Quantization: complex and real case, conic and non conic case. Application to symplectic topology.

We shall briefly recall the notions of quantization in the complex case: microdifferential modules on complex contact manifolds, modules over DQ-algebras on complex symplectic manifolds. Then we will explain how to treat the cotangent bundle to a real manifold by using the microlocal theory of sheaves and its non conic version recently introduced by D. Tamarkin. This last theory gives a new approach to the Arnold's problem of non-displaceability of compact Lagrangian manifolds.