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Discussion of Summary and Contents of "Esquisse d' un Programme"

(The original, in French, is by Alexandre (Alexander) Grothendieck one of the greatest mathematicians of all times)

[This article is now being promptly expanded to explain contents of this important program proposal for scientific and educational purposes.]

This article will present detailed explanations of an important proposed program by Alexandre (Alexander) Grothendieck for advanced mathematical studies in Algebraic Geometry and other areas of Mathematics-- that has been, and still is, being deliberately and very rapidly suppressed from publication-- by narrow interests and manouvering that does not belong to science. Such repeated and deliberate suppression of certain mathematical truths and facts that has occurred in several EU countries is quite harmful for the future progress of Mathematics and the other exact sciences.

Alexander Grothendieck's stated program of mathematical research

A proposed program by Alexandre (Alexander) Grothendieck of advanced mathematical studies in Algebraic Geometry; the original is in French:

===Abstract=== ("Sommaire")

• 1. The Proposal and enterpise ("Envoi").
• 2. "Teichm¨uller's Lego-game and the Galois group of Q over Q" ("Un jeu de “Lego-Teichm¨uller” et le groupe de Galois de Q sur Q").
• 3. Corps de nombres associ´es `a un dessin d’enfant.
• 4. Poly`edres r´eguliers sur les corps finis.

5. Haro sur la topologie dite “g´en´erale”, et r´eflexions heuristiques vers une topologie dite “mod´er´ee”. 6. “Th´eories diff´erentiables” (`a la Nash) et “th´eories mod´er´ees”. 7. A la Poursuite des Champs. 8. Digressions de g´eom´etrie bidimensionnelle. 9. Bilan d’une activit´e enseignante. 10. Epilogue. Notes

Proposed extensions of Galois's theory for groups: Groupoids, categories and functors

Galois has developed a powerful, fundamental algebraic theory in mathematics that provides very efficient computations for certain algebraic problems by utilizing the algebraic concept of groups, which is now known as the theory of groups; such computations were not possible before, and also in many cases are much more effective than the 'direct' calculations without using groups. His fundamental theory in mathematics has been considerably expanded, at first to groupoids as proposed in Alexander Grothendieck's Esquisse d' un Programme (EDP), and now already partially carried out for groupoids, as well as currently being further developed by several mathematicans. Here, we shall focus only on the well-established and fully validated extensions of Galoi's theory. Thus, EDP also proposed and anticipated, along previous Alexander Grothendieck's IHEA seminars (SGA1 to SGA4) held in the 1960s, the development of even more powerful extensions of the original Galois's theory for groups by utilizing categories, functors and natural transformations. Such developments were recently extended in algebraic topology via representable functors and the fundamental groupoid functor.

References and notes

• Alexander Grothendieck. 1971, Revêtements Étales et Groupe Fondamental (SGA1), chapter VI: ''Catégories fibrées et descente'', Lecture Notes in Math. 224, Springer-Verlag: Berlin.

• Alexander Grothendieck. 1957, Sur quelque point d-algébre homologique. , Tohoku Math. J., 9: 119-121.

• Alexander Grothendieck and J. Dieudonné.: 1960, Eléments de geometrie algébrique., Publ. Inst. des Hautes Etudes de Science, 4.

• Alexander Grothendieck et al.,1971. Séminaire de Géométrie Algébrique du Bois-Marie, Vol. 1-7, Berlin: Springer-Verlag.

• Alexander Grothendieck. 1962. Séminaires en Géométrie Algébrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Cohèrents et Théorèmes de Lefschetz Locaux et Globaux. , pp.287. (with an additional contributed exposé by Mme. Michele Raynaud). (Typewritten manuscript available in French; see also a brief summary in English References Cited:
  • J. P. Serre. 1964. Cohomologie Galoisienne, Springer-Verlag: Berlin.
  • J. L. verdier. 1965. Algèbre homologiques et Catégories derivées. North Holland Publ. Cie).

• Alexander Grothendieck. 1957, Sur Quelques Points d'algèbre homologique, Tohoku Mathematics Journal, 9, 119-221.

• Alexander Grothendieck et al. Italic textSéminaires en Géometrie Algèbrique- 4, Tome 1, Exposé 1 (or the Appendix to Exposée 1, by `N. Bourbaki' for more detail and a large number of results. AG4 is freely available in French; also available here is an extensive Abstract in English.

• 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 .

• Alexander Grothendieck, "La longue marche in à travers la théorie de Galois." = "The Long March Towards/Across the Theory of Galois", 1981 manuscript, University of Montpellier preprint series 1996, edited by J. Malgoire.

• Leila Schneps. 1994. The Grothendieck Theory of Dessins d'Enfants. (London Mathematical Society Lecture Note Series), Cambridge University Press, 376 pp.

• David Harbater and Leila Schneps. 2000. Fundamental groups of moduli and the Grothendieck-Teichmüller group, Trans. Amer. Math. Soc. 352 (2000), 3117-3148. MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.