AGrothendiecksMathematicalHeritageEsquisseDunProgramme 2009-03-09 10:32:19 2009-03-11 04:43:53 Topic Esquisse d'un Programme Alexander Grothendieck \subsection{``Esquisse d'un Programme'' (original document was written in French by Alexander Grothendieck)} \subsubsection{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 {\em Esquisse} Grothendieck sketched what he called the ``Galois-Teichm\"uller theory''--a study of the abstract galois Group $Gal(\overline{Q}}/\mathbb{Q})$ via the action of this group on the mapping class (Teichm\"uller) 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. ) \textbf{Abstract of the paper} (In French: ``Sommaire'') \begin{enumerate} \item The Proposal and enterprise (''Envoi''). \item 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''). \item Number fields associated with ``dessin d'enfants''. (or in orig. : ''Corps de nombres associ\'es \`a un dessin d' enfant''). \item Regular polyhedra over finite fields (``Poly\'edres r\'eguliers sur les corps finis''). \item General topology or a `Moderated topology' (``Haro sur la topologie dite 'g\'en\'erale', et r\'eflexions heuristiques vers une topologie dite ``mod\'er\'ee''). \item Differentiable theories and moderated theories (``Th\'eories diff\'erentiables'' (\`{a} la Nash) et ``th\'eories mod\'er\'ees''). \item \PMlinkexternal{Pursuing Stacks (``\`A la Poursuite des Champs'')}{http://www.math.jussieu.fr/~leila/grothendieckcircle/stacks.ps}. \item Digression on two-dimensional geometry (``Digressions de g\'eom\'etrie bidimensionnelle''; now called ``Higher Dimensional Algebra'' that Alexander Grothendieck anticipated by several years). \item A Synthesis of the proposed Research Activity (''Bilan d'une activit\'e enseignante''). \item Epilogue. \item Notes \end{enumerate} \PMlinkexternal{\bf Reference}{http://www.math.jussieu.fr/~leila/grothendieckcircle/EsquisseFr.pdf} 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.