In this talk we consider the classical Monge-Amp´ere equation in two dimensions in a low-regularity regime:   (0.1) det D 2u = f on D ⊂ R2 .   We will assume that f is a given strictly positive, smooth function, but we want to assume as little regularity as possible on u. For instance we are interested in the situation u ∈ C1, 1/2. In particular we don’t make any assumptions on the convexity of u.   First we will recall shortly how (0.1) can interpreted under these assumptions from a topological point of view. Thereafter we will give an idea how this information can be transformed into a regularity results of u i.e. that it (or −u) coincides with the classical Alexandrov solution.   If time permits we will report on its consequence for the rigidity of very weak solutions to the two-dimensional Monge-Amp ´ere equation and two-dimensional isometric embeddings.   The talk is about work in progress.