Homotopy automorphisms of $E_2$-operads & Grothendieck-Teichmüller groups

The little $n$-cubes operads have been introduced to encode operations
acting on $n$-fold loop spaces. Operads homotopy equivalent to little
$n$-cubes (called $E_n$-operads) are also used in algebra, in order to  model a
full scale of homotopy commutative structures, from fully  homotopy
associative but non-commutative ($n=1$) until fully homotopy  associative and
commutative ($n=\infty$).

The main objective of this talk is to explain that the
Grothendieck-Teichmüller group, as defined by Drinfeld in the rational
setting, forms the group of homotopy automorphisms of $E_2$-operads, and  as
such, represents the internal symmetries attached to our first  level of
homotopy commutative structures. The proof of this result  relies on an
interpretation of the classical Drinfeld-Kohno Lie  algebras in terms of
rational models of $E_2$-operads.