Singularity structures in solutions to the Monge-Ampere equation

A celebrated theorem of Jorgens-Calabi-Pogorelov says that global convex solutions to the Monge-Ampere equation det(D^2u) = 1 are quadratic polynomials. On the other hand, an example of Pogorelov shows that local solutions can have line singularities. It is natural to ask what kinds of singular structures can appear in functions that solve the Monge-Ampere equation outside of a small set. We will discuss examples of functions that solve the equation away from finitely many points but exhibit polyhedral and Y-shaped singularities. Along the way we will
discuss geometric and applied motivations for constructing such examples, as well as their connection to a certain obstacle problem for the Monge-Ampere equation.