The Grothendieck Program

The definition of the Grothendieck-Teichmüller group by Drinfeld was motivated by applications to quantum group theory and by Grothendieck’s program aiming to understand a geometric counterpart of absolute Galois groups. The purpose of the this outlook chapter is to give an overview of this program, and to survey the connections between the structures occurring in our study of the homotopy of operads and the objects arising on the arithmetic side of the subject. In Grothendieck’s proposal [101], the fundamental objects are the moduli spaces of marked curves Mgn already considered in §4.3.5 in the genus zero case. We mostly deal with this case g = 0 yet. We then have