Singular solutions to the Monge-Ampere equation

The Monge-Ampere equation det D^2u = 1 arises in several applications. Examples of Pogorelov show that interior regularity is not expected in general. We will discuss optimal estimates on the Hausdorff dimension of the singular set, and sharp integrability conditions for the second derivatives that rule out singularities. Some of this is joint work with T. Collins.