Modern approaches to infinitesimal deformations of algebro-geometric objects (like varieties) use the setting of formal moduli problems, from derived geometry. It allows to prove that all kinds of deformations are governed by a tangent complex equipped with a derived Lie algebra structure. I will use this framework to study equivariant deformations of varieties with respect to the action of an algebraic group. Then, I will explain how this theory of equivariant deformations allows us to prove a dichotomous behaviour for almost all varieties that are homogeneous under a reductive group : either they deform to characteristic 0, or they admit no deformation to any ring of characteristic greater than p.

It is well-known that one can attach Galois representations to modular forms. In the case of cusp forms, the corresponding l-adic Galois representations are irreducible for every prime l, while in the case of Eisenstein series, the corresponding Galois representations are reducible. The Langlands correspondence is expected to generalise this picture, with cuspidal automorphic representations always giving rise to irreducible Galois representations. In the cuspidal, polarized, regular algebraic setting over a CM field, a construction of Galois representations is known, but their irreducibility is still an open problem in general. I will discuss recent joint work with Zachary Feng establishing new instances of irreducibility, and outline how our methods extend some previous approaches to this problem.