Mathematics Colloquium- Giacomo Albi, University of Verona

Shaping Collective Dynamics Across Scales

Large systems of interacting particles or agents arise in many applications, from collective motion, opinion formation and crowd dynamics to physical systems and machine learning. Simple interaction rules at the individual level can generate complex collective behaviour, raising the question of how such dynamics can be influenced or controlled toward desired configurations.

In this talk, we introduce the problem through examples drawn from these different contexts. Starting from a prototypical alignment model, we then illustrate the emergence of collective behaviour and the design of control strategies. As the number of agents grows, direct optimal control quickly becomes computationally prohibitive, naturally leading to a hierarchy of descriptions from microscopic dynamics to kinetic and mean-field models, where the evolution of the system is represented at the level of probability distributions. This change of scale reduces the dependence on the number of agents, while preserving the main collective features of the dynamics. Further challenges arise when the state space itself is high-dimensional, motivating the use of learning-based surrogates to approximate otherwise expensive feedback controls.

Finally, we present some current directions and open challenges, including the control of kinetic plasma models through external magnetic fields and the interpretation of neural networks as interacting dynamical systems.