Collective dynamics can be observed in many situations in our daily lives, for example the motion of bird flocks, the formation of directional lanes in pedestrian flows or opinion dynamics on social media. In all these examples interactions of close individuals lead to the formation of complex phenomena in the entire population.
In this talk I will discuss different mathematical models to describe large interacting particle systems. I will then focus on the corresponding macroscopic equations, which describe the evolution of the entire population. These equations often have a gradient flow structure, which can be used to analyse the large time behaviour. I will discuss the structure of these equilibrium solutions and how they relate to the observed complex behaviour. Finally, I will discuss the application of these ideas in the context of data science, in particular Bayesian inversion.