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.

Contents

Texts by Grothendieck

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

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

En Guise de Programme p I p II, a text written by Grothendieck as a course description while teaching in Montpellier “Introduction à la recherche”.

A chronological bibliography of Grothendieck’s published mathematical writings (pdf).

Texts about Grothendieck

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:

Grothendieck obituary in the Notices AMS (Michael Artin, Allyn Jackson, David Mumford, and John Tate, Coordinating Editors):

Students

See Mathematics Genealogy for Grothendieck

Correspondence

The Grothendieck-Serre correspondence

The Grothendieck-Mumford correspondence

Grothendieck circle

Grothendieck’s Angle by G. Aiello

A. Grothendieck by J. A. Navarro

A. Grothendieck by M. Carmona

A. Grothendieck, una guía a la obra matemática y filosófica (pdf) por F. Zalamea

Quotes

On K-theory:

The way I first visualized a K-group was as a group of “classes of objects” of an abelian (or more generally, additive) category, such as coherent sheaves on an algebraic variety, or vector bundles, etc. I would presumably have called this group C(X)C(X) (XX being a variety or any other kind of “space”), CC the initial letter of ‘class’, but my past in functional analysis may have prevented this, as C(X)C(X) designates also the space of continuous functions on XX (when XX is a topological space). Thus, I reverted to KK instead of CC, since my mother tongue is German, Class = Klasse (in German), and the sounds corresponding to CC and KK are the same.

from Grothendieck’s letter to Bruce Magurn, on 9th February 1985, quoted after:

category: people