Rational Homotopy Theory: Application to the free loop space homology

I will present ideas of Rational Homotopy Theory, which roughly mean "forgetting torsion" in the homotopy groups of spaces. There are two settings to work with. The first one relies on commutative differential graded algebras and is due to D. Sullivan. The second setting, due to D. Quillen, relies on the category of differential graded Lie algebras. From each of the two settings, one can read homotopy groups, cohomology algebras, function spaces and so on. We give an application with the computation of the Gerstenhaber structure of the free loop space homology of  homogeneous spaces.