Propagation in reaction-diffusion equations with obstacles: the role of geometry

This talk is about travelling fronts going through an array of obstacles for reaction-diffusion equations. I will consider the setting of bistable type equations and periodic obstacles. One can also think of a wave going through a perforated wall. We show in general that the wave is either blocked or it invades the domain. This hinges on results related to a celebrated conjecture of De Giorgi regarding stable solutions of elliptic equations in unbounded domains. I will then describe geometric conditions on the obstacles under which there is
either blocking or propagation. I report here on joint work with F. Hamel and H. Matano. I will also recall earlier joint work with L. Caffarelli and L. Nirenberg.