Singularity formation in incompressible fluids

We discuss the problem of singularity formation for some of the basic equations of incompressible fluid mechanics such as the incompressible Euler equation and the surface quasi-geostrophic (SQG) equation. We begin by going over some of the classical model equations which have been proposed to understand the dynamics of these equations such as the models of Constantin-Lax-Majda and De Gregorio. We then explain our recent proof of singularity formation in De Gregorio's model. Finally, we discuss how to use symmetries in the original equations (Euler and SQG) to extract lower dimensional models. These lower-dimensional models satisfy the following nice property: singularity formation for the model implies singularity formation for the original equation. This is based on joint work with In-Jee Jeong.