Wave scattering is ubiquitous in nature, it is one of the most important phenomena in physics. One of the key objects in the study of scattering theory are the so-called resonances. I will briefly talk about classical pictures of obstacle scattering resonances and give a mathematical description of transmission problems. Then I will discuss how geometry and refraction indices influence the distribution of resonances. Obstacles with negative index of refraction exhibit a new type of resonances called plasmons. We will discuss when they are present for non-trapping obstacles and, if time allows, I will briefly go over how Jeff Galkowski and I catch the plasmons.

We consider  Interacting Particle dynamics with Vicsek type interactions, and their macroscopic PDE limit,  in the non-mean-field regime; that is, we consider the case in which each  particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance).  It was observed by Motsch and Tadmore that in this non-mean-field regime the  influence between agents (i.e. the interaction term) can be scaled either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting  at time t  (local scaling). We compare the behaviour of the globally scaled and the locally scaled system in many respects; in particular we observe that, while both models exhibit multiple stationary states,  such equilibria are unstable (for certain parameter regimes) for the globally scaled model, with the instability leading to travelling wave solutions,  while they are always stable for the locally scaled one.   This observation is based on a careful numerical study of the model,   supported by formal analysis.

Based on  work with with P. Butta', B. Goddard, T. Hodgson, K.Painter.