Alexander Grothendieck


Alexander Grothendieck : biography

28 March 1928 –

In 1984 he wrote a proposal to get a position through the Centre National de la Recherche Scientifique, which he held from 1984 to his retirement in 1988. 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 dessins d’enfants and of a new field emerging as anabelian geometry. Esquisse d’un Programme was published in the two-volume proceedings Geometric Galois Actions (Cambridge University Press, 1997).

During this period he also released his work on Bertini type theorems contained in EGA 5, published by the in 2004.

The 1000-page autobiographical manuscript Récoltes et semailles (1986) is now available on the internet in the French original,Alexander Grothendieck, 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 Preludio Carta and the whole La Clef des Songes have been translated into Spanish and Russian.

In the manuscript La Clef des Songes he explains how considering the source of dreams led him to conclude that God exists.Scharlau (2008), p. 940 His growing preoccupation with spiritual matters was also evident in a letter entitled Lettre de la Bonne Nouvelle that he sent to 250 friends in January 1990. In it, he described his encounters with a deity and announced that a "New Age" would commence on 14 October 1996.

Retirement into reclusion

Grothendieck was co-awarded (but declined) the Crafoord Prize with Pierre Deligne in 1988.

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 "Déclaration 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.


Mathematical achievements

Grothendieck’s early mathematical work was in functional analysis. Between 1949 and 1953 he worked on his doctoral thesis in this subject at Nancy, supervised by Jean Dieudonné and Laurent Schwartz. His key contributions include topological tensor products of topological vector spaces, the theory of nuclear spaces as foundational for Schwartz distributions, and the application of Lp spaces in studying linear maps between topological vector spaces. In a few years, he had turned himself into a leading authority on this area of functional analysis — to the extent that Dieudonné compares his impact in this field to that of Banach.

It is, however, in 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, producing the influential "Tôhoku paper" (Sur quelques points d’algèbre homologique, published in 1957) where he introduced Abelian categories and applied their theory 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 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.