These talks will be self-contained presentations of mathematical concepts and theorems. They are intended to be accessible to all mathematics students.

Most phenomena in astrophysics involve turbulence. It is usually omitted from the simulations because no one knows how to deal with it, yet is can dominate the results. In this talk I will describe a new theory of turbulence which makes use of a Lagrangian description which is a particular advantage for astrophysical simulations. One fascinating issue is how large scale modes couple with small scale modes. I will display some non-intuitive results for an infinite chain that relate to issues of chaos in dynamical systems. I will also show some new results on toy stars that are relevant to planet and star systems. Toy stars consist of gas held together by a simpler force than gravity and they have some interesting exact solutions. Not only that, they relate to tidal oscillations of bodies of water. A familiar example is the oscillation of water in a paraboloidal bowl, or wine in a wine glass. We are currently extending the toy star models to give us very useful exact solutions for the dynamics of gas with magnetic fields.