- Alexander Grothendieck
User:Geometry guy/Infobox

**Alexander Grothendieck**(bornMarch 28 ,1928 inBerlin ,Germany ) is considered to be one of the greatestmathematician s of the 20th century. He made major contributions to:algebraic topology ,algebraic geometry ,number theory ,category theory ,Galois theory ,descent theory commutativehomological algebra andfunctional analysis . He was awarded theFields Medal in 1966, and was co-awarded theCrafoord Prize withPierre Deligne in 1988. He declined the latter prize on ethical grounds in an open letter to the media.He is noted for his mastery of abstract approaches to mathematics, and his perfectionism in matters of formulation and presentation. In particular, he demonstrated the ability to derive concrete results using only very general methods. [

*See, for example, harv|Deligne|1998.*] Citation

last =Jackson

first =Allyn

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title = Comme Appelé du Néant — As If Summoned from the Void: The Life of Alexandre Grothendieck I

journal = Notices of the American Mathematical Society

volume = 51

issue = 4

pages = p. 1049

date = 2004

year =

url = http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf

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id = ] cite web

last = Mclarty

first = Colin

title=The Rising Sea: Grothendieck on simplicity and generality I

url=http://www.math.jussieu.fr/~leila/grothendieckcircle/mclarty1.pdf

format =PDF

accessdate=2008-01-13] Relatively little of his work after 1960 was published by the conventional route of thelearned journal , circulating initially in duplicated volumes of seminar notes; his influence was to a considerable extent personal, on French mathematics and the Zariski school atHarvard University . He is the subject of many stories and some misleading rumors concerning his work habits and politics, his confrontations with other mathematicians and the French authorities, his withdrawal from mathematics at age 42, his retirement, and his subsequent lengthy writings.**Mathematical achievements**Homological methods and sheaf theory had already been introduced in algebraic geometry by

Jean-Pierre Serre and others, after sheaves had been described byJean Leray . Grothendieck took them to a higher level of abstraction and turned them into the key organising principle of his theory. He thereby changed the tools and the level of abstraction in algebraic geometry.Amongst his insights, he shifted attention from the study of individual varieties to the "relative point of view" (pairs of varieties related by a

morphism ), allowing a broad generalization of many classical theorems. This he applied first to theRiemann–Roch theorem , around 1956, which had already recently been generalized to any dimension by Hirzebruch. TheGrothendieck–Riemann–Roch theorem was announced by Grothendieck at the initialMathematische Arbeitstagung inBonn , in 1957. It appeared in print in a paper written byArmand Borel with Serre.His foundational work on

algebraic geometry is at a higher level of abstraction than all prior versions. He adapted the use of non-closedgeneric point s, which led to the theory of schemes. He also pioneered the systematic use ofnilpotent s. As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings. His "theory of schemes" has become established as the best universal foundation for this major field, because of its great expressive power as well as technical depth. In that setting one can usebirational geometry , techniques fromnumber theory ,Galois theory andcommutative algebra , and close analogues of the methods ofalgebraic topology , all in an integrated way.Its influence spilled over into many other branches of mathematics, for example the contemporary theory of

D-module s. (It also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems. Grothendieck is one of the few mathematicians who matches the French concept ofmaître à penser ; some go further and call himmaître-penseur .)**EGA and SGA**The bulk of Grothendieck's published work is collected in the monumental, and 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 theSéminaire Bourbaki , also contains important material.Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and

l-adic cohomology theories, which explain an observation ofAndré Weil 's that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation over afinite field reflects the topological nature of its solutions over thecomplex number s. Weil realized that to prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do this until such a theory was found by Grothendieck.This program culminated in the proofs of the

Weil conjectures by Grothendieck's studentPierre Deligne in the early 1970s after Grothendieck had largely withdrawn from mathematics.**Major mathematical topics (from "Récoltes et Semailles")**He wrote a retrospective assessment of his mathematical work (see the external link "La Vision" below). As his main mathematical achievements ("maître-thèmes"), he chose this collection of 12 topics (his chronological order):

#

Topological tensor product s andnuclear space s

#"Continuous" and "discrete"duality (derived categories and "six operations").

#"Yoga" of theGrothendieck–Riemann–Roch theorem (K-theory , relation withintersection theory ).

#Schemes.

#Topoi.

#Étale cohomology includingl-adic cohomology .

#Motives and themotivic Galois group (and Grothendieck categories)

#Crystals andcrystalline cohomology , "yoga" of De Rham and Hodge coefficients.

#Topological algebra, infinity-stacks, 'dérivateurs', cohomological formalism of toposes as an inspiration for a newhomotopic algebra

#Tame topology .

#"Yoga" ofanabelian geometry andGalois–Teichmüller theory .

#Schematic point of view, or "arithmetics" for regular polyhedra andregular configurations of all sorts.He wrote that the central theme of the topics above is that of

topos theory, while the first and last were of the least importance to him.Here the term "yoga" denotes a kind of "meta-theory" that can be used heuristically.Clarifyme|date=March 2008 The word "yoke", meaning "linkage", is derived from the same Indo-European root.

**Life****Family and early life**Born to a Russian father of

Jewish parentage, Alexander Shapiro, and a mother of GermanProtestant origin, Hanka Grothendieck, inBerlin . He was adisplaced person during much of his childhood due to the upheavals ofWorld War II . Alexander lived with his parents both of whom wereanarchist s, until 1933, inBerlin . At the end of that year, Shapiro moved toParis , and Hanka followed him the next year. They left Alexander with a family inHamburg where he went to school. During this time, his parents fought in theSpanish Civil War .**During WWII**In 1939 Alexander came to France and lived in various camps for displaced persons with his mother, first at the

Camp de Rieucros , spending 1942–44 atLe Chambon-sur-Lignon . His father was sent viaDrancy to Auschwitz where he died in 1942.**tudies and contact with research mathematics**After the war, young Grothendieck studied mathematics in

France , initially at theUniversity of Montpellier . He had decided to become a math teacher because he had been told that mathematical research had been completed early in the 20th century and there were no more open problems. [*See Jackson (2004:1). The remark is from the beginning of Récoltes et Semailles (page P4, in the introductory section Prélude en quatre Mouvements)*] However, his talent was noticed, and he was encouraged to go toParis in 1948.Initially, Grothendieck attended

Henri Cartan 's Seminar atÉcole Normale Supérieure , but lacking the necessary background to follow the high-powered seminar, he moved to theUniversity of Nancy where he wrote his dissertation underLaurent Schwartz in functional analysis, from 1950 to 1953. At this time he was a leading expert in the theory oftopological vector space s. By 1957, he set this subject aside in order to work in algebraic geometry andhomological algebra .**The IHÉS years**Installed at the

Institut des Hautes Études Scientifiques (IHÉS), Grothendieck attracted attention, first by his spectacularGrothendieck-Riemann-Roch theorem , and then by an intense and highly productive activity of seminars ("de facto" working groups drafting into foundational work some of the ablest French and other mathematicians of the younger generation). Grothendieck himself practically ceased publication of papers through the conventional,learned journal route. He was, however, able to play a dominant role in mathematics for around a decade, gathering a strong school.During this time he had officially as students

Michel Demazure (who worked on SGA3, ongroup scheme s),Luc Illusie (cotangent complex ),Michel Raynaud ,Jean-Louis Verdier (cofounder of thederived category theory) andPierre Deligne . Collaborators on the SGA projects also includedMike Artin (étale cohomology ) andNick Katz (monodromy theory andLefschetz pencil s). Jean Giraud worked outtorsor theory extensions ofnon-abelian cohomology . Many others were involved.**Politics and retreat from scientific community**Grothendieck's political views were radical left-wing and pacifist. He gave lectures on

category theory in the forests surroundingHanoi while the city was being bombed, to protest against theVietnam War ("The Life and Work of Alexander Grothendieck", American Math. Monthly, vol. 113, no. 9, footnote 6). He retired from scientific life around 1970, after having discovered the partly military funding ofIHÉS (see pp. xii and xiii of SGA1, Springer Lecture Notes 224). He returned to academia a few years later as a professor at the University ofMontpellier , where he stayed until his retirement in 1988. His criticisms of the scientific community are also contained in a [*http://web.archive.org/web/20060106062005/http://www.math.columbia.edu/~lipyan/CrafoordPrize.pdf letter*] , written in 1988, in which he states the reasons for his refusal of theCrafoord Prize .**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. During that period he also released his work on Bertini type theorems contained in EGA 5, published by the [

*http://www.math.jussieu.fr/~leila/grothendieckcircle/index.php Grothendieck Circle*] in 2004."La Longue Marche à travers la théorie de Galois" ("The Long March Through Galois Theory") is an approximately 1600-page handwritten manuscript produced by Grothendieck during the years 1980-1981, containing many of the ideas leading to the "Esquisse d'un programme" [

*http://kolmogorov.unex.es/~navarro/res/esquissefr.pdf*] (see below, and also a more detailed entry [*http://en.wikipedia.org/wiki/Esquisse_d%27un_Programme*] ), and in particular studying the Teichmüller theory.In 1983 he wrote a huge extended manuscript (about 600 pages) entitled "Pursuing Stacks", stimulated by correspondence with

Ronald Brown (mathematician) , (see also [*http://www.bangor.ac.uk/r.brown R.Brown*] and [*http://www.informatics.bangor.ac.uk/~tporter/ Tim Porter*] atUniversity of Bangor in Wales), and starting with a letter addressed toDaniel Quillen . This letter and successive parts were distributed from Bangor (see External Links below): in an informal manner, as a kind of diary, Grothendieck explained and developed his ideas on the relationship betweenalgebraic homotopy theory andalgebraic geometry and prospects for a noncommutative theory of stacks. The manuscript, which is being edited for publication by G. Maltsiniotis, later led to another of his monumental works, "Les Dérivateurs". Written in 1991, this latter opus of about 2000 pages further developed the homotopical ideas begun in "Pursuing Stacks". Much of this work anticipated the subsequent development of the motivic homotopy theory ofF. Morel andV. Voevodsky in the mid 1990s.His "Esquisse d'un programme" [

*http://kolmogorov.unex.es/~navarro/res/esquissefr.pdf*] (1984) is a proposal for a position at theCentre National de la Recherche Scientifique , which he held from 1984 to his retirement in 1988. Ideas from it have proved influential, and have been developed by others, in particular dessins d'enfants and a new field emerging asanabelian geometry . In "La Clef des Songes " he explains how the reality ofdream s convinced him ofGod 's existenceFact|date=August 2008.The 1000-page autobiographical manuscript "Récoltes et semailles" (1986) is now available on the internet in the French original, and an English translation is underway (these parts of Récoltes et semailles have already been [

*http://www.mccme.ru/free-books/grothendieck/RS.html translated into Russian*] and published in Moscow). Some parts of "Récoltes et semailles" [*http://kolmogorov.unex.es/~navarro/res/preludio.pdf*] [*http://kolmogorov.unex.es/~navarro/res/carta.pdf*] and the whole "Le Clef des Songes" [*http://kolmogorov.unex.es/~navarro/res/clef1-6.pdf*] have been translated into Spanish.**Disappearance**In 1991, he left his home and disappeared. He is now said to live in southern France or

Andorra and to entertain no visitors. Though he has been inactive in mathematics for many years, he remains one of the greatest and most influential mathematicians of modern times.**ee also**

*Birkhoff-Grothendieck theorem

*Grothendieck's connectedness theorem

*Grothendieck connection

*Grothendieck's Galois theory

*Grothendieck group

* Grothendieck category [*http://planetmath.org/encyclopedia/GrothendieckCategory.html*]

*Grothendieck inequality orGrothendieck constant

*Grothendieck–Katz p-curvature conjecture

*Grothendieck's relative point of view

*Grothendieck-Riemann-Roch theorem

*Grothendieck's Séminaire de géométrie algébrique

*Grothendieck space

*Grothendieck spectral sequence

*Grothendieck topology

*Grothendieck universe

*Tarski-Grothendieck set theory **Notes****References*** Citation

first = Pierre

last = Cartier

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editor-last =

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contribution = La Folle Journée, de Grothendieck à Connes et Kontsevich — Évolution des Notions d'Espace et de Symétrie

contribution-url =

title = Les Relations entre les Mathématiques et la Physique Théorique — Festschrift for the 40th anniversary of the IHÉS

year = 1998

pages = 11–19

place =

publisher = Institut des Hautes Études Scientifiques

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doi =

id =

* Citation

last = Cartier

first = Pierre

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last2 =

first2 =

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title = A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry

journal = Bull. Amer. Math. Soc.

volume = 38

issue = 4

pages = 389–408

date = 2001

year =

url = http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf

doi =

id = . An English translation of Cartier (1998)

* Citation

first = Pierre

last = Deligne

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editor-last =

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editor2-first =

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

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

place =

publisher = Société Mathématique de France

url =

doi =

id =

*Citation

last =Jackson

first =Allyn

author-link =

last2 =

first2 =

author2-link =

title = Comme Appelé du Néant — As If Summoned from the Void: The Life of Alexandre Grothendieck I

journal = Notices of the American Mathematical Society

volume = 51

issue = 4

pages = 1038–1056

date = 2004

year =

url = http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf

doi =

id =

*Citation

last =Jackson

first =Allyn

author-link =

last2 =

first2 =

author2-link =

title = Comme Appelé du Néant — As If Summoned from the Void: The Life of Alexandre Grothendieck II

journal = Notices of the American Mathematical Society

volume = 51

issue = 10

pages = 1196–1212

date = 2004

year =

url = http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf

doi =

id =

*Citation

last =Rehmeyer

first =Julie

author-link =

last2 =

first2 =

author2-link =

title = Sensitivity to the Harmony of Things

journal = Science News

volume =

issue =

pages =

date = May 9, 2008

year =

url = http://www.sciencenews.org/view/generic/id/31898/title/Sensitivity_to_the_harmony_of_things

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*citation|last=Scharlau|first=Winfred|title=Wer ist Alexander Grothendieck?: Anarchie,Mathematik, Spiritualität|url=http://www.scharlau-online.de/ag_1.html Three-volume biography.

*Citation

last = Scharlau

first = Winifred

year = 2008

date = September 2008

title = Who is Alexander Grothendieck

periodical =Notices of the American Mathematical Society

volume = 55

issue = 8

pages = 930–941

place =Oberwolfach ,Germany

publication-place = Providence, RI

publisher =American Mathematical Society

issn = 1088-9477

doi =

oclc = 34550461

url = http://www.ams.org/notices/200808/tx080800930p.pdf

accessdate = 2008-09-02**External links***

*

* [*http://www.math.jussieu.fr/~leila/grothendieckcircle/index.php Grothendieck Circle*] , collection of mathematical and biographical information, photos, links to his writings

** [*http://gavrilov.akatov.com/Grothendieck Grothendieck Circle discussion Forum*]

* [*http://www.ihes.fr Institut des Hautes Études Scientifiques*]

* [*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 Tim Porter at Bangor, which continued until 1991.

* [*http://acm.math.spbu.ru/RS/ Récoltes et Semailles*] in French.

* [*http://kolmogorov.unex.es/~navarro/res/ Spanish translation*] of "Récoltes et Semailles" et "Le Clef des Songes" and other Grothendieck's texts

* [*http://www.ams.org/notices/200808/tx080800930p.pdf short bio*] fromNotices of the American Mathematical Society

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