nLab
Alexander Grothendieck

The European mathematician Alexander Grothendieck (in French sometimes Alexandre Grothendieck), created a very influential body of work foundational for (algebraic) geometry but also for modern mathematics more generally. He is widely regarded as a singularly important figure of 20th century mathematics and his ideas continue to be highly influential in the 21st century.

Initially working on topological vector spaces and analysis, Grothendieck then made revolutionary advances in algebraic geometry by developing sheaf and topos theory and abelian sheaf cohomology and formulating algebraic geometry in these terms (locally ringed spaces, schemes). Later topos theory further developed independently and today serves as the foundation also for other kinds of geometry. Notably its homotopy theoretic refinement to higher topos theory serves as the foundation for modern derived algebraic geometry.

Grothendieck’s geometric work is documented in texts known as EGA (with Dieudonné), an early account FGA, and the many volume account SGA of the seminars at l’IHÉS, Bures-sur-Yvette, where he was based at the time. (See the wikipedia article for some indication of the story from there until the early 1980s.)

In the late 1970s and early 1980s Grothendieck wrote several documents that have been of outstanding importance in the origins of the theory that underlies the nPOV. These include

  • La Longue Marche à travers la Théorie de Galois (1600 manuscript pages written between January and June 1981, plus addenda etc. which double its length!) (see Long March? for some discussion of the ideas.)

  • Esquisse d'un programme, (January 1984), in which Grothendieck sketches out a vaste programme of research, encorporating many of the ideas from Long March?. A copy is available here. It is discussed in brief at Grothendieck's Esquisse.

  • À la poursuite des Champs (also entitled ”Pursuing Stacks”).
    It starts with a short (12 page) letter to Quillen, dated 19 Feb. 1983, but then discusses a wide ranging vision of homotopy theory and its applicability to problems in algebraic and arithmetic geometry.

  • Les Dérivateurs (another 2000 page manuscript taking up some of the themes in Pursuing Stacks, section 69) Dating from the end of 1990 and the start of 1991.

In the same time he also wrote voluminous intellectual memoirs Recoltes et Semailles.

For an account of his work, including some of the work published in the 1980s, see the English Wikipedia entry.

The video of a talk by W. Scharlau on his life can be seen here.

A recent article in French on Grothendieck is to be found here.

There were two articles on Grothendieck’s life and work in the Notices AMS in 2004:

Allyn Jackson, Comme Appelé du Néant, As If Summoned from the Void: The Life of Alexandre Grothendieck, Part 1,Notices AMS

Allyn Jackson, Comme Appelé du Néant, As If Summoned from the Void: The Life of Alexandre Grothendieck, Part 2,Notices AMS

category: people

Revised on March 11, 2014 23:24:07 by Tim Porter (2.26.40.34)