Atomism and quantization (On work with S. Graffi, A. Knowles, A. Pizzo and S. Schwarz)

By way of examples, it is shown that the atomistic constitution of matter can be understood as arising from a deformation of continuum theories of matter. The first such example concerns the Hamiltonian mechanics of point particles, which can be understood as a deformation of the Vlasov dynamics. The second example concerns quantum (Bose) gases that arise from a deformation of a Hamiltonian system with infinitely many degrees of freedom whose dynamics is described by the Hartree equation. It is then shown how such continuum theories of matter can be recovered from atomistic theories in a mean-field limit. Some novel Egorov-type theorems are described. The talk concludes with a discussion of open problems.