Operads, graph complexes and the rational homotopy of embedding spaces

I will report on joint works with Victor Turchin and Thomas Willwacher on the applications of operads to the study of the rational homotopy type of embedding spaces.
In a first part, I will explain a graph complex description of the rational homotopy of mapping
spaces of $E_n$-operads. Results on the Goodwillie-Weiss calculus implies that this computation gives a description of a delooping of embedding spaces of Euclidean spaces.
Eventually, I will survey an ongoing project where we study operadic models of the rational homotopy of a generalization of these embedding spaces for the open subsets of Euclidean spaces with a bounded complement.