The special Lagrangian equation is a fully nonlinear elliptic PDE whose solutions are potentials for volume-minimizing Lagrangian graphs. There exist continuous viscosity solutions to the Dirichlet problem for this equation, but many basic regularity questions (such as whether these solutions have bounded gradient) remain open. In this talk we will discuss some of these questions. We will use them to motivate the more general question of whether homogeneous functions with nowhere vanishing Hessian determinant can change sign when the degree of homogeneity is between zero and one, and we will answer this question in the negative.