Résumé du cours

Nature and human societies offer many examples of self-organized behavior: ants form colonies, birds flock, mobile networks synchronize, and consensus may emerge from interaction of diverse human opinions. These are simple examples of collective dynamic in which small scale interactions of so-called active particles, lead to emergence of high-order, larger-scale patterns. These lectures will survey recent mathematical developments in collective dynamics, starting with the influential works of Reynolds, Krause, Vicsek and Cucker & Smale. The dynamics is governed by different protocols of pairwise interactions. Collisions are avoided. There is a hierarchy of three levels of descriptions: agent-based description of swarm dynamics on a graph, a Vlasov-type kinetic description, and large-crowd hydrodynamic description. A main question of interest is how different classes of interaction kernels affect the large-time, large-crowd, non-equilibrium dynamics, leading to the emergence of different large-scale patterns.