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Alexander Grothendieck

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Alexander Grothendieck
Alexander Grothendieck in Montreal, 1970
Born (1928-03-28) 28 March 1928 (age 96)
Berlin, Germany
Alma materUniversity of Montpellier
University of Nancy
AwardsFields Medal (1966), Crafoord Prize(1988, declined)
Scientific career
FieldsMathematics
Doctoral advisorLaurent Schwartz
Doctoral studentsJean-Louis Verdier
Pierre Deligne
Michel Raynaud
Luc Illusie

Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most influential mathematicians of the twentieth century. He is known principally for his revolutionary advances in algebraic geometry, and also for major contributions to algebraic topology, number theory, category theory, Galois theory, descent theory, commutative and homological algebra, and functional analysis. He was awarded the Fields Medal in 1966. He was co-awarded the Crafoord Prize, with Pierre Deligne, in 1988, but Grothendieck declined this.

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. 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, on French mathematics and the Zariski school at Harvard University. He retired in 1988 and within a few years became reclusive.

Mathematical achievements

Grothendieck's early mathematical work was done in functional analysis between 1949 and 1953 working on (what became) his doctoral thesis in Nancy, supervised by Jean Dieudonné and Laurent Schwartz. His key contributions include topological tensor products of vector spaces, the theory of nuclear spaces and the application of Lp spaces in studying linear maps between topological vector spaces. In the space of a few years, he had turned himself into a leading authority on the theory of topological vector spaces — to the extent that Dieudonné compares his impact in this field to that of Banach.[1]

However, it is algebraic geometry and related fields where Grothendieck did his most important and influential work. From about 1955 he started to work on sheaf theory and homological algebra, rapidly producing the very influential "Tôhoku paper" (Sur quelques points d'algèbre homologique, published in 1957) where he introduced Abelian categories and applied it to show that sheaf cohomology can be defined as certain derived functors in this context.

Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and others, after sheaves had been defined by Jean Leray. Grothendieck took them to a higher level of abstraction and turned them into a key organising principle of his theory. He thereby changed the tools and the level of abstraction in algebraic geometry. 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. The first major application was the relative version of Serre's theorem showing that the cohomology of a 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.

Next, in 1956, he applied the same thinking to the Riemann–Roch theorem, which had already recently been generalized to any dimension by Hirzebruch. The Grothendieck–Riemann–Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957. It appeared in print in a paper written by Armand Borel with Serre. This result was his first major achievement in algebraic geometry. He went on to plan and execute a major foundational programme for rebuilding the foundations of algebraic geometry; he exposed the main outlines of this 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 points, which led to the theory of schemes. He also pioneered the systematic use of nilpotents. 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 use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.[2][3][4]

His influence spilled over into many other branches of mathematics, for example the contemporary theory of D-modules. (It also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems.)[5][6]

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 the Séminaire 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 of André 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 a finite field reflects the topological nature of its solutions over the complex numbers. 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, the last of which was settled by Grothendieck's student Pierre 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):

  1. Topological tensor products and nuclear spaces
  2. "Continuous" and "discrete" duality (derived categories and "six operations").
  3. Yoga of the Grothendieck–Riemann–Roch theorem (K-theory, relation with intersection theory).
  4. Schemes.
  5. Topoi.
  6. Étale cohomology including l-adic cohomology.
  7. Motives and the motivic Galois group (and Grothendieck categories)
  8. Crystals[disambiguation needed] and crystalline cohomology, yoga of De Rham and Hodge coefficients.
  9. Topological algebra, infinity-stacks, 'dérivateurs', cohomological formalism of toposes as an inspiration for a new homotopic algebra
  10. Tame topology.
  11. Yoga of anabelian geometry and Galois–Teichmüller theory.
  12. Schematic point of view, or "arithmetics" for regular polyhedra and regular 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.[clarification needed] The word yoke, meaning "linkage", is derived from the same Indo-European root.

Life

Family and early life

Alexander Grothendieck was born in Berlin by the anarchist parents: a Russian father from an ultimately Hassidic family, Alexander "Sascha" Shapiro aka Tanaroff, and a mother from a German Protestant family, Johanna "Hanka" Grothendieck; both of his parents had broken away from their early backgrounds in their teens.[7] At the time of his birth Grothendieck's mother was married to Johannes Raddatz, a German journalist, and his birthname was initially recorded as Alexander Raddatz. The marriage was dissolved in 1929 and Shapiro/Tanaroff acknowledged his paternity, but never married Hanka Grothendieck.[7]

Grothendieck lived with his parents until 1933 in Berlin. At the end of that year, Shapiro moved to Paris, and Hanka followed him the next year. They left Grothendieck in the care of Wilhelm Heydorn, a Lutheran Pastor and teacher[8] in Hamburg where he went to school. During this time, his parents fought in the Spanish Civil War.

During WWII

In 1939 Grothendieck came to France and lived in various camps for displaced persons with his mother, first at the Camp de Rieucros, and subsequently lived for the remainder of the war in the village of Le Chambon-sur-Lignon, where he was sheltered and hidden in local boarding-houses or pensions. His father was sent via Drancy to Auschwitz where he died in 1942. While Grothendieck lived in Chambon, he 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 being hidden in Chambon attended Cévenol and it was at this school that Grothendieck apparently first became fascinated with mathematics.

Studies and contact with research mathematics

After the war, the young Grothendieck studied mathematics in France, initially at the University 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.[9] However, his talent was noticed, and he was encouraged to go to Paris in 1948.

Initially, Grothendieck attended Henri Cartan's Seminar at École Normale Supérieure, but lacking the necessary background to follow the high-powered seminar, he moved to the University of Nancy where he wrote his dissertation under Laurent Schwartz in functional analysis, from 1950 to 1953. At this time he was a leading expert in the theory of topological vector spaces. By 1957, he set this subject aside in order to work in algebraic geometry and homological algebra.

The IHÉS years

Installed at the Institut des Hautes Études Scientifiques (IHÉS), Grothendieck attracted attention 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, on group schemes), Luc Illusie (cotangent complex), Michel Raynaud, Jean-Louis Verdier (cofounder of the derived category theory) and Pierre Deligne. Collaborators on the SGA projects also included Mike Artin (étale cohomology) and Nick Katz (monodromy theory and Lefschetz pencils). Jean Giraud worked out torsor theory extensions of non-abelian cohomology. Many others were involved.

The 'Golden Age'

Alexander Grothendieck's work during the `Golden Age' period at IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory and complex analysis. His first (pre-IHÉS) breakthrough in algebraic geometry was the Grothendieck–Hirzebruch–Riemann–Roch theorem, a far-reaching 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 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. He went on to introduce the étale cohomology theory of schemes, providing the key tools for proving the Weil conjectures, as well as crystalline cohomology and algebraic de Rham cohomology to complement it. Closely linked to these cohomology theories, he originated topos theory as a generalisation of topology (relevant also in mathematical logic, category theory, and also to computer software programming and institutional ontology classification and bioinformatics[citation needed]). He also provided an algebraic definition of fundamental groups of schemes and more generally the main structures of a categorical Galois theory. As a framework for his coherent duality theory he also introduced derived categories, which were further developed by Verdier.

The results of 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 IHES.

Politics and retreat from scientific community

Grothendieck's political views were radical and pacifist. Thus he strongly opposed both United States aggression in Vietnam as well as Soviet military expansionism. He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam 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 of IHÉS (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 of Montpellier, where he stayed until his retirement in 1988. His criticisms of the scientific community, and especially of several mathematics circles, are also contained in a letter, written in 1988, in which he states the reasons for his refusal of the Crafoord Prize.[10] He declined the prize on ethical grounds in an open letter to the media.[11]

While the issue of military funding was perhaps the most obvious explanation for Grothendieck's departure from IHÉS, those who knew him say that the causes of the rupture ran deeper. 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. The Grothendieck Festschrift was a three-volume collection of research papers to mark his sixtieth birthday (falling in 1988), and published in 1990.[12]

In it 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 "une cage dorée" ("a golden 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. 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 of mathematics. David Ruelle, a physicist who joined the IHÉS faculty in 1964, said that Grothendieck came to talk to him a few times about physics. (In the 1970s Ruelle and the Dutch mathematician Floris Takens produced a new model for turbulence, and it was Ruelle who invented the concept of a strange attractor in a dynamical system.) Biology interested Grothendieck much more than physics, and he organized some seminars on biological topics.[13]

After leaving the IHÉS, Grothendieck tried but failed to get a position at the Collège de France.

He then went to Université de Montpellier, where he became increasingly estranged from the mathematical community. Around this time, he founded a group called Survivre, which was dedicated to antimilitary and ecological issues. His mathematical career, for the most part, ended when he left the IHÉS. In 1984 he wrote a proposal to get a position through the Centre National de la Recherche Scientifique. The proposal, entitled Esquisse d'un Programme ("Program Sketch") describes new ideas for studying the moduli space of complex curves. Although Grothendieck himself never published his work in this area, the proposal became the inspiration for work by other mathematicians and the source of the theory of dessin d'enfants. Esquisse d’un Programme was published in the two-volume proceedings Geometric Galois Actions (Cambridge University Press, 1997)."[14]

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 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[15] (see below, and also a more detailed entry), and in particular studying the Teichmüller theory. (For an English translation of the tables of contents of these manuscripts, see Esquisse d'un programme.)

In 1983 he wrote a huge extended manuscript (about 600 pages) entitled Pursuing Stacks, stimulated by correspondence with Ronald Brown, (see also R.Brown and Tim Porter at University of Bangor in Wales), and starting with a letter addressed to Daniel 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 between algebraic homotopy theory and algebraic 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 of F. Morel and V. Voevodsky in the mid 1990s.

His Esquisse d'un programme[15] (1984) is a proposal for a position at the Centre 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 as anabelian geometry. In La Clef des Songes he explains how the reality of dreams convinced him of God's existence[citation needed].

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 translated into Russian] and published in Moscow[16]). Some parts of Récoltes et semailles[17][18] and the whole La Clef des Songes [19] have been translated into Spanish.

Retirement into reclusion

In 1991, Grothendieck moved to an address he did not provide to his previous contacts in the mathematical community. He is now said to live in southern France or Andorra and to be reclusive.

In January 2010, Grothendieck wrote a letter to Luc Illusie. In this "Declaration d’intention de non-publication", he states that essentially all materials that have been published in his absence have been done without his permission. He asks that none of his work should be reproduced in whole or in part, and even further that libraries containing such copies of his work remove them.[20]

See also

Notes

  1. ^ (Dieudonné 1990)
  2. ^ See, for example, (Deligne 1998).
  3. ^ Jackson, Allyn (2004), "Comme Appelé du Néant — As If Summoned from the Void: The Life of Alexandre Grothendieck I" (PDF), Notices of the American Mathematical Society, 51 (4): 1049
  4. ^ Mclarty, Colin. "The Rising Sea: Grothendieck on simplicity and generality I" (PDF). Retrieved 2008-01-13.
  5. ^ Peck, Morgen, Equality of Mathematicians, Alexandre Grothendieck is arguably the most important mathematician of the 20th century...
  6. ^ Leith, Sam (March 20, 2004), "The Einstein of maths", The Spectator, [A] mathematician of staggering accomplishment ... a legendary figure in the mathematical world.
  7. ^ a b Society for Industrial and Applied Mathematics
  8. ^ Allyn Jackson, The Life of Alexander Grothendieck, p. 1040
  9. ^ See Jackson (2004:1). The remark is from the beginning of Récoltes et Semailles (page P4, in the introductory section Prélude en quatre Mouvements)
  10. ^ Crafoord Prize letter
  11. ^ Matthews, Robert (20 August 2006). "Mathematics, where nothing is ever as simple as it seems". Daily Telegraph. Retrieved 5 July 2009. [dead link]
  12. ^ The editors were Pierre Cartier, Luc Illusie, Nick Katz, Gérard Laumon, Yuri Manin, and Ken Ribet. A second edition has been printed (2007) by Birkhauser.
  13. ^ The IHÉS at Forty Allyn Jackson, March 1999, Noticed of the AMS pp. 329-337
  14. ^ Google book link
  15. ^ a b ESQUISSE D’UN PROGRAMME par Alexandre Grothendieck
  16. ^ In Russian
  17. ^ COSECHAS Y SIEMBRAS: Reflexiones y testimonios sobre un pasado de matem´atico Preludio
  18. ^ COSECHAS Y SIEMBRAS: Reflexiones y testimonios sobre un pasado de matem´atico Carta
  19. ^ La Clef des Songes
  20. ^ http://sbseminar.wordpress.com/2010/02/09/grothendiecks-letter
  21. ^ Generator, Generator Family and Cogenerator

References

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