Colloquium

We present an overview of recent results on localized pattern formation in non-local PDEs that arise in swarming and particle self-assembly models. We begin by computing the non-local stability analysis for particles which bifurcate away from radially symmetric states such as rings and spheres. The linear theory accurately characterizes the rich array of localized patterns which have been observed in the fully nonlinear problem in two and three dimensions. This aspect of the theory allows us to solve the inverse problem of designing specified potentials which assemble into targeted patterns. Time permitting, we will also show how to leverage this mathematical theory to provide, for the first time, a purely isotropic physical model that produces spherical assemblies at the nano scale, such as viral capids.