Sergei Merkulov: A line in the plane and the Grothendieck-Teichmueller group

The Grothendieck-Teichmueller group (GT) appears in many different parts of mathematics: in the theory of moduli spaces of algebraic curves, in number theory, in the theory of motives, in the theory of defomtaion quantization etc. We argue that GT controls deformation theory of a line in the complex plane when one understands these geometric structures via their associated  (compactified) configuration spaces. This result is an elementary but deep corollary of some recent breakthrough theorems by Thomas Willwacher. My purpose is to give an intuitive and pictorial introduction to that theorems for non-specialisits.