The algebraic fundamental group

Summary

An apology. - On 12-IV-1996 a “Grothendieck day” was organized in Utrecht. For non-specialists we tried to explain some of the basic ideas of Grothendieck and their impact on modern mathematics. The text below is the contents of one of the talks. It is written for a general mathematical audience. Simplifications were made in an attempt to make these ideas accessible for a general audience, with the effect that it certainly dilutes the ideas discussed. - The reader should not expect anything more than just an informal exposé, which perhaps would suit better the coffee table than any serious publication.

Introduction.

In 1984 Alexandre Grothendieck wrote:

..aujourd'hui je ne suis plus, comme naguère,

le prisonnier volontaire de tâches interminables,

qui si souvent m'avaient interdit de m'élancer dans l'inconnu,

mathématique ou non.

See [2], page 51.

(1.1) [EGA] J. Dieudonné and A. Grothendieck published in 1960 - 1967 eight volumes:

Elements de la géométrie algébrique.

There are 4 chapters, in 8 volumes, together more than 1800 pp. All published as volumes in the series Publ. Math, at the IHES.

In 1960 Grothendieck had planned 12 chapters, at that moment he already had a rather clear picture of what should be contained in the various volumes.

(1.2) [SGA] In 1960 - 1969 Grothendieck, together with many collaborators, had seminars:

Séminaire de géométrie algébrique du Bois Marie.

There are 12 volumes, together more than 6200 pp. Eleven volumes are published as Lecture Notes Math., Springer - Verlag; one volume is published by the North - Holland Publ. Cy.