I have been fascinated how even innocent symmetry forces many systems to be classifiable, and satisfy other unexpected properties. Case in point are commuting expanding maps of compact manifolds. Either they have common powers or they arise by taking products or are smoothly equivalent to a linear expanding on a torus or nilmanifold. The extra symmetry here refers one map commuting with anther nontrivially.
I will explain underlying ideas, some results and applications. Speaker(s): Ralf Spatzier (Michigan)