# Mathematics Colloquia and Seminars

### Topological Galois Actions

Colloquium

 Speaker: Dr. Leila Schneps, CERN, France and MSRI, Berkeley Location: 693 Kerr Start time: Mon, Nov 29 1999, 4:10PM

Let \$K\$ be a field and let \$overline K\$ be its algebraic closure, i.e. the field formed by adding to \$K\$ all roots of polynomials defined over \$K\$. The absolute Galois group \$G_K={ m Aut}_K(overline K)\$ is defined to be the group of automorphisms of \$overline K\$ which fix \$K\$. But \$G_K\$ acts on many kinds of objects associated to \$K\$, not just algebraic numbers. \$G_K\$ is a profinite group which is sometimes extremely mysterious, and never more so than when \$K\$ is the field \${sf Q}\$ of rational numbers, the case considered in this talk. Our approach to \$G_{sf Q}\$ is to consider its action on two kinds of topological objects:

* {it dessins d'enfants}, which are cellular graphs embedded in topological surfaces, and

* {it diffeomorphisms} of topological surfaces (modulo those isotopic to the identity).

In studying the Galois action on dessins d'enfants, we ask ourselves above all the following question: {it can we find Galois invariants of dessins of a topological nature ? and can we find enough to characterize the Galois action?} In studying the action on diffeomorphisms, we obtain a parametrization of each element \$sigmain G_{sf Q}\$ as a pair \$(lambda,f)\$ with \$lambda\$ in the profinite completion of \${sf Z}\$ and \$f\$ in the profinite free group \$F_2\$ on two generators. Here the main question is: {it can we find properties of pairs \$(lambda,f)\$ ensuring that they come from an element of \$G_{sf Q}\$?} In some sense, these questions come down to asking if the language of topology is rich enough to use for the study of arithmetic.