Esquisse d'un Programme

"Esquisse d'un Programme" is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck [Scharlau, Winifred (September 2008), written at Oberwolfach, Germany, "Who is Alexander Grothendieck", "Notices of the American Mathematical Society" (Providence, RI: American Mathematical Society) 55(8): 930–941, ISSN 1088-9477, OCLC 34550461, http://www.ams.org/notices/200808/tx080800930p.pdf] . He pursued the sequence of logically linked ideas in his important project proposal from 1984 until 1988, but his proposed research continues to date to be of major interest in several branches of advanced mathematics. Grothendieck's vision provides inspiration today for several developments in mathematics such as the extension and generalization of Galois theory, which is currently being extended based on his original proposal.

Outline of paper's content

Submitted in 1984, the "Esquisse d'un Programme" [Alexander Grothendieck, 1984. "Esquisse d'un Programme", (1984 manuscript), finally published in Schneps and Lochak (1997, I), pp.5-48; English transl., ibid., pp. 243-283. MR 99c:14034 ] was a successful proposal submitted by Alexander Grothendieck for a position at the Centre National de la Recherche Scientifique, which Grothendieck held from 1984 till 1988. [Rehmeyer, Julie (May 9, 2008), "Sensitivity to the Harmony of Things", "Science News"] This proposal was however not formally published until 1997, because the author "could not be found, much less his permission requested". [Schneps and Lochak (1997, I) p.1] The dessins d'enfants, or "children's drawings" and "anabelian algebraic geometry" — non-Abelian algebraic topology and noncommutative geometry — that are contained in this long-term program, continue even today to inspire extensive mathematical studies.

Abstract of the paper

("Sommaire")

*1. The Proposal and enterpise ("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". ("Corps de nombres associés à un dessin d’enfant").
*4. Regular polyhedra over finite fields ("Polyèdres 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"} [http://www.bangor.ac.uk/r.brown/pstacks.htm] .
*8. Two-dimensional geometry ("Digressions de géométrie bidimensionnelle" [Cartier, Pierre (2001), "A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry", "Bull. Amer. Math. Soc." 38(4): 389–408, . An English translation of Cartier (1998)] .
*9. Summary of proposed studies ("Bilan d’une activité enseignante").
*10. Epilogue.
*Notes

Suggested further reading for the interested mathematical reader is providedin the "References" section.

Extensions of Galois's theory for groups: Galois 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 Galois groups; such computations were not possible before, and also in many cases are much more effective than the 'direct' calculations without using groups [Cartier, Pierre (1998), "La Folle Journée, de Grothendieck à Connes et Kontsevich — Évolution des Notions d'Espace et de Symétrie", "Les Relations entre les Mathématiques et la Physique Théorique — Festschrift for the 40th anniversary of the "IHÉS", Institut des Hautes Études Scientifiques", pp. 11–19] . To begin with, Alexander Grothendieck stated in his proposal:" "Thus, the group of Galois is realized as the automorphism group of a concrete, pro-finite group which respects certain structures that are essential to this group." This fundamental, Galois group 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; the latter are now further developed beyond groupoids to categories by several groups of mathematicians. Here, we shall focus only on the well-established and fully validated extensions of Galois' theory. Thus, EdP also proposed and anticipated, along previous Alexander Grothendieck's "IHÉS" 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, as well as further expansion of the manifold of ideas presented in Alexander Grothendieck's "Descent Theory". The notion of motive has also been pursued actively [http://en.wikipedia.org/wiki/Motive_(algebraic_geometry) ] . This was developed into the Motives and the motivic Galois group, Grothendieck topology and Grothendieck category [http://planetmath.org/encyclopedia/GrothendieckCategory.html] . Such developments were recently extended in algebraic topology "via" representable functors and the fundamental groupoid functor.

Notes

References

Related works by Alexander Grothendieck

*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 Jean Dieudonné.: 1960," Éléments de géométrie algébrique"., Publ. "Inst. des Hautes Etudes de Science," "(IHÉS)", 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:
**Jean-Pierre 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. Séminaires 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 is an extensive Abstract in English.

*Alexander Grothendieck, 1984. [http://people.math.jussieu.fr/~leila/grothendieckcircle/EsquisseFr.pdf "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.

Relating to the "Esquisse"

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

*Citation | editor1-last=Schneps | editor1-first=Leila | editor2-last=Lochak | editor2-first=Pierre | title=Geometric Galois Actions I: Around Grothendieck's Esquisse D'un Programme | publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series | volume=242 | isbn=978-0-521-59642-8 | year=1997

*Citation | editor1-last=Schneps | editor1-first=Leila | editor2-last=Lochak | editor2-first=Pierre | title=Geometric Galois Actions II: The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups | publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series | volume=243 | isbn=978-0521596411 | year=1997

*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.