Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles, the resulting material is superconducting and the so-called energy bands are flat. In 2011, Bistritzer and MacDonald proposed a model that is experimentally very accurate in predicting magic angles. In this talk, I will introduce some recent mathematical progress on the Bistritzer--MacDonald's model, including the mathematical characterization of magic angles and flat bands, and generic existence of Dirac cones.

I will present two complementary small-data stability theorems for the 3D irrotational and isentropic compressible Euler equations in spherical symmetry. We consider the Cauchy problem with initial data that are perturbations of a non-vacuum constant state.