Editing Alexander Grothendieck (section)

=== 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>