The Monge-Ampere equation det(D^2 u) = 1 arises in prescribed
curvature problems and in optimal transport. An interesting feature of the
equation is that it admits singular solutions. We will discuss new examples
of convex functions on R^n that solve the Monge-Ampere equation away from
finitely many points, but contain polyhedral and Y-shaped singular
structures. Along the way we will discuss geometric and applied motivations
for constructing such examples, as well as their connection to a certain
obstacle problem.