Following my colloquium talk earlier this week, I will present the convex integration scheme (a la Nash & Kuiper) and details of the rigidity arguments for the Monge-Ampere equation on a two-dimensional set Ω. In particular, we will see that for any f ∈ L²(Ω), any continuous function v₀ in Ω, and any exponent α < 1/7, there exists a sequence of exact (weak) solutions v_n ∈ C^1,α to the equation det(∇v) = f in Ω, that converges uniformly to v₀. On the other hand, this is largely impossible when α > 2/3.