What are dessins d'enfants? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T06:34:31Z http://mathoverflow.net/feeds/question/1909 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1909/what-are-dessins-denfants What are dessins d'enfants? Ilya Nikokoshev 2009-10-22T18:31:24Z 2013-03-22T09:58:49Z <p>There was an observation that any algebraic curve over <code>Q</code> can be rationally mapped to <code>P^1</code> without three points and this led <a href="http://en.wikipedia.org/wiki/Grothendieck" rel="nofollow">Grothendieck</a> to define a special class of these mappings, called the <em>Children's Drawings</em>, or, in French, <em>Dessins d'Enfants</em> (his quote was something like "things as simple as the drawings...").</p> <p>I'm not an expert in this field, so could somebody please write more about those dessins, and what things they are related to? What's their importance? How does the cartographic group act on these?</p> http://mathoverflow.net/questions/1909/what-are-dessins-denfants/1917#1917 Answer by javier for What are dessins d'enfants? javier 2009-10-22T18:49:09Z 2009-10-22T18:49:09Z <p>You can find a nice introduction to them in <a href="http://www.neverendingbooks.org/index.php/the-best-rejected-proposal-ever.html" rel="nofollow">The best rejected proposal ever</a>, followed up by some discussions about the cartographer's groups and more.</p> http://mathoverflow.net/questions/1909/what-are-dessins-denfants/1927#1927 Answer by Michael Lugo for What are dessins d'enfants? Michael Lugo 2009-10-22T19:36:53Z 2009-10-22T19:36:53Z <p>This is not my area at all, but the <i>Notices</i> published a piece a few years ago called <a href="http://www.ams.org/notices/200307/what-is.pdf" rel="nofollow">"What is a dessin d'enfant?</a></p> http://mathoverflow.net/questions/1909/what-are-dessins-denfants/1929#1929 Answer by Fabian Dreher for What are dessins d'enfants? Fabian Dreher 2009-10-22T19:40:32Z 2009-10-22T19:40:32Z <p>In <a href="http://people.math.jussieu.fr/~leila/Fschneps.pdf" rel="nofollow">Leila Schneps - Dessins d'enfants on the Riemann Sphere</a> you can find a definition of dessins and the Grothendieck correspondence between Belyi morphisms and dessins. It also has pictures of how the cartographic group acts on the flag set of a dessin.<br> Grothendieck correspondence means that there is a bijection between isomorphism classes of dessins and isomorphism classes of Belyi morphisms (morphisms f:X->P<sup>1</sup>C which are ramified only over three points).</p> http://mathoverflow.net/questions/1909/what-are-dessins-denfants/1938#1938 Answer by David Eppstein for What are dessins d'enfants? David Eppstein 2009-10-22T20:22:35Z 2009-10-22T20:22:35Z <p>There's also a Wikipedia article that attempts to answer this question.</p> <p>By the way, the original invention of these things was much earlier than Grothendieck. See <a href="http://www.neverendingbooks.org/index.php/kleins-dessins-denfant-and-the-buckyball.html" rel="nofollow">Klein’s dessins d’enfant and the buckyball</a> on lieven le bruyn's blog.</p> http://mathoverflow.net/questions/1909/what-are-dessins-denfants/24236#24236 Answer by ogerard for What are dessins d'enfants? ogerard 2010-05-11T12:39:50Z 2010-05-11T12:39:50Z <p>There is a french <a href="http://www.math.polytechnique.fr/xups/xups04-03.pdf" rel="nofollow">talk</a> by Alexander Zvonkin which can be a good introduction to this subject as well.</p> <p>If readers are interested I can translate parts of it in english.</p> http://mathoverflow.net/questions/1909/what-are-dessins-denfants/49503#49503 Answer by Dr Shello for What are dessins d'enfants? Dr Shello 2010-12-15T09:41:32Z 2010-12-15T09:41:32Z <p>A very modern compendium of thoughts on this topic can be found here:</p> <p><em>Theory of motives, homotopy theory of varieties, and dessins d'enfants</em>: <a href="http://www.aimath.org/WWN/motivesdessins/motivesdessins.pdf" rel="nofollow">http://www.aimath.org/WWN/motivesdessins/motivesdessins.pdf</a></p> http://mathoverflow.net/questions/1909/what-are-dessins-denfants/63687#63687 Answer by Leonardo for What are dessins d'enfants? Leonardo 2011-05-02T07:52:32Z 2011-05-02T07:52:32Z <p>Historically, one of the first papers on th subject is <em>Drawing curves over number fields</em>, by G.B. Shabat and V.A. Voevodsky (The Grothendieck Festschrift, Vol. III, 199–227, Progr. Math., 88), which I strongly recommend. Another nice and historical paper is <em>Triangulations</em>, by M. Bauer and C. Itzykson (many references: R.C.P. 25, Vol. 44 (1992), Discrete Math. 156 (1996), no. 1-3 or L. Schneps's book below). Both papers are also concerned with the (combinatorial) cellular decomposition of moduli spaces of curves. They appeared before L. Schneps book <em>The Grothendieck theory of dessins d'enfants</em> (London Mathematical Society Lecture Note Series, 200), which is now the main reference on the subject.</p> http://mathoverflow.net/questions/1909/what-are-dessins-denfants/63747#63747 Answer by Luis H Gallardo for What are dessins d'enfants? Luis H Gallardo 2011-05-02T22:11:50Z 2013-03-22T09:58:49Z <p>Also might be interesting:</p> <p>Zbl 1076.14040 Oesterle, Joseph<br> Dessins d'enfants. (Dessins d'enfants.) (French)</p> <p>Bourbaki seminar. Volume 2001/2002. Exposes 894–908. Paris: Societe Mathematique de France (ISBN 2-85629-149-X/pbk). Astérisque 290, 285-305, Exp. No. 907 (2003).</p> <p>Grothendieck's dessins d'enfants are closely connected to the study of coverings of the three times punctured sphere, and such coverings can be considered from many different points of view. In this survey it is shown how all of them are equivalent, and how the absolute Galois group acts on these objects.</p> <p>Reviewer: Ernesto Girondo (Madrid]</p> <p>MR2074061 (2006c:14031) Oesterle, Joseph(F-PARIS6-IMJ) Dessins d'enfants. (French. French summary) Seminaire Bourbaki. Vol. 2001/2002. Asterisque No. 290 (2003), Exp. No. 907, ix, 285–305. 14G32 (14E20 14H30)</p> <p>From the text (translated from the French): "In 1984, A. Grothendieck presented a research program, entitled <code>Esquisse d'un programme' (published in 1997 [in Geometric Galois actions, 1, 5--48, Cambridge Univ. Press, Cambridge, 1997; MR1483107 (99c:14034)]), as part of his application for a position at the CNRS (a position that he would hold until his retirement in 1988). In his program Grothendieck used the term</code>dessin d'enfant' (in its ordinary sense) as a visual analogue of certain cell maps; he explained that <code>every finite oriented map is realized canonically over a complex algebraic curve' and that </code>the Galois group of $\overline{\bf Q}$ over $\bf Q$ acts on the category of these maps in a natural way': one derives this by comparing various approaches to the study of coverings of $\bf P_1 - {0,1,\infty}$. Since then, the term `dessin d'enfant' has been used often, by various authors in various mathematical senses, to denote objects (or isomorphism classes of objects) arising in those approaches. In this paper we do not try to define the term; we content ourselves with using it to denote the theory as a whole.</p> <p>"Here are some reasons why one should pay particular attention to finite coverings of the curve $\bf P_1 - {0,1,\infty}$: "(a) It is the simplest algebraic curve whose fundamental group is not commutative. "(b) It has many coverings over $\overline{\bf Q}$: according to a theorem of Belyi(, every integral algebraic curve over $\overline{\bf Q}$ has an open Zariski set that is realized as such a covering. "(c) It is identified with the moduli space $M_{0,4}$ of genus-0 curves equipped with four marked points. The study of the action of ${\rm Gal}(\overline{\bf Q}/\bf Q)$ on its $\pi_1$ is the starting point for the study of the Grothendieck-Teichmüller tower (consisting of the fundamental groupoids of all the moduli spaces $M_{g,n}$ on which ${\rm Gal}(\overline{\bf Q}/\bf Q)$ acts).'' </p> http://mathoverflow.net/questions/1909/what-are-dessins-denfants/125257#125257 Answer by Nick Gill for What are dessins d'enfants? Nick Gill 2013-03-22T09:51:51Z 2013-03-22T09:51:51Z <p>There are many good answers to this question already. However it seems important to me that the contribution of Jones and Singerman to this subject is noted. These two British mathematicians from the University of Southampton wrote an important paper on this subject some time <strong>before</strong> Grothendieck wrote his <em>Esquisse</em>.</p> <p>The paper in question is:</p> <blockquote> <p><strong>MR0505721 Zbl0391.05024</strong> Jones, Gareth A.; Singerman, David <em>Theory of maps on orientable surfaces</em>. Proc. London Math. Soc. (3) 37 (1978), no. 2, 273–307. </p> </blockquote> <p>The paper is beautifully written, and outlines the correspondence between maps on topological surfaces, maps on Riemann surfaces, and groups with certain distinguished generators. They do not consider the Galois action, this being the aspect of the area that so excited Grothendieck. Their notion of a <em>map</em> is a particular instance of a <em>dessin d'enfant</em> (these days a map is also known as a <em>clean dessin</em>), the more general notion of <em>hypermap</em> which was considered subsequently corresponds to the general dessin d'enfant.</p> <p>A later paper, by Bryant and Singerman, extended the treatment to surfaces with boundary.</p>