$L_\infty$ algebras governing simultaneous deformations in algebra and geometry

We will explain how simultaneous deformations problems in algebra and geometry are naturally governed by non quadratic $L_\infty$ algebras and how such algebras can be constructed by super geometry or operad theory, depending on the applications.

 This will be illustrated with examples. We will consider simultaneous deformations of pairs such as couples of algebras/morphisms, coisotropic/Poisson, Dirac/Courant, generalized complex/Courant (if time permits).