The art of anabelian geometry, towards a combinatorial approach to arithmetic and geometry.

In his pioneering work, Évariste Galois related polynomial equations (in one variable) to group theory via Galois theory and the study of Galois groups. Polynomial equations in several variables define algebraic varieties: the main objects of study in algebraic geometry. Inspired by classical Galois theory and the theory of topological fundamental groups, Alexandre Grothendieck introduced the theory of étale fundamental groups in algebraic geometry in the early 1960’s. The Galois group of the field of rational numbers, as well as the fundamental groups of the moduli spaces of curves; the so-called mapping class groups, are major topics of investigation in number theory and arithmetic geometry.  

Four decades ago, Grothendieck initiated a major research programme, the so-called anabelian programme, which can be roughly summarised by the slogan: "Arithmetic and Geometry are encoded in Group theory". One of the major goals of the anabelian programme is to provide a purely combinatorial group theoretic description of the Galois group of the field of rational numbers, and the mapping class groups. In my talk, I will review the state of the art of anabelian geometry, and explain how extremely close we are of achieving the above goal as a culmination of major recent accomplishments in combinatorial anabelian geometry.