Operad of little disks, differential topology and Galois theory

The operad of little n-disks is a fundamental object in algebraic topology that was introduced as a way of recognizing n-fold loop spaces. I will recall its definition and then survey some recent work of Dwyer--Hess and Boavida--Weiss relating mapping spaces between the operads of little disks and spaces of knots and higher dimensional knotted objects. I will then describe a faithful action of the absolute Galois group of Q on the profinite completion of the operad of little 2-disks.