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Alexander Grothendieck

Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is considered to be one of the greatest mathematicians of the 20th century. He made major contributions to: algebraic topology, algebraic geometry, number theory, category theory, Galois theory,descent theory commutative homological algebra and functional analysis. He was awarded the Fields Medal in 1966, and was co-awarded the Crafoord Prize with Pierre 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. 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 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 by Jean 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 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 Mathematische 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.)

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 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 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. 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 German Protestant origin, Hanka Grothendieck, in Berlin. He was a displaced person during much of his childhood due to the upheavals of World War II. Alexander lived with his parents both of whom were anarchists, until 1933, 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.

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 at Le Chambon-sur-Lignon. His father was sent via Drancy to Auschwitz where he died in 1942.

Studies and contact with research mathematics

After the war, 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. 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, first by his spectacular Grothendieck-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, 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.

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 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 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. 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 (see below, and also a more detailed entry), 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 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 (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.

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). Some parts of Récoltes et semailles and the whole Le Clef des Songes 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.

See also

Notes

References

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