A Line in the Plane and the Grothendiek-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 deformation quantization etc. Using recent breakthrough theorems by Thomas Willwacher, we argue that GT controls the deformation theory of a line in the complex plane when one understands these geometric structures via their associated operads of (compactified) configuration spaces. Applications to Poisson geometry, deformation quantization, and Batalin–Vilkovisky formalism are discussed.