Talk:PlanetPhysics/A Grothendiecks Mathematical Heritage Esquisse Dun Programme

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 \subsection{``Esquisse d'un Programme'' (``Sketch of a Program''--the 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 \htmladdnormallink{Alexander Grothendieck}{http://planetphysics.us/encyclopedia/AlexanderGrothendieck.html} in 1984 that continues to inspire even today several related areas of mathematics. Of considerable interest to many mathematicians are the recent Galois \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} and categorical generalizations of Galois theory initiated by Alexander Grothendieck, now developed towards maturity by several other seasoned mathematicians.

In the second \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} of the {\em Esquisse} Grothendieck sketched what he called the ``Galois-Teichm\"uller theory''--a study of the abstract Galois \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $Gal(\overline{Q}/ Q)$ via the action of this group on the mapping class (Teichm\"uller) groups; the latter are the \htmladdnormallink{fundamental groups}{http://planetphysics.us/encyclopedia/HomotopyCategory.html} of the moduli spaces of \htmladdnormallink{Riemann surfaces}{http://planetphysics.us/encyclopedia/RiemannSurface.html} 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 \htmladdnormallink{fundamental group}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} of this space--which is the profinite completion of the free group on two \htmladdnormallink{generators}{http://planetphysics.us/encyclopedia/Generator.html} 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 \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} associated with ``dessin d'enfants''.
(or in orig. : ''Corps de nombres associ\'es \`a un dessin d' enfant'').
\item \htmladdnormallink{regular}{http://planetphysics.us/encyclopedia/CoIntersections.html} 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 \htmladdnormallink{Pursuing Stacks (``\`A la Poursuite des Champs'')}{http://www.math.jussieu.fr/~leila/grothendieckcircle/stacks.ps}.
\item Digression on \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} 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}

\htmladdnormallink{\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.

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