Sketch of a programme. (Esquisse d’un programme.) (French, English) Zbl 0901.14001

Grothendieck, Alexandre

Schneps, Leila (ed.) et al., Geometric Galois actions. 1. Around Grothendieck’s “Esquisse d’un programme”. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 5-48; English translation: 243-283 (1997).

Alexandre Grothendieck, whose impact on the development of mathematics during the last fifty years can barely be overestimated, quit his career as university teacher in the mid 1980’s. At that time, his decision was meant as a protest against the general tendency at universities, of which he thought that it would make the possibility of teaching mathematics at the (necessarily high) research level more and more illusory. As a further consequence, A. Grothendieck resolved to apply for admission to the French National Centre for Scientific Research (CNRS), in order to devote his energy to the development of projects and perspectives “for which it is becoming clear that no student (nor even, as it seems, any colleague in mathematics) will be found to develop them in my stead”.
Although A. Grothendieck had ceased any publication of scientific articles as early as in 1970, apparently so for the same reason of protest, his ingenious mathematical mind has produced, in the meantime, a wealth of innovating and prospective ideas of gradually programmatic character. His mathematical reflections on the principal themes of his interest, carried out during the period 1970-1983, had materialized over the years, in a pile of handwritten notes. The present notes, entitled “Esquisse d’un programme-(Sketch of a programme)” and written up in 1984, provide a summarizing sketch of some of the themes occurring in Grothendieck’s handwritten “Mathematical reflections”. Their purpose was, as Grothendieck pointed out, quite multifarious: revealing the progress of his thoughts in a condensed form, giving a sketch of his program of work for the coming years, attaching a research proposal to his application for admission to the CNRS, and suggesting some of his research themes to a few colleagues and former students for kind attention.
Grothendieck’s “Sketch of a programme”, written on his portable typewriter, circulated amongst a few algebro-geometric insiders during the last fifteen years, serving in such a way as a spring of inspiration for some of them. Unfortunately, it could not be published earlier, because Alexandre Grothendieck had untraceably disappeared for many years, and his permission could not be requested until 1995.
The inclusion of his famous “Sketch of a programme” in these conference proceedings, whose articles are devoted to topics that actually arose from Grothendieck’s circulating “Esquisse”, is therefore a very gratifying, and certainly overdue event in the domain of public mathematical communication.
Without any doubt, and as the contributions to these conference proceedings vividly demonstrate, A. Grothendieck’s “Sketch of a programme” is, now as before, much more than a historical delicacy donated by a legendary great man in mathematics. In fact, it still is what its title strikingly says: a sketch of masterly, extremely far-sighted, here and there visionary, thoroughly challenging, and multi-sidedly inspiring programme for the further development of algebraic geometry (and its related areas) in various directions. The vast amount of ideas presented by A. Grothendieck in these sketchy notes makes it absolutely impossible to review them separately, and much less so in an appropriate manner. May it therefore suffice to recommend the individual and thorough study of A. Grothendieck’s notes to every interested (and ambitious) algebraic geometer.
Grothendieck’s “Sketch of a programme” is not a mathematical paper in the usual sense, and not whatsoever a treatise written in his unimitable “EGA-style”. These notes are written in a rather narrative style, reflecting the master’s countless, spontaneously flashing ideas and visions in their progressing, and being profusely amalgamated with biographical reminiscences, non-mathematical digressions, and educational expressions of opinion.
For the entire collection see [Zbl 0868.00041].

Reviewer: W.Kleinert (Berlin)

MSC:

14-03	History of algebraic geometry
14A20	Generalizations (algebraic spaces, stacks)
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
01A67	Future perspectives in mathematics
01A65	Development of contemporary mathematics
14H30	Coverings of curves, fundamental group

Keywords:

algebraic varieties; moduli spaces; Galois group actions; fundamental groups; Teichmüller theory; anabelian algebraic geometry; differentiable algebraic geometry; Nash functions; algebraic stacks; two-dimensional geometry