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Encyclopedia > Grothendieck

Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966 and coawarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter prize on ethical grounds.


Because of his mastery of abstract approaches to mathematics, but also because of the many stories told about his retirement and his alleged mental disorders, he is one of the most intriguing scientific personalities of the 20th century.

Contents

Mathematical achievements

Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre, after sheaves had been invented by Kiyoshi Oka and Jean Leray. Grothendieck took them to a higher level, changing the tools and the level of abstraction.


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 the Riemann-Roch theorem, around 1956, which had already recently been generalized to any dimension by Hirzebruch). The Grothendieck-Riemann-Roch theorem was announced by Grothendieck at the initial Arbeitstagung in Bonn, in 1957. It appeared in print in a paper written by Armand 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-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.


Its 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. Grothendieck is one of the few mathematicians who matches the French concept of maître à penser; some go further and call him maître-penseur.)


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). This is considered to have gone a long way to answering Kronecker's wish for a foundational theory based on the integers alone, by using the relative point of view based on finitely-generated commutative rings. The style of the mathematics is very distant from Kronecker's, though. On the EGA project Grothendieck collaborated with Jean Dieudonné. It is axiomatic, and claims descent (according to Dieudonné) from David Hilbert's approach; as interpreted by Nicolas Bourbaki. On the other hand Grothendieck himself applied an intuitive approach, as well as a generalising one. There were many other contributors to the SGA series.


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 by Grothendieck's student Pierre Deligne in the early 1970s after Grothendieck had 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. "Continual" 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. Toposes.
  6. Étale cohomology including l-adic cohomology.
  7. Motives and the motivic Galois group (and Grothendieck categories)
  8. Crystals 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 above is that of topos theory, while the first and last were of the least importance to him.


Here the usage of yoga means a kind of 'meta-theory' that can be used heuristically.


Life

Childhood and studies

Born to a Russian Jewish father and German Jewish mother, he was a displaced person during much of his childhood due to the upheavals of World War II. Alexander lived with his father, Alexander Shapiro, and his mother, Hanka Grothendieck, both of whom were socialist revolutionaries. Until 1933 they lived together in Berlin. At the end of that year, Shapiro moved to Paris, and Hanka followed him the next year. They left Alexander with a family in Hamburg where he went to school. During this time, his parents fought in the Spanish Civil War. In 1939 Alexander came to France and lived in various camps for displaced persons with his mother. His father was sent to Auschwitz where he died in 1942. After the war, young Grothendieck studied mathematics in France, initially at 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. However, his talent was noticed, and he was encouraged to go to Paris in 1948. He wrote his dissertation under Laurent Schwartz in functional analysis, from 1950. At this time he was a leading expert in the theory of topological vector spaces. However he set this subject aside by 1957 in order to work in algebraic geometry and homological algebra.


Politics and retreat from scientific community

Grothendieck's radical left-wing and pacifist politics were doubtless born of his wartime experiences. He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam war. He retired from scientific life around 1970, after having discovered the partly military funding of IHES (see pp. xii and xiii of SGA1, Springer Lecture Notes 224). He returned to academics 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 are also contained in a letter written in 1988, in which he states the reasons for his refusal of the Crafoord 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.


La longue marche a travers la theorie de Galois (roughly The Long Walk Through Galois Theory) is an approximately 1600-page handwritten manuscript produced by Grothendieck during the years 1980-1981 and contains many of the ideas leading to the Esquisse d'un Programme (see below) and in particular studies the Teichmüller theory.


In 1983 he wrote a huge extended letter (about 600 pages) titled Pursuing Stacks addressed to Daniel Quillen and others, in which he explained his ideas on the relationship between algebraic homotopy theory and algebraic geometry and prospects for a noncommutative theory of stacks. This later led to another of his monumental works Les Derivateurs. 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 Morel and Voevodsky in the mid 1990s.


His Esquisse d'un programme (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 in 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.


The 2000-page autobiographical manuscript Récoltes et Semailles (1986) is now partly available on the internet in the French original, and an English translation is underway (these parts of Récoltes et Semailles are already translated to Russian (http://imperium.lenin.ru/%7Everbit/Grothendieck/Grothendieck.html) and published in Moscow).


Disappearance

In 1991, he left his home and disappeared. He is said to now live in the South of France and to entertain no visitors. Various false rumors have him living in Ardèche, herding goats and entertaining radical ecological theories. Though he has been inactive in mathematics for many years, he remains one of the greatest and most influential mathematicians of modern times.


External links

  • MacTutor biography of Grothendieck (http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Grothendieck.html)
  • Grothendieck Circle (http://www.grothendieck-circle.org), collection of mathematical and biographical information, photos, links to his writings
  • Institut des Hautes Études Scientifiques (http://www.ihes.fr)
  • Récoltes et Semailles (original text) (http://mapage.noos.fr/recoltesetsemailles/TdM.htm)
    • La Vision (http://mapage.noos.fr/recoltesetsemailles/8%20La%20vision.htm)
  • Grothendieck biography project (http://www.fermentmagazine.org/home5.html) by Roy Lisker. English translation of some parts of Récoltes et Semailles can be found here.
  • Grothendieck biography (Part 1 (http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf), Part 2 (http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf)) published in AMS Notices

  Results from FactBites:
 
Alexander Grothendieck - Wikipedia, the free encyclopedia (1972 words)
Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century.
The Grothendieck-Riemann-Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957.
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.
Grothendieck biography (535 words)
It is no exaggeration to speak of Grothendieck's years 1959-70 at the IHES as a 'Golden Age', during which a whole new school of mathematics flourished under Grothendieck's charismatic leadership.
In looking back at this period, one marvels at the generosity with which Grothendieck shared his ideas with colleagues and students, the energy he and his collaborators devoted to meticulous redaction, the excitement with which they set out to explore a new land.
Grothendieck was always strongly pacifist in his views and campaigned against the military built-up of the 1960s.
  More results at FactBites »

 
 

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