un dessin d’enfance

Posted by on Dec 1, 2008 in Manin2008, news • 1 comment

Last week I gave a talk at the 60th birthday conference for Jacques Alev. If you are interested in the slides, here they are. The official title was supposed to be “dessins d’enfants” with this summary

I will try to convince you that Grothendieck’s ‘dessins d’enfant’ form an example of a noncommutative manifold over the mythical field with one element (in the sense of Soule and Connes-Consani).

However, dessins only appear at the final slide. The main part of the talk consisted in explaining one sentence in Manin’s recent paper (page 4, line 3):

Soule’s definition of an \mathbb{F}_1-scheme X involves besides X_{\mathbb{PZ}}, a \C-algebra \mathcal{A}_X, and each cyclotomic point of X_{\mathbb{Z}} coming from X must assign ‘values’ to the elements of \mathcal{A}_X. His choice of \mathcal{A}_X for the multiplicative group \mathbb{G}_m is that of continuous functions on the unit circle in \C… We suggest to consider the ring of Habiro’s analytic functions…

I promised Jacques to do a proper write-up of the talk (and include some more details on the final slide) so I might as well do a couple of posts on it, later.

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  1. Kea on Dec 2, 2008, 3:09 am:

    Wonderful! I especially liked the example of C(C3), which is to say the matrix of the 3×3 quantum Fourier transform. Completely coincidentally, I also posted this matrix today, although in connection with neutrino mixing matrices, as usual. Hmmm. I have been studying the BRT polynomial as a dessins invariant. Do you think it would be helpful to think of this polynomial of such a manifold invariant? I would like to understand the quantum Fourier/BRT connection better.

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