I will briefly introduce the main objects of two different theories: multiple zeta values, which are special values at integers of many-variable ζ-functions that satisfy a family of "double shuffle" algebraic relations, and the Grothendieck-Teichmüller group on the other hand, which arose from Grothendieck's attempt to describe the absolute Galois group by its action on fundamental groups of varieties defined over Q, particularly moduli spaces of curves. There is a natural way to associate a Lie algebra to each of these two theories, via the double shuffle Hopf algebra on the one hand and the pro-unipotent version of the Grothendieck-Teichmüller group on the other. Furusho showed in 2011 that the GT Lie algebra is contained inside the double shuffle one. We will indicate a new, elementary proof of this surprising inclusion and discuss the possibility of an isomorphism between the two.
