Higher structures in Kontsevich's deformation quantization

To a smooth manifold one can associate the Lie algebras of multi-vector fields and multi-differential operators, where one can encode classical data (Poisson structures) and quantum data (star products). Relating these two led Kontsevich to his famous formality theorem that establishes the deformation quantization of Poisson manifolds. In this talk I will give an introduction to the subject and we will see that there are richer algebraic structures in the relevant objects, namely of BV algebras (or of Gravity algebras in the equivariant case). These structures can be neatly encoded using certain operads which allow us to prove "homotopy BV-Gravity"  extensions of Kontsevich formality theorem.