Abstract or Additional Information
A wide range of phenomena, such as electrostatics, swarm behavior, and cell interactions, can be modeled using a locally repulsive pairwise potential (one that causes particles to repel one another when sufficiently close), and possibly with some external field. The minimizers of the resulting system's energy then provide steady states of its dynamics. When local repulsion is sufficiently strong, one expects particles to be well distributed in some sense. However, for sufficiently weak local repulsion, it has been observed in several works that particles often concentrate on or around lower dimensional sets, though the exact nature of this phenomenon is a mystery. This talk will be focused on certain types of repulsive-attractive potentials (for which particles repel one another at close range but attract each other when sufficiently far apart) and strictly repulsive potentials with radially symmetric attractive external fields. In both cases, we will discuss our recent results determining when particles concentrate on a sphere. The work in this presentation is joint with Djalil Chafaï, Rupert Frank, Edward Saff, Minh Vu, and Robert Womersley.