Profinite completion of operads and the absolute Galois group of Q
I will first talk about the profinite completion of spaces. This is a construction originally due to Artin and Mazur which approximates a space by truncated spaces with finite homotopy groups. Using a good point-set level model of this construction, one can form the profinite completion of operads in spaces. The main theorem I want to talk about states that the profinite completion of the operad E2 is acted on by the absolute Galois group of Q and that this action is faithful. This suggest that the operad E2 has an algebro-geometric origin.