"We present an extension of the KAM invariant torus theorem to infinite-dimensional systems consisting of infinitely many weakly interacting particles.

We show that, despite the infinitely many degrees of freedom and the presence of long-range, “all-to-all” interactions, there exist many initial states for which the system exhibits stable almost-periodic motions, corresponding to infinite-dimensional invariant tori.

In the case of singular potentials, such as the Newtonian potential, we introduce the notion of collective motion, which provides a mechanism for excluding collisions. Under the influence of the first, more massive particles, individual particles may undergo displacements much larger than their mutual separations, while these separations tend to zero. Nevertheless, the collective nature of the motion ensures that the particles remain sufficiently separated from one another.

This notion also plays an essential role in the study of normally elliptic tori.

Joint work with Dmitry Dolgopyat and Jaime Paradela."