The spontaneous formation of patterns in nature is a fascinating and ubiquitous phenomenon of great importance both from the theoretical and the applied point of view.
Self-assembled patterns (namely ordered geometric structures forming on a mesoscopic scale without the aid of external forces) are physically believed to stem from the competition between short range attractive interactions and long range repulsive interactions. However, the mechanisms at the base of the emergence of such structures are still poorly understood in most physical cases and they are the widely open goals of an active research field.
The aim of the course is to provide an overview of the main results in the literature about the formation of patterns for prototypical physical interactions, with a focus on the most recent results in which striped pattern formation was rigorously proved in suitable regimes, through techniques of calculus of variations and geometric measure theory.
