Alexander Grothendieck

Alexander Grothendieck (1970)

Alexander Grothendieck (born March 28, 1928 in Berlin , † November 13, 2014 in Saint-Lizier near Saint-Girons , Département Ariège ) was a French mathematician of German origin. He was the founder of his own school of algebraic geometry , the development of which he significantly influenced in the 1960s. In 1966 he was awarded the Fields Medal , recognized as the highest honor in mathematics . Influenced by political ideas in France in May 1968 , he largely withdrew from his central position in the mathematical life of Paris by 1970. In 1991 he completely disappeared from the public eye; his last whereabouts in the Pyrenees were known only to a few friends.

Alexander Grothendieck's mathematical publications cover the areas of topology , algebraic geometry and functional analysis . His later works include thesis papers and meditation writings from the fields of ecology , philosophy , religion and, above all, esotericism .

Because a large part of his life and work took place in France , his name is often given as Alexandre Grothendieck , while he himself occasionally emphasized that he kept his original first name. However, on his handwritten letter, which he wrote on January 3, 2010 on the occasion of the removal of his writings himself, he signed with the French name form.

life and work

Origin and youth

Stumbling block at the house, Brunnenstrasse 165, in Berlin-Mitte

Alexander Grothendieck was born in Berlin, where his mother, the north German journalist and writer Johanna "Hanka" Grothendieck (1900–1957), lived and was still married to a man who was not his father (a half-sister Maida came from the relationship) . His father, Alexander Schapiro (1890–1942), lived illegally since 1921 under the name "Alexander Tanarow". He was of Hasidic origin, became an anarchist and, because he had been active in the Ukrainian Machno movement , had to leave Ukraine after the Russian October Revolution and go to Berlin. There he earned his living as a street photographer and met Hanka Grothendieck.

"Schurik", as Grothendieck was called as a child, spent his early childhood in Berlin with his parents. In 1933 the father fled to Paris from the National Socialists . The mother followed him a few months later and placed the son in the care of foster parents : Dagmar and Wilhelm Heydorn in Hamburg . Wilhelm Heydorn, a former Protestant theologian who had already joined the atheistic monist union in 1913 , remained politically active under the Nazi regime. Schurik attended elementary school and then high school in Hamburg-Blankenese .

His birth parents, Hanka Grothendieck and Alexander Schapiro, were involved in the Spanish civil war on the side of the anarcho-syndicalist groups .

After Grothendieck was brought to France by his parents in 1939, the family were interned in a concentration camp by the Vichy government in 1940. Alexander Schapiro was brought to Auschwitz-Birkenau concentration camp in 1942 and was murdered there as one of the first victims.

In 1942, Alexander Grothendieck escaped from the camp and went to Le Chambon-sur-Lignon in the Cevennes - the Protestant village that sheltered Jews during the German occupation of France . There he met André Trocmé . Grothendieck attended the Cevenol College there and graduated with a baccalauréat in 1945 . After the liberation by the Allies , mother and son were reunited. They remained closely connected until their mother's death - she died in 1957 of tuberculosis that she contracted during internment. In 1947 Grothendieck was co-founder of the International of Opponents of War (IDK) together with Theodor Michaltscheff .

Studies and functional analysis

From 1945 to 1948 Grothendieck studied mathematics in Montpellier , where he rediscovered the results of measure theory and the Lebesgue integral for himself . Then he changed the place of study, first to Paris to the École normal supérieure , where he attended the famous seminar of Henri Cartan . Since Grothendieck specialized in functional analysis , Cartan advised him at the end of 1949 to go to Jean Dieudonné and Laurent Schwartz in Nancy.

Grothendieck completed his influential dissertation on topological vector spaces in Nancy in 1953 , in which he solved many open problems with abstract algebraic (homological) methods ( tensor products and nuclear spaces, published in the Memoirs of the American Mathematical Society 1955). It is even said that he solved all of the problems on a list of 14 problems in one year, considered pioneering by distribution theory pioneer and Fields medalist Laurent Schwartz.

As in France at that time no posts in view were for Grothendieck - he remained until the summer of 1971, stateless, adoption of French citizenship would have meant military service - which complicated his candidacy, he went on the recommendation of friends in São Paulo and the University of Kansas , where he stayed until 1956. There he continued his series of fundamental works in functional analysis.

Algebraic Geometry

From 1955 Alexander Grothendieck turned to algebraic geometry . First he wrote an influential work in Kansas on the theory of Abelian categories used in the Tohoku Mathem. Journal appeared. He familiarized himself with the subject in Claude Chevalley's seminar in Paris and had intensive discussions with Jean-Pierre Serre , whose broad knowledge, including classical results, he repeatedly relied on (the correspondence between the two of them from this period was published in 2003). Here, too, he first tried to abstract the theory as far as possible: Within the framework of category theory, sentences about algebraic varieties were converted into those about mappings (morphisms) between categories of objects such as varieties and groups. His most impressive success for the mathematical world of that time was the abstract formulation of the Hirzebruch-Riemann-Roch theorem , which deals with the dimension of the space of the vector bundles over a variety (in the classic case of a Riemann surface ). Serre had already given a formulation as the alternating sum of the dimensions of the associated cohomology groups on the one hand, which is expressed in the theorem by topological invariants. The theorem was proved by Friedrich Hirzebruch using complicated topological methods. Grothendieck formulated and proved it in an abstract algebraic framework. The result was published in a work by Jean-Pierre Serre and Armand Borel in 1957 (allegedly Grothendieck himself was not abstract enough). This work also contains the origins of the topological K-theory of the 1960s, developed by Michael Atiyah and Hirzebruch , among others, in connection with the Atiyah-Singer index theorem . Grothendieck made a major breakthrough in this area and was asked at the ICM in Edinburgh in 1958 to give one of the plenary lectures (title: The Cohomology theory of abstract algebraic varieties ). Here he also sketched his later program to formulate an abstract topological homology theory in algebraic geometry, which is so general that it can be used simultaneously for fields such as complex and real numbers (classical algebraic geometry) as well as for finite and p - formulated adic bodies (number theory). Analogies between number fields and function fields (algebraic geometry) that have been known since the 19th century (e.g. Richard Dedekind , Heinrich Weber , Leopold Kronecker ) could find an explanation in this way in a natural way (it is still the case that sentences whose Proof for number fields is too difficult, can only be proven in the simpler case of “function fields ”).

Entrance of the IHES

Grothendieck worked on it intensively for the next twelve years (often twelve hours a day) in the center of a large school of algebraic geometricians such as Luc Illusie , Michael Artin , Jean-Louis Verdier , Michel Raynaud , Michèle Raynaud , Jacob Murre , Michel Demazure , Jean Giraud , Pierre Deligne , Pierre Gabriel , William Messing , Hoàng Xuân Sính , Monique Hakim and others who pushed his program forward. For several years (until 1960) he was also active in the Bourbaki circle. From 1959 he was at the Institut des Hautes Études Scientifiques (short: IHES) in Bures-sur-Yvette near Paris, but the seminar initially took place in Paris, and only from 1963 in Bures-sur-Yvette (at the same time Grothendieck moved there from Paris ). In the USA, too, where he lectured regularly at Harvard University from 1960 at the invitation of Oscar Zariski , a school was formed: Robin Hartshorne , who wrote a widespread textbook on Grothendieck's schema access to algebraic geometry, Barry Mazur , Nicholas Katz and other. Algebraic geometry was rebuilt around the concept of the schema , an idea originally from Pierre Cartier (1957). These are ring spaces, locally isomorphic to “Spec  (A) ”, the spectrum of a ring (set of prime ideals), which take the place of algebraic varieties. According to Grothendieck, the reason for the introduction of schemes was not the urge to generalize as much as possible, but rather that they contain nilpotent elements as natural objects of algebraic geometry and contribute to a better understanding of varieties. Special schemes are used for the different varieties occurring in classical algebraic geometry. In order to achieve his long-term goal, the proof of the Weil conjectures , Grothendieck also invented a new type of topology in algebraic geometry, which does not formalize algebraic sub-varieties like the already used Zariski topology , but the idea of ​​superimposition manifold over a base space as in the theory of Riemann surfaces or with algebraic number fields in the class field theory. He called this topology Étale Topologie ( étale French for expanded). By transferring ideas from Solomon Lefschetz from classical theory, Grothendieck succeeded in proving part of the Weil conjectures (rationality of the zeta function, functional equation). He formulated a series of “standard conjectures” about algebraic cycles from which these follow. While these are still unproven to this day, his colleague and student Pierre Deligne succeeded in 1974 in proving the last and most difficult of the Weil conjectures, the analogue to the Riemann conjecture, on the theoretical building erected by Grothendieck. In doing so, he used a trick from the classical theory of module functions that Grothendieck would not have thought of because of his limited knowledge of the literature. When Grothendieck had the evidence explained to him, he was disappointed that he was not being led along the route he had mapped out, and he lost all interest.

The fruits of this work from the 1960s are the Éléments de géométrie algébrique (EGA), written with Jean Dieudonné , and the extensive Seminaires de Geometry Algebrique du Bois Marie (SGA) (Bois Marie is the name of the forest in which the IHES is located) different authors. When asked why so few books were found in his seminar at IHES, Grothendieck replied that they would write them there themselves. From statements by Grothendieck himself it can be seen that when his intensive efforts to prove the Weil conjectures encountered obstacles towards the end of the 1960s, he was "burned out" around this time . Even today (2008) Grothendieck's conjectures developed at the end of the 1960s about connections between the various cohomology theories investigated by the Grothendieck School ( L- adic cohomology, crystalline cohomology and others) in algebraic geometry, the motifs (e.g. in conversations with Yuri Manin , who wrote an essay about this in 1968). An example from the classical algebraic geometry of the curves would be the assignment of special Abelian varieties , the Jacobi varieties , to the curve and its Riemann surface; In the Langlands program, connections to automorphic representations are assumed as motifs.

Turning away from math

In 1965 Grothendieck was elected to the American Academy of Arts and Sciences , and in 1966 he was honored with the Fields Medal , the highest distinction of the mathematical research community. For political reasons, however, he refused to travel to Moscow for the official ceremony . Grothendieck was already considered a promising candidate in 1958, but at that time few of his works were published and it was expected that he would receive the award in 1962, which may not have happened because he was already considered too established at the time. The criteria were changed again in 1966 when the general age limit of 40 years was introduced.

As early as the 1950s, Grothendieck had shaved a bald head like his father, whom he admired, and wore Russian peasant clothes. The student movement at the end of the 1960s made him politically active (in some cases probably also the model of his politically highly committed teacher Laurent Schwartz ), in 1967 he visited Hanoi and gave lectures there. His house in Paris was open to everyone. From 1970 Grothendieck began his retreat from mathematics and increasingly turned to ecology , philosophy and esotericism . He also resigned from his position at IHES when he learned that the institution had received funding from the French Ministry of Defense. He explored the religions , especially Buddhism (from 1974) and from the early 1980s Christian, mystical and esoteric ideas. In 1970 he and his two friends, the mathematicians Claude Chevalley and Pierre Samuel , founded the group Survivre et vivre , which advocated pacifist and ecological ideas against the background of the 1968 movement. In the years that followed, he became more and more committed to the alternative way of life of the 1960s and 1970s: at times he lived like in a commune . From the late 1960s he lived in his home in Massy .

He used the lectures held in Paris at the beginning of the 1970s at the Collège de France (where, after leaving IHES, he was professor for two years thanks to the use of Jean-Pierre Serre) and Orsay in Paris to talk about environmental protection and peace theory , and got into trouble with his superiors. At the International Congress of Mathematicians in Nice in 1970 (at which he was invited speaker with the lecture Groupes de Barsotti-Tate et cristaux ) he sold his group's newspaper and offended the organizer of the Congress Jean Dieudonné , in 1973 he opposed the Antwerp Conference on modular forms against funding from NATO and angered his long-time friend Jean-Pierre Serre . Grothendieck saw an ecological catastrophe approaching humanity that would make it impossible to deal with mathematics in the future as well, and consequently turned to non-mathematical questions.

In 1973 he moved to the small village of Villecun on the southern edge of the Cevennes and from then on he lived only in the countryside in small towns. In 1974 he became a professor at the University of Montpellier and from 1984 until his retirement in 1988 he held a position at the National Center for Scientific Research ( CNRS ). For example, his former colleague at IHES David Ruelle later saw the difficulties Grothendieck had after leaving IHES in obtaining a position appropriate for a mathematician in France as a scandal and ascribed it to the peculiarities of the French university and research system, where attending certain elite universities was and is of great importance to one's career and where Grothendieck remained an outsider. He held lectures in Montpellier until 1984, albeit not through his previous research program, but at an elementary level - and, according to former students, successfully. Mathematical memoranda of his, which he had circulated, also in the hope of leading a new research group at the CNRS, continued to cause a stir, for example his Esquisse d'un program (sketch of a program) from 1983, which consists of simple graphs on Riemann surfaces ( Dessins d'enfants, "children's drawings") and the effects of Galois groups (especially the absolute Galois group above the rational numbers) on these. He wrote an open letter to Gerd Faltings and propagated anabelian geometry, a new kind of synthesis around the modular spaces of algebraic curves. In a “letter” of almost 600 pages ( Pursuing stacks, A la poursuite des champs, 1983) to Daniel Gray Quillen , who was instrumental in developing the K theory initiated by Grothendieck, he showed interest in his theory of higher categories (his Book Homotopical algebra from 1967), on the basis of which he also saw a new basis for topology (including his own vision of a generalization from the 1960s, the topos theory).

On the other hand, there are rumors of irritating statements (golden age after a new Holocaust, small deviations in the natural constants are the work of the devil, criticism about former colleagues, etc.) in his writings Säen und Ernten (1983-1986) and The Key of Dreams, in which he pursued with the idea that God would talk to him in his dreams. Sowing and harvesting was originally intended as an introduction to pursuing stacks and was supposed to explain his new working style more intuitively, but then developed into a complex diary-like collection of thoughts on a wide variety of topics. In a 1000-page digression (The Burial) , he accused former students and employees of having buried his work and work style by stealing his ideas and not developing the “building sites” he left behind in 1970. In a La Lettre de la Bonne Nouvelle (Letter of Good News) to his friends in 1990, he announced the imminent approach of a “New Age of Liberation”, only to take back the visions in a letter shortly afterwards. Also in 1990, he underwent a religiously motivated, strict 45-day fasting period, as a result of which he almost died.

When he was about to be awarded the prestigious Swedish Crafoord Prize in 1988 , he turned down the prize. He justified this with a criticism of the politics of François Mitterrand as well as the lack of ethics and widespread moral corruption among his colleagues (letter to Le Monde , May 4, 1988, also in Mathematical Intelligencer 1989). The majority of his mathematician colleagues did not understand this behavior.

In 1991 Grothendieck disappeared from public life. He moved from his place of residence, Les Aumettes , and from then on lived in complete isolation, his exact whereabouts were known only to a few confidants. Earlier, he handed over in 1991 to his pupil Malgoire several boxes containing about 20,000 pages records from him his Sudelschriften called (gribouillis), which today are stored in the University of Montpellier, but are not determined by the will of Grothendieck for publication. According to Yves André , who briefly got a glimpse in 2004, this also includes mathematical writings.

In 1995 he gave the mathematician Malgoire a 2,000-page manuscript Les Derivateurs on the fundamentals of homotopy theory . He had already given his student Malgoire the 1,300-page manuscript The Long March through Galois Theory , which was written in a seminar in 1981 in Montpellier that he and Malgoire consisted of.

At the beginning of 2010, Grothendieck stated in a letter that he wished that his writings, which he had not agreed to publish, would no longer be published. The Grothendieck Circle website complied with this request and removed all of Grothendieck's writings from its offer. A project for a new edition of the SGA volumes also seems to have been suspended for the time being. By agreement with the heirs of Grothendieck, the University of Montpellier, where he last worked, puts around 18,000 pages from the Grothendieck estate online (which in turn is only part of the much more extensive written estate).

After his mother's death (1957), Grothendieck lived with Mireille Dufour, whom he married a few years later and with whom he had three children (born 1959, 1961, 1965). After the divorce in the early 1970s, he was in a relationship with the American Justine Skalba for two years, which resulted in a child.

He died on November 13, 2014 at the age of 86 in the Saint-Girons hospital in Saint-Lizier in the Ariège department (France), near the village of Lasserre , where he had retired since the early 1990s.

On March 22, 2017, stumbling blocks for him and his family were laid in front of his former home at Brunnenstrasse 165 in Berlin-Mitte .

Appreciation

Grothendieck was a theory builder par excellence. He always pushed for the greatest possible abstraction using homological algebra , but then made it fruitful for the proof of theorems. One example is his proof of his version of the Riemann-Roch theorem in the 1950s. Grothendieck himself had little knowledge of many areas of classical mathematics (even in algebraic geometry), as he himself admitted, but obtained the necessary information in discussions from friends like Jean-Pierre Serre . The long-term goal of his developments of algebraic geometry, which he abstracted until it was manageable on the same level as number theory, was to prove the Weil conjectures , in which his student and colleague Pierre Deligne was only successful in 1974.

In his autobiographical work Récoltes et Semailles , he summarizes his main results in 12 points in chronological order:

1. Topological Tensor Products and Nuclear Spaces
2. Continuous and discrete duality (derivative categories and "six operations" )
3. "Yoga" of the Riemann-Roch-Grothendieck theorem (K theory, connection to the intersection theory)
4. Schemes
5. Topos
6. Étale cohomology and l -adic cohomology
7. Motifs and motivic Galois group (and Grothendieck categories)
8. Crystals and crystalline cohomology, "Yoga" the "De Rham coefficient", "Hodge coefficient", ...
9. Topological algebra: infinity stacks. Dérivateurs. Cohomological formalism of Topoi, as inspiration for a new homotopic algebra
10. Tame topology (topology modérée, Tame topology)
11. "Yoga" Anabelian geometry , Galois-Teichmüller theory
12. Scheme - view or arithmetic view of regular polyhedra and regular configurations of all kinds

By “yoga” he understood the basic conception or the heuristic use of an idea in the context of a theory that is still partially unknown. Grothendieck assigned the first and last area of ​​the above list the least importance and considered the area of ​​topos theory to be the most important in terms of the development of a future theory that combines topology, number theory and algebraic geometry, together with the theory of schemes and most extensive of the twelve topics. According to Grothendieck, the schema theory forms the basis of the other topics (except 1, 5 and 10) and had been expanded relatively far by his school by 1970, which, according to Grothendieck, only made up a modest part of the potential of this theoretical building.

See also

• Grothendieck topology
• Grothendieck Universe

Mathematical publications

• Bibliography by Grothendieck. At: Leila.Schneps / GrothendieckCircle. PDF.

The main fonts are:

• Produits tensoriels topologiques et espaces nucléaires. In: Memoirs of the American Mathematical Society. Volume 16, 1955, pp. 1-140.
  • Summary Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires. In: Ann. Inst. Fourier. Volume 4, 1952, pp. 73-112, online (dissertation 1952).
• Résumé de la théorie métrique des produits tensoriels topologiques. In: Bol. Soc. Math. Sao Paulo. Volume 8, 1956, pp. 1-79 (from 1953), PDF.
• Sur quelques points d'algèbre homologique. In: Tôhoku Math J .. Volume 9, 1957, No. 2, pp. 119-221.
• Armand Borel, Jean-Pierre Serre: Le théorème de Riemann-Roch. In: Bulletin de la Société mathématique de France. Volume 86, 1958, pp. 97-136, PDF (presentation of the results by Grothendieck).
• Fondements de la Géometrie Algébrique (FGA), a series of contributions to the Séminaire Nicolas Bourbaki from 1957 to 1962 (Nos. 149, 182, 190, 195, 212, 221, 232, 236 and complement in Sèminaire Bourbaki, Volume 14, 1961 / 62)
• With Jean Dieudonné : Éléments de géométrie algébrique . (EGA). Volume 1 to 4. In: Publications mathématiques de l'IHES. 1960 to 1967:
  • EGA 1: Le Langage des schémas. In: Publications mathématiques de l'IHÉS. Volume 4, 1960, pp. 5-228, numdam.org.
  • EGA 2: Étude global élémentaire de quelques classes de morphismes. In: Publications mathématiques de l'IHÉS. Volume 8, 1961, pp. 5-22, numdam.org.
  • EGA 3: Étude cohomologique des faisceaux cohérents. In: Publications mathématiques de l'IHÉS. Volume 11, 1964, pp. 5-167, Volume 17, 1967, pp. 5-91, Volume 1 , Volume 2.
  • EGA 4: Étude locale des schémas et des morphismes de schémas. In: Publications mathématiques de l'IHÉS. Volume 20, 1964, pp. 5-259, Volume 24, 1964, pp. 5-231, Volume 28, 1966, pp. 5-255, Volume 32, 1967, pp. 5-361, Volume 1 , Volume 2 , Volume 3 , Volume 4.
• Séminaire de géométrie algébrique du Bois Marie . (SGA). 1960 to 1969, online :
  • Michèle Raynaud : SGA 1: Revêtements étales et groupe fondamental. 1960–1961, Springer Verlag. In: Lecture Notes in Mathematics. Volume 224, 1971, pp. 1-447 and SMF: Collection Documents Mathématiques. Paris 2003.
  • Michèle Raynaud: SGA 2: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. 1961–1962, In: Advanced Studies in Pure Mathematics. North Holland, 1968, pp. 1-287, as well as SMF: Collection Documents Mathématiques. Paris 2005.
  • Michel Demazure : SGA 3: Schémas en groupes. 1962–1964, Springer Verlag. In: Lecture Notes in Mathematics. Volume 151, 1970, pp. 1-564, Volume 152, 1970, pp. 1-654, Volume 153, 1970, pp. 1-529.
  • Michael Artin , Jean-Louis Verdier and others: SGA 4: Théorie des topos et cohomologie étale des schémas. 1963–1964, Springer Verlag. In: Lecture Notes in Mathematics. Volume 269, 1972, pp. 1-535, Volume 270, 1972, pp. 1-418, Volume 305, 1973, pp. 1-640.
  • Pierre Deligne (Ed.): SGA 4 1/2: Cohomologie étale. Springer publishing house. In: Lecture Notes in Mathematics. Volume 569, 1977, pp. 1-312.
  • Luc Illusie , Jean-Pierre Serre et al .: SGA 5: Cohomologie l-adique et fonctions L. 1965–1966. In: Lecture Notes in Mathematics. Volume 589, 1977, pp. 1-484.
  • Luc Illusie, Pierre Berthelot and others: SGA 6: Théorie des intersections et théorème de Riemann-Roch. 1966–1967, Springer Verlag. In: Lecture Notes in Mathematics. Volume 225, 1971, pp. 1-700.
  • Pierre Deligne, Nicholas Katz : SGA 7: Groupes de monodromie en géométrie algébrique. 1967–1969, Springer Verlag. In: Lecture Notes in Mathematics. Volume 288, 340, 1972-1973, pp. 1-523, 1-438.
• Standard conjectures on algebraic cycles. In: Algebraic Geometry (International Colloquium Tata Institute, Bombay, 1968). Oxford University Press, 1969, pp. 193-199.
• Pursuing stacks. 1983 manuscript.
• Esquisse d'un programs. 1984, manuscript, 44 pages, PDF.
• Les Dérivateurs. 1991 (manuscript, 1976 pages), online, edited by M. Künzer, J. Malgoire, G. Maltsiniotis.

Other publications, including the contributions by Grothendieck to the Cartan, Chevalley and Bourbaki seminars , are available online (PDF format) in the numdam.org database . There were various projects to make Grothendieck's writings accessible online, for example a modern transcription by SGA ( Bas Edixhoven ) and in the Grothendieck Circle , which were temporarily suspended or restricted in 2010 after the letter from Grothendieck, who did not want this, became known .

Meditation writings

Alexander Grothendieck wrote various unpublished meditation writings . Its main ones include:

• Eloge de l'inceste. 1981 (poetic work, lost).
• Récoltes et Semailles, Réflexions et témoignage sur un passé de mathématicien. (“Harvesting and sowing”), University of Montpellier and CNRS 1985, PDF.
• La clef des songes - ou dialogue avec le Bon Dieu. 1986 ("The key to dreams - a dialogue with God").
• Notes pour La clef des songes. 1987 (notes to "Key to Dreams").

literature

• Winfried Scharlau : Who is Alexander Grothendieck? Anarchy, math, spirituality, loneliness. A biography. Of the four planned volumes that have been published so far:
  • Part 1: anarchy. Books-on-Demand 2009, ISBN 978-3-8423-7147-7 .
  • Part 3: Spirituality. Books-on-Demand 2010, ISBN 978-3-8391-4939-3 .
• Winfried Scharlau: From world star to hermit. In: Spectrum of Science . 2015, issue 7, pp. 52–60.
• Jean-Pierre Serre and Pierre Colmez (eds.): Grothendieck-Serre correspondence. AMS 2003.
• Pierre Cartier, Luc Illusie, Nicholas Katz et al. (Eds.): Grothendieck Festschrift. 3 vols., Birkhäuser, Basel 1990, ISBN 3-7643-3429-0 (for the complete edition); with a bibliography of his writings from 1950–1973, pp. XIII – XX.
• Pierre Cartier: A mad days work - from Grothendieck to Connes and Kontsevich. Bulletin AMS 2001, online here: Bulletin of the AMS - Volume 38, Number 4.
• Pierre Cartier: Grothendieck et les motifs. IHES 2000 preprint, online here: Pierre Cartier Prepublications.
• Pierre Cartier: Alexander Grothendieck. A country known only by name. Notices AMS, Volume 62, 2015, No. 4, p. 373, PDF. , Inference 2020
• Pierre Cartier: A country of which one knows nothing except the name: Grothendieck and “Motive”. e-enterprise, 2016.
• Pierre Deligne: Quelques idées maîtresses de l'œuvre de A. Grothendieck. , in: Michele Audin (ed.), Matériaux pour l'histoire des mathématiques au XXe siècle Actes du colloque à la mémoire de Jean Dieudonné (Nice 1996), SMF 1998, PDF.
• Leila Schneps , Pierre Lochak (Eds.): Geometric Galois actions - around Grothendiecks Esquisse d'un program. London Math. Society Lecture Notes, Cambridge 1997 (with Grothendiecks Esquisse ).
• Leila Schneps (Ed.): Alexandre Grothendieck: A Mathematical Portrait. International Press of Boston, 2014, ISBN 978-1-57146-282-4 .
• Pragacz: The life and work of Alexander Grothendieck. American Mathematical Monthly, November 2006.
• Robin Hartshorne: Algebraic geometry. Springer 1997 (standard textbook on Grothendieck's approach).

In 2013, a French documentary was made about Grothendieck by Catherine Aira (Alexander Grothendieck, sur les routes d'un génie).

Web links

Commons : Alexander Grothendieck  - Collection of images, videos and audio files

• Literature by and about Alexander Grothendieck in the catalog of the German National Library
• John J. O'Connor, Edmund F. Robertson :  Alexander Grothendieck. In: MacTutor History of Mathematics archive .
• Alexander Grothendieck in the Mathematics Genealogy Project (English)
• University of Montpellier, Grothendieck Archives
• Grothendieck Circle , there also links to online work (e.g. EGA, SGA, Esquisse, Recoltes et semailles).
• Allyn Jackson: Comme Appelé du Néant - As If Summoned from the Void: The Life of Alexandre Grothendieck. PDF, PDF.
• Scharlau to Grothendieck. Notices AMS, September 2008.
• Artin et al: Grothendieck. Notices AMS, March 2016 (and Colin McLarty: How Alexander Grothendieck simplified algebraic geometry ) and Notices AMS, April 2016.
• Barry Mazur: What is a motive? Notices AMS, November 2004 (English, PDF).
• Luc Illusie: What is a topos? Notices AMS, October 2004 (English, PDF).
• Honoring Scharlau on the occasion of his 80th birthday. In: Zeit.de. March 27, 2008.
• Scharlau u. a. with materials on his planned Grothendieck biography.
• Deutschlandradio: Culture on Grothendieck.
• Philippe Douroux: Alexandre Grothendieck, ou la mort d'un génie qui voulait se faire oublier. In: Liberation. November 13, 2014.
• Lonely genius: Mathematics monk Alexander Grothendieck is dead. In: Spiegel.de. November 14, 2014.
• Jacob Murre: Memories of Grothendieck. Nieuw Archief voor Wiskunde, March 2016, PDF.
• A. Grothendieck , Mateo Carmona

Individual evidence

1. ↑ David Mumford, John Tate: Alexander Grothendieck (1928-2014). In: Nature . Volume 517, No. 7534, 2015, p. 272, doi: 10.1038 / 517272a .
2. ↑ ^{a b} Grothendieck's letter . Sbseminar.wordpress.com. February 9, 2010. Retrieved September 16, 2010.
3. ↑ Scharlau: "Spirituality." 2010, p. 75.
4. ↑ In Recoltes et Semailles , chapter 8.1, he lists (his "twelve apostles") as a student from the very beginning: Yves Ladegaillerie, Michel and Michèle Raynaud, Demazure, Verdier, Illusie, P. Joanolou, Sinh, Hakim, Giraud, Berthelot , N. Saavedra, six of whom only received their doctorate after 1970, when Grothendieck was already working on other topics. His students in Montpellier included Carlos Contou-Carrère, Sinh, Hakim, Ladegaillerie (all of whom did their PhDs with him), Jean Malgoire and Christine Voisin.
5. ^ Jacob Murre: Interview and memories of Grothendieck. Nieuw Archief voor Wiskunde, March 2016.
6. ↑ Michael Barany, The Fields Medal should return to its roots , Nature, January 12, 2018
7. ↑ For example Sylvia Nasar in her John Nash biography Beautiful mind.
8. ↑ Alexandre Grothendieck: mort d'un génie des maths. In: Le Monde. 15th November 2014.
9. ↑ Murre: Interview. Nieuw Archief voor Wiskunde, March 2016.
10. ↑ Source: The mathematicians brain. Princeton University Press 2007, p. 40.
11. ↑ Winfried Scharlau : Who is Alexander Grothendieck?
12. ↑ Philippe Douroux : Le trésor oublié du genie of maths. In: Liberation. July 1, 2012.
13. ↑ Nicolas Bourbaki : La disparation. 2009, PDF.
14. ↑ Pierre Ropert, Les notes manuscrites de Grothendieck, un trésor des mathématiques maintenant en libre accès , France Culture, May 11, 2017
15. ↑ recoltes et Semailles. Section 2.8 (La Vision, ou douze thèmes pour une harmonie), P21.
16. ^ Six Operations, n-Lab.
17. ↑ Grothendieck writes: -catégories de Grothendieck ${\displaystyle \otimes }$
18. ↑ Literally: champs. ${\displaystyle \infty }$
19. ↑ Recoltes et Semailles. P21.
20. ↑ Ronald Brown: The origins of Alexander Grothendieck's 'Pursuing Stacks'.
21. ^ Alexander Grothendieck in the database of numdam.org . Mathdoc. Retrieved January 18, 2019.
22. ↑ Grothendieck Circle , on grothendieckcircle.org
23. ↑ Website for the Grothendieck film.