Effective dynamics of interacting classical, quantum, and wave systems

Interacting systems of particles and waves are foundational in many natural phenomena. This talk will outline mathematical approaches for deriving effective, statistical descriptions of such many-body dynamics by connecting them to solutions of nonlinear partial differential equations. Key examples include the Boltzmann equation, which emerges as a limit of interacting hard spheres; the nonlinear Schrödinger equation, which describes quantum particle dynamics initialized near a Bose-Einstein condensate; the Vlasov equation, which is an effective model for both non-collisional particles evolving under Newtonian dynamics or as a semiclassical limit of fermionic interactions; and the kinetic wave equations, which model the behavior of interacting waves. I will discuss my joint work on each of these equations, highlighting how to frame these PDEs as limits of the underlying particle or wave dynamics.