Grothendieck's mathematical diagram. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:33:12Z http://mathoverflow.net/feeds/question/29811 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29811/grothendiecks-mathematical-diagram Grothendieck's mathematical diagram. Koundinya Vajjha 2010-06-28T17:04:57Z 2011-03-14T17:45:58Z <p>I was going through <a href="http://www.ihes.fr/document?id=1723&amp;id_attribute=48" rel="nofollow"> this article</a> which appeared in the Notices of the AMS, and in it, there's a picture which shows a mathematical diagram drawn by Grothendieck. I would be delighted if anyone could explain what that is. Thanks in advance.</p> <p>EDIT: The comments below suggest that the diagram is a dessin d'enfant. And even though I am by no means an expert, I somehow feel that it may not be one. I'd love some clarification on how one may call it so. Questions I want to ask now, are</p> <p>1) Is the diagram really a dessin? Or is it something else?</p> <p>2) Do the shaded parts of the diagram signify something? Are there any special types of dessins which have a similar a structure?</p> http://mathoverflow.net/questions/29811/grothendiecks-mathematical-diagram/58391#58391 Answer by Sándor Kovács for Grothendieck's mathematical diagram. Sándor Kovács 2011-03-14T07:18:15Z 2011-03-14T17:45:58Z <p>First of all, I believe that according to Grothendieck's definition of dessins d’enfants the picture (if I am looking at the right one) indeed seems to show one. At the same time you have a point that this is not one of the more interesting ones. </p> <p>On the other hand it might be the very first one Grothendieck ever drew and then one could make a wild guess as to what it shows. I would venture to say that the diagram in question shows the <em>complex conjugation of the Riemann sphere</em>.</p> <p>If I understand correctly, Grothendieck's inventing and studying dessins d’enfants was motivated by his goal of finding non-trivial elements of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb Q)$. An obvious (and the only obvious) non-trivial element is complex conjugation of $\mathbb C$, which actually extends to $\mathbb P^1_{\mathbb C}$, a.k.a. the Riemann sphere.</p> <p>My guess is that this picture shows that and was perhaps the visual clue that led Grothendieck to make the definition of dessins d’enfants.</p> <p><strong>Addendum</strong>: To answer the question raised in the comments: The picture clearly shows a reflection. Complex conjugation is a reflection. I did not claim I have a proof for this, indeed, notice the words <em>wild guess</em> above. The only argument I can offer is that </p> <p>i) it is reasonable to assume that this picture is or at least has something to do with <em>dessins d’enfants</em>.</p> <p>ii) it is a very simple drawing for that</p> <p>iii) there should still be some significance for someone to have put it in the article</p> <p>iv) it is reasonable to assume that it is an early drawing of <em>dessins d’enfants</em></p> <p>v) it is clearly a reflection</p> <p>vi) complex conjugation is a reflection <em>and</em> has a lot to do with the birth of <em>dessins d’enfants</em>.</p> <p>As I said, this is a guess, but I wonder if anyone can offer anything other than a guess.</p>