# LPTMS PhD Proposal: Phase transition in Mean Field Games

### Responsable: Denis ULLMO 01 69 15 74 76

Mean field games present a new area of research at the boundary between applied mathematics, social sciences, engineering sciences and physics. It has been initiated a decade ago by Pierre-Louis Lions (recipient of the 94 Fields medal) and Jean-Michel Lasry as a new and promising tool to study many problem of social sciences, and with an explicit mention of the influence of concepts coming from physics (the notion of “mean field approximation”). This field has since then grown significantly, and after a period where mainly stylized models where introduced, we witness now the appearance of (necessarily more involved) mean field game models closer to practical applications in finance, vaccination policies, or energy management through smart electronics. Up to now, the development of Mean Field Games has mainly originated from the mathematics and economic communities. Mean Field Games theory is, however, by essence a multi-disciplinary field for which the input of physicists is much needed. Indeed, as important as they are, the studies of internal consistency and the numerical schemes developed by mathematicians cannot replace the deeper understanding of the behavior of these models, obtained in particular through powerful approximation schemes, that physicists (and essentially only them) know how to provide.

In this general context, the goal of this internship will be to study "phase transitions" in MFG, that is a discontinuous changes of behavior as a parameter is varied. During the internship, this study will be limited to a class of Mean Field Games for which there exist a formal, but deep, connection with the non-linear Schrödinger equation, which is making their analysis, and in particular the origin of these phase transitions, more transparent.