Complex Geometry and the Hele-Shaw flow

The goal of this talk is to discuss a link between the Homogeneous Monge Ampere Equation in complex geometry, and a certain flow in the plane motivated by some fluid mechanics.   After discussing and motivating the Dirichlet problem for this equation I will focus to what is probably the first non-trivial case that one can consider, and prove that it is possible to understand regularity of the solution in terms of what is known as the Hele-Shaw flow in the plane. As such we get, essentially explicit, examples of boundary data for which there is no regular solution, contrary to previous expectation.  All of this is joint work with David Witt Nystrom.