Collective behaviour in systems with non-local interactions.

Animal groups often form striking aggregation patterns such as fish schools, locust swarms and bacterial cultures. This collective behaviour is a captivating natural phenomenon that inspires evolutionary and biological questions as well as structural and physical ones. Emerging behaviour usually serve a biological purpose such as predator avoidance or collective foraging for food. Related models also arise in other important applications such as self-assembly of nanoparticles, theory of granular gases, and molecular dynamics simulations of matter.Mathematically, a common feature of many swarming models is the presence of long-range interaction within the population. This is often modelled using either particle ODE models with global coupling or partial differential equations with non-local kernels. Due to nonlocal nature, these systems often exhibit complex and novel phenomena that pose challenging questions and motivate the developement of new mathematical tools. In the last decade, driven by experimental observations, swarming models have become popular in biological and broader scientific literature. On the other hand, computer simulations have revealed a plethora of complex behaviour. However, the mathematical understanding has been lagging. It is the goal of this proposal to advance the mathematical understanding of collective behaviour.