Charting a New Paradigm for Large Non-Exchangeable Multi-Agent and Many-Particle Systems

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The project will develop a new framework for the study of large systems of interacting "agents" or "particles". The main goal is to obtain a reduction in complexity for systems with a very large number of agents, by replacing exact interactions with a notion of mean field. Existing theories, to justify this approximation, typically assume that the agents are identical or at least indistinguishable. This assumption can be valid when considering classical applications in Physics, such as the dynamics of electrons under an electrostatic potential. The investigator will introduce new approaches to remove the assumption of indistinguishable particles or agents while still involving realistic singular interactions: This will allow to consider more complex applications such as multi-species models in Physics and models for the dynamics of networks of biological neurons. The project provides training research opportunities for graduate students. Mathematically speaking having non-identical particles or agents changes the structure of the systems. First, it makes them non-symmetric and non-exchangeable, which has deep consequences when trying to define so-called observables, such as the joint law of a sub-system. It also breaks some of the cancellation effects, for example the anti-symmetry of the interaction, that many current methods relied on to handle singular interactions. The project introduces novel methods to bypass those issues such as new duality formulations and an extended notion of marginal.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.