

{{Short description|French mathematician (1928–2014)}}
{{Infobox scientist
| name              = Alexander Grothendieck
| image             = Alexander Grothendieck.jpg
| caption           = Alexander Grothendieck in Montreal, 1970
| birth_date        = {{birth date|1928|03|28|df=yes}}
| birth_place       = [[Place:Berlin|Berlin]], Prussia, Germany
| death_date        = {{nowrap|{{death date and age|2014|11|13|1928|03|28|df=yes}}}}
| death_place       = Saint-Lizier, Ariège, France
| nationality       = {{ublist|class=nowrap |French since 1971{{sfn|Cartier|2004}}{{sfn|Douroux|2012}}  |Stateless (post 1945 until 1971){{sfn|Douroux|2012}}{{sfn|Cartier|2004|loc=p. 10, footnote 12}}{{sfn|Kleinert|2007}}}}
| workplaces        = {{ublist|class=nowrap |[[Organization:Institut des Hautes Études Scientifiques|Institut des hautes études <br />scientifiques]] (IHÉS) |[[Organization:University of Montpellier|University of Montpellier]]{{sfn|Scharlau|2008}}|University of São Paulo{{sfn|Cartier|Illusie|Katz|Laumon|2007|p=[https://books.google.com/books?id=_BnSoQSKnNUC&pg=PR7 7]}}}}
| alma_mater        = {{ublist|class=nowrap |[[Organization:University of Montpellier|University of Montpellier]] |University of Nancy}}
| thesis_title      = Produits tensoriels topologiques et espaces nucléaires
| thesis_url        = https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/PTT.pdf
| thesis_year       = 1953
| doctoral_advisors = {{ublist|class=nowrap |[[Biography:Laurent Schwartz|Laurent Schwartz]] |[[Biography:Jean Dieudonné|Jean Dieudonné]]}}
| doctoral_students = {{ublist|class=nowrap |[[Biography:Pierre Berthelot|Pierre Berthelot]] |[[Biography:Pierre Deligne|Pierre Deligne]] |[[Biography:Michel Demazure|Michel Demazure]] |[[Biography:Pierre Gabriel|Pierre Gabriel]] |Jean Giraud |[[Biography:Luc Illusie|Luc Illusie]] |[[Biography:William Messing|William Messing]] |[[Biography:Michel Raynaud|Michel Raynaud]] |[[Biography:Hoàng Xuân Sính|Hoàng Xuân Sính]] |[[Biography:Jean-Louis Verdier|Jean-Louis Verdier]]}}
| fields            = [[Mathematics]] – [[Functional analysis|functional analysis]], [[Algebraic geometry|algebraic geometry]], [[Homological algebra|homological algebra]]
| awards            = {{ublist|class=nowrap |{{smaller|1966}}&nbsp;Fields Medal |{{smaller|1977}}&nbsp;Émile Picard Medal |{{smaller|1988}}&nbsp;Crafoord Prize 
{{smaller|(declined)}}}}
| known_for         = Renewing [[Algebraic geometry|algebraic geometry]] and synthesis between it and [[Number theory|number theory]] and [[Topology|topology]]<br />List of things named after Alexander Grothendieck
}}

'''Alexander Grothendieck''' ({{IPAc-en|ˈ|g|r|oʊ|t|ən|d|iː|k}}; {{IPA-de|ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk|-|De-Alexander Grothendieck.ogg}}; {{IPA-fr|ɡʁɔtɛndik|lang}}; 28 March 1928&nbsp;– 13 November 2014) was a French [[Biography:Mathematician|mathematician]] who became the leading figure in the creation of modern [[Algebraic geometry|algebraic geometry]].{{sfn|Jackson|2004b}}<ref name="nytob">{{cite news |title=Alexander Grothendieck, Math Enigma, Dies at 86 |date=14 November 2014 |author1=Bruce Weber |author2=Julie Rehmeyer |newspaper=The New York Times |url=https://www.nytimes.com/2014/11/16/world/europe/alexander-grothendieck-math-enigma-dies-at-86.html |url-access=limited |archive-url=https://ghostarchive.org/archive/20220101/https://www.nytimes.com/2014/11/16/world/europe/alexander-grothendieck-math-enigma-dies-at-86.html |archive-date=2022-01-01}}{{cbignore}}</ref> His research extended the scope of the field and added elements of [[Commutative algebra|commutative algebra]], [[Homological algebra|homological algebra]], sheaf theory, and [[Category theory|category theory]] to its foundations, while his so-called [[Grothendieck's relative point of view|"relative" perspective]] led to revolutionary advances in many areas of [[Pure mathematics|pure mathematics]].{{sfn|Jackson|2004b}}<ref name="MumfordTate2015">{{cite journal|last1=Mumford|first1=David |last2=Tate|first2=John |author-link2=John Tate (mathematician) |title=Alexander Grothendieck (1928–2014) Mathematician who rebuilt algebraic geometry|journal=Nature |volume=517|issue=7534|year=2015|pages=272|issn=0028-0836|doi=10.1038/517272a |pmid=25592527 |bibcode=2015Natur.517..272M|doi-access=free}}</ref> He is considered by many to be the greatest mathematician of the twentieth century.<ref>{{cite web |title=Guardian obituary |website=Independent.co.uk |url=https://www.independent.co.uk/news/world/europe/greatest-mathematician-of-the-20th-century-and-the-most-important-scientific-mind-you-ve-never-heard-of-alexander-grothendieck-dies-aged-86-9862306.html}}</ref><ref name=obit/>

Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the [[Organization:Institut des Hautes Études Scientifiques|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|algebraic geometry]], [[Homological algebra|homological algebra]], and [[K-theory]].<ref>{{cite web |url=https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1966/index.html |title=Fields Medals 1966 |website=mathunion.org |access-date=5 January 2022 |url-status=live|archive-url=https://web.archive.org/web/20190322225223/https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1966/index.html |archive-date=22 March 2019 }}</ref> He later became professor at the [[Organization:University of Montpellier|University of Montpellier]]<!--(see "Politics and retreat from scientific community" section below, ¶1)-->{{sfn|Scharlau|2008}} 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).<ref>{{cite web |last1=Scharlau |first1=Winfried |title=Who is Alexander Grothendieck? Anarchy, Mathematics, Spirituality, Solitude |url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Spirituality/SpiritualityForeword.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Spirituality/SpiritualityForeword.pdf |archive-date=2022-10-09 |url-status=live}}</ref> 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.{{sfn|Ruelle|2007|p=40}}

== Life ==

=== Family and childhood ===
Grothendieck was born in [[Place:Berlin|Berlin]] to [[Social:Anarchism|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 [[Place:Hamburg|Hamburg]] and worked as a journalist.{{efn|Testimony by 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 [[Place:Paris|Paris]] to evade [[Social:Nazism|Nazism]]. His mother followed soon thereafter. Grothendieck was left in the care of Wilhelm Heydorn, a Lutheran [[Religion:Pastor|pastor]] and teacher in [[Place:Hamburg|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 [[Biography:Winfried Scharlau|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 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 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.<ref name="Aczel" /> Later, his mother Hanka was transferred to the Gurs internment camp for the remainder of [[Social:World War II|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 pensions, 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 [[Religion:Vichy France|French Vichy government]] to the Germans to be sent to be murdered at the [[Social:Auschwitz concentration camp|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 [[Organization:University of Montpellier|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 [[Biography:Henri Cartan|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 [[Biography:André Weil|André Weil]], he moved to the University of Nancy where two leading experts were working on Grothendieck's area of interest, [[Topological vector space|topological vector space]]s: [[Biography:Jean Dieudonné|Jean Dieudonné]] and [[Biography:Laurent Schwartz|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.<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>

In Nancy, he wrote his dissertation under those two professors on [[Functional analysis|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 - 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|algebraic topology]] and [[Homological algebra|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|sheaf cohomology]] based on them, leading to the very influential "[[Grothendieck's Tôhoku paper|Tôhoku paper]]".<ref>{{Citation |last1=Grothendieck |first1=Alexander |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 by [[Biography:Oscar Zariski|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 ([[Biography:Pierre Samuel|Pierre Samuel]], Roger Godement, [[Biography:René Thom|René Thom]], [[Biography:Jacques Dixmier|Jacques Dixmier]], [[Biography:Jean Cerf|Jean Cerf]], Yvonne Bruhat, [[Biography:Jean-Pierre Serre|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 [[Physics: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 [[Organization:Institut des Hautes Études Scientifiques|Institut des hautes études scientifiques]] (IHÉS), a new privately funded research institute that, in effect, had been created for [[Biography:Jean Dieudonné|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 [[Biography:Michel Demazure|Michel Demazure]] (who worked on SGA3, on [[Group scheme|group scheme]]s), [[Biography:Luc Illusie|Luc Illusie]] (cotangent complex), [[Biography:Michel Raynaud|Michel Raynaud]], [[Biography:Jean-Louis Verdier|Jean-Louis Verdier]] (co-founder of the [[Derived category|derived category]] theory), and [[Biography:Pierre Deligne|Pierre Deligne]]. Collaborators on the SGA projects also included [[Biography:Michael Artin|Michael Artin]] ([[Étale cohomology|étale cohomology]]), [[Biography:Nick Katz|Nick Katz]] (monodromy theory, and [[Lefschetz pencil]]s). Jean Giraud worked out torsor theory extensions of [[Nonabelian cohomology|nonabelian cohomology]] there as well. Many others such as [[Biography:David Mumford|David Mumford]], [[Biography:Robin Hartshorne|Robin Hartshorne]], [[Biography:Barry Mazur|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|algebraic geometry]], [[Number theory|number theory]], [[Topology|topology]], [[Category theory|category theory]], and [[Complex analysis|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|étale cohomology]] theory of schemes, providing the key tools for proving the [[Weil conjectures]], as well as [[Crystalline cohomology|crystalline cohomology]] and algebraic de Rham cohomology to complement it. Closely linked to these cohomology theories, he originated [[Topos|topos]] theory as a generalisation of topology (relevant also in [[Categorical logic|categorical logic]]). He also provided, by means of a categorical [[Galois theory]], an algebraic definition of [[Fundamental group|fundamental group]]s 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|fundamental group scheme]]. As a framework for his [[Coherent duality|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|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 [[Philosophy:Political radicalism|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|category theory]] in the forests surrounding [[Place:Hanoi|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 [[Organization:University of Montpellier|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.<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. [[Biography:David Ruelle|David Ruelle]], a physicist who joined the IHÉS faculty in 1964, said that Grothendieck came to talk to him a few times about [[Physics:Physics|physics]].{{efn|Ruelle<!--not Lorenz?--> invented the concept of a strange attractor in a [[Dynamical system|dynamical system]] and, with the Dutch mathematician [[Biography:Floris Takens|Floris Takens]], produced a new model for [[Physics:Turbulence|turbulence]] during the 1970s.}} [[Biology:Biology|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, [[Biography:Claude Chevalley|Claude Chevalley]] and [[Biography:Pierre Samuel|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 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 [[Biography:Ronald Brown (mathematician)|Ronald Brown]] and Tim Porter at [[Organization:Bangor University|Bangor University]], Grothendieck wrote a 600-page manuscript entitled ''[[Pursuing Stacks]]''. It began with a letter addressed to [[Biography:Daniel Quillen|Daniel Quillen]]. This letter and successive parts were distributed from Bangor (see External links below). Within these, in an informal, diary-like manner, Grothendieck explained and developed his ideas on the relationship between algebraic homotopy theory and [[Algebraic geometry|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 motivic homotopy theory of [[Biography:Fabien Morel|Fabien Morel]] and [[Biography:Vladimir Voevodsky|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|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|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 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.{{sfn|Jackson|2004a}} The ''Récoltes et semailles'' work is now available on the internet in the French original,{{sfn|Grothendieck|1986}} 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".<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> 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.<ref>{{cite web|url= https://www.ihes.fr/parution-recoltes-semailles-grothendieck/|title= Parution de « Récoltes et semailles » d'Alexandre Grothendieck|work= IHES|date= 13 January 2022|access-date= 23 January 2022|language=French}}</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 [[Philosophy: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 [[Biography:Bernhard Riemann|Bernhard Riemann]].<ref>{{Citation |last=Scharlau |first=Winfried |title=Die Mutanten – Les Mutants – eine Meditation von Alexander Grothendieck |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>{{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 [[Biology:Taraxacum|dandelion soup]].<ref>{{cite book |author=[[Biography:John Derbyshire|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 |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 [[Biography:Luc Illusie|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>

On 13 November 2014, aged 86, Grothendieck died in the hospital of 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 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.{{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 exempted him from military service.{{sfn|Cartier|2004}}

=== Family ===
Grothendieck was very close to his mother to whom he dedicated his dissertation. She died in 1957 from the [[Medicine:Tuberculosis|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 commune in the early 1970s.{{sfn|Scharlau|2008}}

== Mathematical work ==
Grothendieck's early mathematical work was in [[Functional analysis|functional analysis]]. Between 1949 and 1953 he worked on his doctoral thesis in this subject at Nancy, supervised by [[Biography:Jean Dieudonné|Jean Dieudonné]] and [[Biography:Laurent Schwartz|Laurent Schwartz]]. His key contributions include [[Topological tensor product|topological tensor product]]s of [[Topological vector space|topological vector space]]s, the theory of [[Nuclear space|nuclear space]]s as foundational for Schwartz distributions, and the application of [[Lp space|L<sup>p</sup> 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 [[Biography:Stefan Banach|Banach]].{{sfn|Dieudonné|2007}}

It is, however, in [[Algebraic geometry|algebraic geometry]] and related fields where Grothendieck did his most important and influential work. From approximately 1955 he started to work on [[Sheaf (mathematics)|sheaf]] theory and [[Homological algebra|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 category|abelian categories]] and applied their theory to show that [[Sheaf cohomology|sheaf cohomology]] may be defined as certain [[Derived functor|derived functor]]s in this context.{{sfn|Jackson|2004a}}

Homological methods and sheaf theory had already been introduced in algebraic geometry by [[Biography:Jean-Pierre Serre|Jean-Pierre Serre]]{{sfn|Serre|1955}} and others, after sheaves had been defined by [[Biography:Jean Leray|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 ''[[Grothendieck's relative point of view|relative point of view]]'' (pairs of varieties related by a [[Morphism|morphism]]), allowing a broad generalization of many classical theorems.{{sfn|Pragacz|2005}} The first major application was the relative version of Serre's theorem showing that the cohomology of a [[Coherent sheaf|coherent sheaf]] on a complete variety is finite-dimensional; 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 one-point space.

In 1956, he applied the same thinking to the [[Riemann–Roch theorem]], which recently had been generalized to any dimension by [[Biography:Friedrich Hirzebruch|Hirzebruch]]. The [[Grothendieck–Riemann–Roch theorem]] was announced by Grothendieck at the initial [[Mathematische Arbeitstagung]] in [[Place:Bonn|Bonn]], in 1957.{{sfn|Pragacz|2005}} It appeared in print in a paper written by [[Biography:Armand Borel|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 [[Biography:Claude Chevalley|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 point|generic point]]s, which led to the theory of [[Scheme (mathematics)|schemes]]. Grothendieck also pioneered the systematic use of [[Nilpotent|nilpotent]]s. As 'functions' these can take only the value 0, but they carry [[Infinitesimal|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|birational geometry]], techniques from [[Number theory|number theory]], [[Galois theory]], [[Commutative algebra|commutative algebra]], and close analogues of the methods of [[Algebraic topology|algebraic topology]], all in an integrated way.{{sfn|Jackson|2004a}}{{sfn|Deligne|1998}}<ref name="McLarty">{{cite web |last=McLarty |first=Colin |title=The Rising Sea: Grothendieck on simplicity and generality I |url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathbiographies/mclarty1.pdf |access-date=29 April 2020 |archive-url=https://ghostarchive.org/archive/20221009/https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathbiographies/mclarty1.pdf |archive-date=2022-10-09 |url-status=live}}</ref>

Grothendieck is noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation.<ref name="artist"/> 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 [[D-module]]s. Although lauded as "the Einstein of mathematics", his work also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems.<ref name="scienceline">{{cite web |url=http://scienceline.org/2007/01/math_controversy_peck/ |last=Peck |first=Morgen |title=Equality of Mathematicians |date=31 January 2007 |website=ScienceLine |quote=Alexandre Grothendieck is arguably the most important mathematician of the 20th century...}}</ref><ref>{{harvnb|Leith|2004}}: "[A] mathematician of staggering accomplishment... a legendary figure in the mathematical world."</ref>

===''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.{{sfn|Jackson|2004a}}

Grothendieck's work includes the invention of the [[Étale cohomology|étale]] and l-adic cohomology theories, which explain an observation made by [[Biography:André Weil|André Weil]] that argued for a connection between the topological characteristics of a variety and its diophantine (number theoretic) properties.{{sfn|Pragacz|2005}} For example, the number of solutions of an equation over a [[Finite field|finite field]] reflects the topological nature of its solutions over the [[Complex number|complex number]]s. 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 [[Biography:Pierre Deligne|Pierre Deligne]] in the early 1970s after Grothendieck had largely withdrawn from mathematics.{{sfn|Jackson|2004a}}

=== Major mathematical contributions ===
In Grothendieck's retrospective ''Récoltes et Semailles'', he identified twelve of his contributions that he believed qualified as "great ideas".{{sfn|Grothendieck|1986|p=21}} In chronological order, they are:
# [[Topological tensor product]]s and [[Nuclear space|nuclear space]]s
# "Continuous" and "discrete" [[Duality (mathematics)|duality]] ([[Derived category|derived categories]], "six operations")
# Yoga of the [[Grothendieck–Riemann–Roch theorem]] [[K-theory]] relation with [[Intersection theory|intersection theory]]
# [[Scheme (mathematics)|Scheme]]s
# [[Topos|Topoi]]
# [[Étale cohomology]] and l-adic cohomology
# [[Motive (algebraic geometry)|Motive]]s and the motivic Galois group (Grothendieck ⊗-categories)
# Crystals and [[Crystalline cohomology|crystalline cohomology]], yoga of "de Rham coefficients", "Hodge coefficients"...
# "Topological algebra": ∞-stacks, [[Derivator|derivator]]s; cohomological formalism of topoi as inspiration for a new [[Homotopical algebra|homotopical algebra]]
# [[Tame topology]]
# Yoga of [[Anabelian geometry|anabelian algebraic geometry]], Galois–Teichmüller theory
# "Schematic" or "arithmetic" point of view for [[Regular polyhedron|regular polyhedra]] and regular configurations of all kinds

Here the term ''yoga'' denotes a kind of "meta-theory" that may be used heuristically; [[Biography:Michel Raynaud|Michel Raynaud]] writes the other terms "Ariadne's thread" and "philosophy" as effective equivalents.<ref>{{cite journal| url=https://www.ams.org/notices/200309/rev-raynaud.pdf |title=Correspondance Grothendieck-Serre |author=Michel Raynaud |date=October 2003 |journal=Notices of the AMS |department=Book Review |volume=50 |issue=9 |page=1086 |url-status=dead |archive-date=2003-10-03 |archive-url=https://web.archive.org/web/20031003104125/https://www.ams.org/notices/200309/rev-raynaud.pdf}}</ref>

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.{{sfn|Grothendieck|1986|p=22}}

== Influence ==
Grothendieck is considered by many to be the greatest mathematician of the twentieth century.<ref name=obit/> In an obituary [[Biography:David Mumford|David Mumford]] and John Tate wrote:
<blockquote>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.<ref name=obit>[http://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html 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</ref></blockquote>

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,<ref name=M&M>{{cite book|author1=Saunders Mac Lane|author2=Ieke Moerdijk|title=Sheaves in Geometry and Logic: a First Introduction to Topos Theory|publisher=Springer-Verlag New York Inc.|date=1992|isbn=0-387-97710-4|author1-link=Saunders Mac Lane}}</ref> but influenced logic, in the field of categorical logic.<ref>{{cite book|author1=Dov M. Gabbay|author2=Akihiro Kanamori|author3=John Woods, Jr.|title=Sets and Extensions in the Twentieth Century|url=https://books.google.com/books?id=ZF_QckMFy-oC&pg=PA733|year=2012|publisher=Elsevier|isbn=978-0-444-51621-3|page=733}}</ref>

=== 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 curve|algebraic curve]]s 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|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 [[Scheme (mathematics)|schemes]] that informally can be thought of as [[Topological space|topological space]]s on which a [[Commutative ring|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.<ref>{{cite book|author=Miles Reid|title=Undergraduate Algebraic Geometry|url=https://archive.org/details/undergraduatealg0000reid|url-access=registration|date=15 December 1988|publisher=Cambridge University Press|isbn=978-0-521-35662-6|page=[https://archive.org/details/undergraduatealg0000reid/page/115 115]}}</ref>

His generalization of the classical [[Riemann–Roch theorem]] related topological properties of complex [[Algebraic curve|algebraic curve]]s 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 K-theory|algebraic]] and [[Topological K-theory|topological K-theory]], which explores the topological properties of objects by associating them with rings.<ref name="Hartshorne AG">{{Hartshorne AG}}</ref> After direct contact with Grothendieck's ideas at the Bonn Arbeitstagung, topological K-theory was founded by [[Biography:Michael Atiyah|Michael Atiyah]] and [[Biography:Friedrich Hirzebruch|Friedrich Hirzebruch]].<ref name="Atiyah2014">{{cite book|author=Michael Atiyah|title=Michael Atiyah Collected Works: Volume 7: 2002–2013|url=https://books.google.com/books?id=Rm6VAwAAQBAJ&pg=PA383|date=3 April 2014|publisher=Oxford University Press|isbn=978-0-19-968926-2|pages=383–}}</ref>

=== Cohomology theories ===
Grothendieck's construction of new [[Cohomology|cohomology]] theories, which use algebraic techniques to study topological objects, has influenced the development of [[Algebraic number theory|algebraic number theory]], [[Algebraic topology|algebraic topology]], and [[Representation theory|representation theory]]. As part of this project, his creation of topos theory, a category-theoretic generalization of point-set topology, has influenced the fields of [[Set theory|set theory]] and [[Mathematical logic|mathematical logic]].<ref name=M&M />

The [[Weil conjectures]] were formulated in the later 1940s as a set of mathematical problems in [[Arithmetic geometry|arithmetic geometry]]. They describe properties of analytic invariants, called [[Local zeta function|local zeta function]]s, of the number of points on an algebraic curve or variety of higher dimension.  Grothendieck's discovery of the [[Étale cohomology|ℓ-adic 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 [[Biography:Pierre Deligne|Pierre Deligne]].<ref name="Hartshorne AG"/> Grothendieck's large-scale approach has been called a "visionary program".<ref>{{cite book|author1=M. Ram Murty|author2=V. Kumar Murty|title=The Mathematical Legacy of Srinivasa Ramanujan|url=https://books.google.com/books?id=r4tGFgO_nc8C&pg=PA156|date=6 October 2012|publisher=Springer Science & Business Media|isbn=978-81-322-0769-6|pages=156–}}</ref> The ℓ-adic cohomology then became a fundamental tool for number theorists, with applications to the [[Langlands program]].<ref>R. P. Langlands, Modular forms and l-adic representations, Lecture Notes in Math. 349. (1973), 361—500</ref>

Grothendieck's conjectural theory of [[Motive (algebraic geometry)|motive]]s 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|algebraic K-theory]], [[Motivic cohomology|motivic homotopy theory]], and [[Motivic integration|motivic integration]].<ref>{{cite book |author=J.S. Milne |title=Étale cohomology |publisher=Princeton University Press |year=1980}}</ref> This theory, [[Biography:Daniel Quillen|Daniel Quillen]]'s work, and Grothendieck's theory of [[Chern class]]es, are considered the background to the theory of [[Algebraic cobordism|algebraic cobordism]], another algebraic analogue of topological ideas.<ref>{{cite book|author1=Marc Levine|author2=Fabien Morel|title=Algebraic Cobordism|url=https://books.google.com/books?id=obVAAAAAQBAJ&pg=PR8|date=23 February 2007|publisher=Springer Science & Business Media|isbn=978-3-540-36824-3|page=viii}}</ref>

=== Category theory ===
Grothendieck's emphasis on the role of [[Universal property|universal properties]] across varied mathematical structures brought [[Category theory|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.<ref>{{Cite book|url=https://plato.stanford.edu/archives/win2015/entries/category-theory/|title=The Stanford Encyclopedia of Philosophy|last=Marquis|first=Jean-Pierre|date=2015|publisher=Metaphysics Research Lab, Stanford University|editor-last=Zalta|editor-first=Edward N.|edition=Winter 2015}}</ref> His notion of [[Abelian category|abelian category]] is now the basic object of study in [[Homological algebra|homological algebra]].<ref>{{cite book |author1=S. Gelfand |author2=Yuri Manin |title=Methods of homological algebra |year=1988 |publisher=Springer}}</ref> The emergence of a separate mathematical discipline of category theory has been attributed to Grothendieck's influence, although unintentional.<ref>{{cite book|author=Ralph Krömer|title=Tool and Object: A History and Philosophy of Category Theory|url=https://books.google.com/books?id=41bHxtHxjUAC&pg=PA158|date=25 June 2007|publisher=Springer Science & Business Media|isbn=978-3-7643-7524-9|pages=158–}}</ref>

=== 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.<ref>{{Cite web|url=http://www.restlessbooks.com/bookstore/colonel-lagrimas|title=Colonel Lágrimas|website=Restless Books|language=en-US|access-date=2017-09-12}}</ref>

The band ''Stone Hill All Stars'' have a song named after Alexander Grothendieck.<ref>{{Cite web|url=https://youtube.com/watch?v=rfAuCOT5z6U | title= Alexander Grothendieck | website= YouTube | language=en-US | access-date=2021-11-15}}</ref>

In the novel ''When We Cease to Understand the World'', Benjamin Labatut dedicates one chapter to the story of Grothendieck.<ref>{{cite book |last1=Labatut |first1=Benjamín |title=When we cease to understand the world |date=2020 |location=New York, NY |isbn=978-1-68137-566-3}}</ref>

In the novel ''The Passenger'' and its sequel ''Stella Maris'' by Cormac McCarthy, one of the main characters is a student of Grothendieck. <ref>{{cite web|title=CORMAC MCCARTHY HAS NEVER BEEN BETTER |website=The Atlantic |url=https://www.theatlantic.com/magazine/archive/2023/01/cormac-mccarthy-the-passenger-stella-maris-book-review/672234/|access-date=2022-12-05}}</ref>

== Publications ==

* {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}} <!-- {{sfn | Grothendieck | 1966 | p=}} -->
* {{Grothendieck Topological Vector Spaces}} <!-- {{sfn | Grothendieck | 1973 | p=}} -->

== See also ==
{{div col |colwidth=22em}}
* [[∞-groupoid|∞-groupoid]]
* [[Λ-ring|λ-ring]]
* [[AB5 category]]
* [[Abelian category]]
* [[Accessible category]]
* [[Algebraic geometry]]
* [[Algebraic stack]]
* {{annotated link|Approximation property}}
* [[Barsotti–Tate group]]
* [[Chern class]]
* {{annotated link|Crystal (mathematics)}}
* {{annotated link|Crystalline cohomology}}
* {{annotated link|Delta-functor}}
* {{annotated link|Derivator}}
* {{annotated link|Derived category}}
* [[Descent (mathematics)]]
* [[Dévissage]]
* {{annotated link|DF-space}}
* [[Dunford–Pettis property]]
* {{annotated link|Effaceable functor}}
* [[Excellent ring]]
* {{annotated link|Fibred category}}
* [[Formally smooth map]]
* {{annotated link|Fundamental groupoid}}
* [[Fundamental group scheme]]
* {{annotated link|Gorenstein ring}}
* {{annotated link|Grothendieck's Tôhoku paper}}
* [[K-theory]]
* [[Hilbert scheme]]
* [[Homotopy hypothesis]]
* {{annotated link|Infinitesimal cohomology}}
* List of things named after Alexander Grothendieck
* {{annotated link|Local cohomology}}
* [[Nakai conjecture]]
* {{annotated link|Motive (algebraic geometry)}}
* [[Nuclear operator]]
* [[Nuclear space]]
* [[Parafactorial local ring]]
* [[Projective tensor product]]
* {{annotated link|Pursuing Stacks}}
* Quasi-finite morphism
* [[Quot scheme]]
* {{annotated link|Ramanujam–Samuel theorem}}
* [[Scheme (mathematics)]]
* [[Section conjecture]]
* [[Semistable abelian variety]]
* [[Sheaf cohomology]]
* [[Stack (mathematics)]]
* [[Standard conjectures on algebraic cycles]]
* [[Esquisse d'un Programme|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]]
{{div col end}}

== Notes ==
{{notelist}}

== References ==
{{reflist}}

== Sources and further reading ==
{{refbegin}}
* {{cite book |last1=Grothendieck |first1=Alexander |title=Récoltes et semailles: réflexions et témoignage sur un passé de mathématicien |language=fr |date=1986 |publisher=Gallimard |location=Paris |url=http://lipn.univ-paris13.fr/~duchamp/Books&more/Grothendieck/RS/pdf/RetS.pdf |archive-date=18 August 2017 |archive-url=https://web.archive.org/web/20170818053357/http://lipn.univ-paris13.fr/~duchamp/Books%26more/Grothendieck/RS/pdf/RetS.pdf |url-status=dead |isbn=978-2-07-288980-6 }}
* {{cite journal|date=March 2016|title=Alexandre Grothendieck 1928–2014, Part 1|url=https://www.ams.org/publications/journals/notices/201603/rnoti-p242.pdf|archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/publications/journals/notices/201603/rnoti-p242.pdf|archive-date=2022-10-09|url-status=live|journal=[[Organization:Notices of the American Mathematical Society|Notices of the American Mathematical Society]]|volume=63|issue=3|pages=242–255|author1=[[Biography:Michael Artin|Michael Artin]]|author2=Allyn Jackson|author3=[[Biography:David Mumford|David Mumford]]|author4=John Tate|author5=((Coordinating Editors))|doi=10.1090/noti1336|doi-access=free}}
* {{cite journal|date=April 2016|title=Alexandre Grothendieck 1928–2014, Part 2|url=https://www.ams.org/publications/journals/notices/201604/rnoti-p401.pdf|archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/publications/journals/notices/201604/rnoti-p401.pdf|archive-date=2022-10-09|url-status=live|journal=[[Organization:Notices of the American Mathematical Society|Notices of the American Mathematical Society]]|volume=63|issue=4|pages=401–2413|author1=[[Biography:Michael Artin|Michael Artin]]|author2=Allyn Jackson|author3=[[Biography:David Mumford|David Mumford]]|author4=John Tate|author5=((Coordinating Editors))|doi=10.1090/noti1361|doi-access=free}}
* {{Citation
  | last = Cartier
  | first = Pierre
  | title = Les relations entre les mathématiques et la physique théorique – Festschrift for the 40th anniversary of the IHÉS
  | url = http://www.numdam.org/item?id=PMIHES_1998__S88__23_0
  | year = 1998
  | volume = S88
  | pages = 11–19
  | contribution = La folle journée, de Grothendieck à Connes et Kontsevich – Évolution des notions d'espace et de symétrie
  | publisher = Institut des Hautes Études Scientifiques
  }}
** {{Cite journal
| last = Cartier
| first = Pierre
| year = 2001
| title = A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry
| url = https://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf
| journal = [[Organization:Bulletin of the American Mathematical Society|Bulletin of the American Mathematical Society]]
| volume = 38
| issue = 4
| pages = 389–408
| doi = 10.1090/S0273-0979-01-00913-2
| doi-access = free
}} English translation of {{harvp|Cartier|1998}}.
* {{Citation
 |last          = Cartier
 |first         = Pierre
 |title         = Réel en mathématiques-psychanalyse et mathématiques
 |date          = 2004
 |chapter       = Un pays dont on ne connaîtrait que le nom (Grothendieck et les " motifs ")
 |chapter-url   = http://lipn.univ-paris13.fr/~duchamp/Books&more/Grothendieck/CartierHerreman2.pdf
 |publisher     = Editions Agalma
 |editor1-last  = Cartier
 |editor2-last  = Charraud
 |editor1-first = Pierre
 |editor2-first = Nathalie
 |language      = fr
 |url-status    = dead
 |archive-url   = https://web.archive.org/web/20160304001144/http://lipn.univ-paris13.fr/~duchamp/Books%26more/Grothendieck/CartierHerreman2.pdf
 |archive-date  = 4 March 2016
 |df            = dmy-all
}}
** English translation: {{cite web |url=http://xahlee.info/math/i/Alexander_Grothendieck_cartier.pdf |title=A country of which nothing is known but the name: Grothendieck and "motives" }}
*{{cite book |editor-last1=Cartier |editor-first1=Pierre |editor-last2=Illusie |editor-first2=Luc |editor-last3=Katz |editor-first3=Nicholas M. |editor-last4=Laumon |editor-first4=Gérard |editor-last5=Manin |editor-first5=Yuri I. |editor-last6=Ribet |editor-first6=Kenneth A. |editor-link2=Luc Illusie |editor-link3=Nick Katz |editor-link4=Gérard Laumon |editor-link5=Yuri Manin |editor-link6=Ken Ribet |title=The Grothendieck Festschrift, Volume I: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck |year=2007 |orig-year=1990 |publisher=Birkhäuser |isbn=978-0-8176-4566-3 |url=https://books.google.com/books?id=_BnSoQSKnNUC }}
**{{harvc |last=Dieudonné |first=Jean Alexandre |year=2007 |c=De L'analyse fonctionelle aux fondements de la géométrie algébrique |in1=Cartier |in2=Illusie |in3=Katz |in4=Laumon |in5=Manin |in6=Ribet}}
*{{cite book |editor1-last=Colmez |editor1-first=Pierre |editor2-last=Jean-Pierre |editor2-first=Serre |title=Grothendieck-Serre Correspondence: Bilingual Edition |publisher=AMS and the Société Mathématique de France |isbn=978-1-4704-6939-9 |page=600|year=2004|ref={{SfnRef|Colmez|Serre|2004}}}}
* {{Citation
  | last = Deligne
  | first = Pierre
  | title = Matériaux pour l'histoire des mathématiques au XXe siècle – Actes du colloque à la mémoire de Jean Dieudonné (Nice 1996)
  | year = 1998
  | pages = 11–19
  | contribution = Quelques idées maîtresses de l'œuvre de A. Grothendieck
  | contribution-url = http://smf.emath.fr/Publications/SeminairesCongres/1998/3/pdf/smf_sem-cong_3_11-19.pdf
  | publisher = Société Mathématique de France
  }}
*{{cite web
|url=http://images.math.cnrs.fr/Alexandre-Grothendieck.html
|title=Alexandre Grothendieck
|last=Douroux
|first=Philippe
|date=2012-02-08
|language=fr
|access-date=2014-04-02
}}
* {{cite book |last1=Hersh |first1=Reuben |last2=John-Steiner |first2=Vera |year=2011 |title=Loving and Hating Mathematics: Challenging the Myths of Mathematical Life |publisher=Princeton University Press |isbn=978-1400836116 |url=https://books.google.com/books?id=gvsHAnAuIp4C&pg=PA109 }}
*{{Citation
  | last = Jackson
  | first = Allyn
  | title = Comme Appelé du Néant – As If Summoned from the Void: The Life of Alexandre Grothendieck I
  | url = https://www.ams.org/notices/200409/fea-grothendieck-part1.pdf
  | year = 2004a
  | journal = Notices of the American Mathematical Society
  | volume = 51
  | issue = 4
  | pages = 1038–1056
  }}
*{{Citation
  | last = Jackson
  | first = Allyn
  | title = Comme Appelé du Néant – As If Summoned from the Void: The Life of Alexandre Grothendieck II
  | url = https://www.ams.org/notices/200410/fea-grothendieck-part2.pdf
  | year = 2004b
  | journal = Notices of the American Mathematical Society
  | volume = 51
  | issue = 10
  | pages = 1196–1212
  }}
*{{Citation
 |last         = Pragacz
 |first        = Piotr
 |title        = Topics in Cohomological Studies of Algebraic Varieties: Impanga Lecture Notes
 |year         = 2005
 |chapter      = Notes on the life and work of Alexander Grothendieck
 |chapter-url  = http://www.landsburg.com/grothendieck/pragasz.pdf
 |archive-url  = https://ghostarchive.org/archive/20221009/http://www.landsburg.com/grothendieck/pragasz.pdf
 |archive-date = 2022-10-09
 |url-status   = live
 |publisher    = Birkhäuser
 |editor-last  = Pragracz
 |editor-first = Piotr
}}
*{{Citation
  | last = Rehmeyer
  | first = Julie
  | title = Sensitivity to the Harmony of Things
  | date = 9 May 2008
  | url = http://www.sciencenews.org/view/generic/id/31898/title/Sensitivity_to_the_harmony_of_things
  | journal = Science News
  | access-date = 5 August 2008
  | archive-date = 16 February 2012
  | archive-url = https://web.archive.org/web/20120216201212/http://www.sciencenews.org/view/generic/id/31898/title/Sensitivity_to_the_harmony_of_things
  | url-status = dead
  }}
*{{Cite journal |last= Ribenboim |first= Paulo |title= Excerpt from ''The Grothendieck I Knew: Telling, Not Hiding, Not Judging'' By Paulo Ribenboim |journal= Notices of the American Mathematical Society |issue= August 2019 |pages= 1069–1077 |doi= 10.1090/noti1909 |ref= {{harvid|Ribenboim2019}} |doi-access= free }}
*{{cite book 
|last=Ruelle 
|first=David 
|title=The Mathematician's Brain 
|year=2007 
|url=https://books.google.com/books?id=B3A1bjOkOaEC 
|publisher=Princeton University Press 
|isbn=978-0691129822 
}}
* {{citation |last=Scharlau |first=Winfried |title=Wer ist Alexander Grothendieck?: Anarchie, Mathematik, Spiritualität |year=2007 |url=http://www.scharlau-online.de/ag_1.html }} First part of planned four-volume biography. 
** {{cite book |last=Scharlau |first=Winfried |year=2011 |title=Who is Alexander Grothendieck? Part 1: Anarchy |publisher=Books on Demand |oclc=801767784 |isbn=978-3842340923}} English version.
** {{cite web |last=Scharlau |first=Winfried |date=n.d. |title=Wer ist Alexander Grothendieck? |pages=1–22 |language=de |url=http://www.scharlau-online.de/DOKS/Wer%20ist%20AG.pdf }}
** {{cite journal
|last=Kleinert
|first=Werner
|year=2007<!-- year of *review* unclear; using year of book -->
|title=Wer ist Alexander Grothendieck? Anarchie, Mathematik, Spiritualität. Eine Biographie. Teil 1: Anarchie.
|language=en<!-- review is in English -->
|journal=zbMATH Open
|zbl=1129.01018
|trans-title=Who is Alexander Grothendieck? Anarchy, mathematics, spirituality. A biography. Part 1: Anarchy.
}} A review of the German edition
*{{Cite journal
 | last = Scharlau
 | first = Winfried
 | title = Who is Alexander Grothendieck ?
 | date = September 2008
 | url = https://www.ams.org/notices/200808/tx080800930p.pdf
 | journal = [[Organization:Notices of the American Mathematical Society|Notices of the American Mathematical Society]]
 | volume = 55
 | issue = 8
 | pages = 930–941
 | place = Providence, RI
 | publisher = American Mathematical Society
 | issn = 1088-9477
 | oclc = 34550461
 | access-date = 1 September 2011
 }}
* {{cite web
|url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Spirituality.html
|last=Scharlau
|first=Winfried
|year=2010
|title=Who is Alexander Grothendieck? Anarchy, Mathematics, Spirituality, Solitude. Part 3: Spirituality
|language=en
}} Third part of planned four-volume biography; crowd-financed translation into English.
* {{cite web
|url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathematics/
|last=Schneps
|first=Leila
|year=n.d.
|title=Who is Alexander Grothendieck? Anarchy, Mathematics, Spirituality, Solitude. Part 2: Mathematics
|language=en
}} First 4 chapters from the incomplete second part of planned four-volume biography.
*{{Citation
 | title = Alexandre Grothendieck: A Mathematical Portrait
 | date = 2014
 | location = Somerville Massachusetts
 | publisher = International Press of Boston, Inc.
 | editor-last = Schneps
 | editor-first = Leila
 | isbn = 978-1-57146-282-4
}}
*{{cite journal|last1=Serre|first1=Jean-Pierre|title=Faisceaux algébriques cohérents|journal=Annals of Mathematics|volume=61|pages=197–278|year=1955|issue=2|doi=10.2307/1969915|jstor=1969915|mr=0068874|url=https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf}}
{{refend}}

== External links ==
* https://www.grothendieckcircle.org is a website devoted to the life and works of Alexandre Grothendieck.
* {{MacTutor Biography|id=Grothendieck}}
* {{MathGenealogy|id=31245}}
* [http://users.unimi.it/barbieri/SG.html Séminaire Grothendieck] is a peripatetic seminar on Grothendieck view not just on mathematics 
* [https://web.archive.org/web/20221006020023/http://www.grothendieckcircle.org/ Grothendieck Circle], collection of mathematical and biographical information, photos, links to his writings
* [https://archive.today/20120722054515/http://www.bangor.ac.uk/r.brown/pstacks.htm 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 [https://web.archive.org/web/20100503001504/http://www.bangor.ac.uk/~mas013/ Tim Porter] at Bangor, which continued until 1991. See also [http://education.lms.ac.uk/2014/12/alexander-grothendieck-some-recollections/ Alexander Grothendieck: some recollections].
* [http://acm.math.spbu.ru/RS/ Récoltes et Semailles]
* [http://matematicas.unex.es/~navarro/res/ "Récoltes et Semailles" et "La Clef des Songes"], French originals and Spanish translations
* [http://www.landsburg.com/grothendieck/clef.pdf English summary of "La Clef des Songes"]
* [http://www.dailymotion.com/video/x8juek_colloque-grothendieck-winfried-scha_tech Video of a lecture] with photos from Grothendieck's life, given by Winfried Scharlau at IHES in 2009
* [http://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html Can one explain schemes to biologists] —biographical sketch of Grothendieck by [[Biography:David Mumford|David Mumford]] & John Tate
* [https://grothendieck.umontpellier.fr/archives-grothendieck/ Archives Grothendieck]
* "[https://www.ams.org/notices/200808/tx080800930p.pdf Who Is Alexander Grothendieck?], Winfried Scharlau, Notices of the AMS 55(8), 2008.
* "[https://www.ams.org/notices/201504/rnoti-p373.pdf Alexander Grothendieck: A Country Known Only by Name], Pierre Cartier, Notices of the AMS 62(4), 2015.
* [http://matematicas.unex.es/~navarro/res/notices201603.pdf Alexandre Grothendieck 1928–2014, Part 1], Notices of the AMS 63(3), 2016.
* [https://agrothendieck.github.io/ A. Grothendieck] by Mateo Carmona
* [https://www.lemonde.fr/sciences/article/2019/05/06/les-archives-insaisissables-d-alexandre-grothendieck_5459049_1650684.html Les-archives-insaisissables-d-alexandre-grothendieck]
* Kutateladze S.S. [http://www.math.nsc.ru/LBRT/g2/english/ssk/rebellious_genius_e.html Rebellious Genius: In Memory of Alexander Grothendieck]
* [https://www.lemonde.fr/sciences/article/2019/05/06/alexandre-grothendieck-une-mathematique-en-cathedrale-gothique_5459035_1650684.html Alexandre-Grothendieck-une-mathematique-en-cathedrale-gothique]
* [https://www.lemonde.fr/sciences/article/2019/05/06/les-archives-insaisissables-d-alexandre-grothendieck_5459049_1650684.html Les-archives-insaisissables-d-alexandre-grothendieck]
{{Fields medalists}}


{{DEFAULTSORT:Grothendieck, Alexander}}
[[Category:Algebraic geometers]]
[[Category:Algebraists]]
[[Category:Functional analysts]]

{{Sourceattribution|Alexander Grothendieck}}