Organizer lead-in

Lead-in to ``Theory of motives, homotopy theory of varieties, and dessins d'enfants", to be held at the American Institute of Mathematics, April 23-26, 2004

We are writing to give you further detail about the meeting, and to ask you to give some thought to its subject matter.

The goal of this workshop is to bring together mathematicians working in two distinct areas, namely the area surrounding the ideas of (1) Dessins d'enfants, the Grothendieck-Teichmüller groups, and moduli stacks for curves on the one hand, and (2) the area of homotopy theory of varieties, including etale homotopy theory and the motivic (or $\mathbb{A}^1$-homotopy theory) on the other. One of the major goals of area (2) is to develop topological techniques which are applicable to arithmetic and algebraic geometric problems, and it is therefore natural that those working in that area would benefit from exposure to a quickly developing area within number theory, which already has a distinct geometric flavor. There is also a sense that many of those working in area (1) may not be aware of the recent developments in area (2), and that their insight may provide a useful guide for the development of the ``technology'' in area (2). We would like the program to provide workers in each area to the techniques and results of the other, and to encourage the development of new approaches and problems in both areas.

The program of the workshop will include a number of expository talks, which will introduce all participants to the language and techniques of both areas. It will also include a number of discussion sessions, planned around some particular questions. While we will suggest some specific questions below, we ask for your help in determining the subject matter for the sessions by proposing your own questions, or suggesting directions of research, to us. To help you in thinking about these ideas, we add a collection of suggested reading material at the end of this note. Here is a list of three suggested general areas of discussion.

• How can the techniques of motivic homotopy theory best be used to study actual arithmetic and algebraic geometric problems? Or, perhaps better, what technology should be built into the homotopy theory of varieties in order to allow it to study questions about the existence of rational points in varieties or moduli spaces, the existence of sections of vector bundles or algebraic maps, or the cohomology of moduli spaces in the algebraic setting? What are the problems which currently seem most amenable to attack by motivic methods?
• Can one compute the motivic homotopy type for curves $/\overline{\mathbb{Q}}$ given its presentation in terms of a dessin? In particular, can one begin to understand the motivic $\pi _0$ (an extremely complicated object) or $\pi _1$ in terms of the dessin description of the curve?
• As described in [Dessins], there are conjectures that $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ can be identified as the automorphism group of a certain Teichmüller tower, obtained by considering a family of moduli spaces of curves of varying genus together with certain identifications or relations among them. There has also been recent work by Madsen and Weiss [MW] which computes the cohomology (indeed, the homotopy type) of the stabilized moduli spaces of curves, where the moduli spaces are ``glued together'' using topological methods. We would then like to ask what are the natural algebraic methods for assembling different moduli spaces into towers, or for stabilizing moduli spaces, and do any of them correspond to the topological version given by Madsen-Weiss?

We view these questions as starting points for thinking about these areas. We strongly encourage you to propose your own questions, or refining and making more specific the questions posed above.

We are very much looking forward to this meeting, and look forward to seeing all of you in April.

Gunnar Carlsson
Rick Jardine

Bibliography

[Dessins]The Grothendieck Theory of Dessins d'Enfants, edited by Leila Schneps, London Mathematical Society Lecture Notes Series, vol. 200, Cambridge University Press, 1994.

[Galoisactions] Geometric Galois Actions, I and II, edited by Leila Schneps and Pierre Lochak, London Mathematical Society Lecture Note Series, vols. 242-243, Cambridge University Press, 1997.

[DI] D. Dugger and D. Isaksen, The Hopf condition for bilinear forms over arbitrary fields, Preprint (2003), Hopf Topology Archive, http://hopf.math.purdue.edu

[J2] J.F. Jardine, Generalised sheaf cohomology theories, Axiomatic, Enriched and Motivic Homotopy Theory, NATO Science Series II, Vol. 131 (2004), 29-68, also available online at http://www.math.uwo.ca/$\sim$jardine/papers.

[M1] F. Morel, An introduction to $\mathbb{A}^{1}$-homotopy theory, Contemporary Developments in Algebraic Topology, M. Karoubi, A.O.Kuku, C. Pedrini (ed.), ICTP Lecture Notes 15 (2003), 357-441, also available online at http://www.math.jussieu.fr/$\sim$morel/listepublications.html.

[MV] F. Morel and V. Voevodsky, $\mathbb{A}^{1}$-homotopy theory of schemes, Publ. Math. IHES 90 (1999), 45-143.

[MW] I. Madsen and M. Weiss, The stable moduli of Riemann surfaces: Mumford's conjecture, Preprint (2002), Hopf Topology Archive http://hopf.math.purdue.edu/ .

[V2] V. Voevodsky, $\mathbb{A}^{1}$-Homotopy Theory, Doc. Math., Extra Volume ICM 1998 I, 579-604, http://www.math.uiuc.edu/documenta.

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