0.1 “Esquisse d’un Programme” (“Sketch of a Program”–the original document was written in French by Alexander Grothendieck)

0.1.1 A Concise Summary and Outline of “Esquisse d’un Programme”:

An influential research proposal submitted by Alexander Grothendieck in 1984 that continues to inspire even today several related areas of mathematics. Of considerable interest to many mathematicians are the recent Galois groupoid and categorical generalizations of Galois theory initiated by Alexander Grothendieck, now developed towards maturity by several other seasoned mathematicians.

In the second section of the Esquisse Grothendieck sketched what he called the “Galois-Teichmüller theory”–a study of the abstract Galois group Gal(Q∕Q) via the action of this group on the mapping class (Teichmüller) groups; the latter are the fundamental groups of the moduli spaces of Riemann surfaces with marked points. Then, in the third section he focuses on the ‘simple’ but non-trivial case of the smallest moduli space of spheres with four ordered marked points. The Galois action on the fundamental group of this space–which is the profinite completion of the free group on two generators leads to the “dessin d’ enfants”. The generalization of this theme to all moduli spaces discussed in the second section was the subject of a 1995 mathematics conference published as the “Geometric Galois Actions: The inverse Galois.” (London Mathematical Series No. 243, Cambridge University Press., Leila Schneps and Pierre Lochak, Eds. )

Abstract of the paper (In French: “Sommaire”)

1. The Proposal and enterprise (”Envoi”).
2. Teichmüller’s Lego-game and the Galois group of Q over Q (“Un jeu de ”Lego-Teichmüller” et le groupe de Galois de Q sur Q”).
3. Number fields associated with “dessin d’enfants”. (or in orig. : ”Corps de nombres associés à un dessin d’ enfant”).
4. regular polyhedra over finite fields (“Polyédres réguliers sur les corps finis”).
5. General topology or a ‘Moderated topology’ (“Haro sur la topologie dite ’générale’, et réflexions heuristiques vers une topologie dite “modérée”).
6. Differentiable theories and moderated theories (“Théories différentiables” (à la Nash) et “théories modérées”).
7. Pursuing Stacks (“À la Poursuite des Champs”).
8. Digression on two-dimensional geometry (“Digressions de géométrie bidimensionnelle”; now called “Higher Dimensional Algebra” that Alexander Grothendieck anticipated by several years).
9. A Synthesis of the proposed Research Activity (”Bilan d’une activité enseignante”).
10. Epilogue.
11. Notes

Reference

Alexander Grothendieck, 1984. “Esquisse d’un Programme”, (1984 manuscript), finally published in “Geometric Galois Actions”, L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242,Cambridge University Press, 1997, pp.5-48; English transl., ibid., pp. 243-283. MR 99c:14034.