How to solve polynomial equations? A famous ancient problem in mathematics is the solubility of polynomials by radicals, which was a major theme in mathematics for a couple of millennia. Two centuries ago, this problem was solved (negatively in general) by Abel-Ruffini and Evariste Galois. The work of Galois, then a teenager, revolutionised algebra, and indeed mathematics. The basic idea of Galois is to think about the symmetries of roots of polynomials. This led him to define what nowadays are called Galois groups, and initiate group theory, the latter became a central topic in mathematics and beyond. The major discovery of Galois was that the group of symmetries of the roots of a polynomial equation encodes the information on the solubility of the polynomial by radicals, which is a property of arithmetic and geometric nature one would argue.

During the second half of last century, Grothendieck further pushed and refined the ideas of Galois, by introducing the theory of fundamental groups in arithmetic and algebraic geometry. The idea/fact, already observed by Galois, that some geometric and arithmetic aspects of (systems of) polynomial equations, are (should be) encoded in (combinatorial) group theory, was central to Grothendieck's mathematical thinking. He put forward, few decades ago, a vast and far reaching research programme: the so-called anabelian programme. The ultimate goal of this programme is to establish a purely combinatorial group-theoretic framework in which both
arithemtic and geometry are encoded, very much in the spirit of the work of Galois.

In my talk I will review the historical context which led to the birth of anabelian geometry and discuss some developments and results in recent decades, mostly established at RIMS Kyoto! and which are heading towards achieving Grothendieck's aims and beyond. These developments are concerned with the description of the absolute Galois group of the field of rational
numbers; a major object of study in number theory, and the (arithmetic version of) mapping class group.

16:15- tea