Long time dynamics of Hamiltonian nearest-neighbour chains fifty years after Fermi, Pasta and Ulam

I will discuss some history and recent progress in the understanding of the long-time dynamics of Fermi-Pasta-Ulam (FPU) chains, i.e. 1D infinite chains of particles with general anharmonic nearest-neighbour interaction. While a huge body of rigorous work is available on analogous infinite-dimensional Hamiltonian PDE's, discrete systems are much less well understood. But some rigorous results are beginning to emerge on the many interesting effects in the discrete case.In particular, a fundamental question by Fermi, Pasta and Ulam (whether for generic initial data and interaction potentials, energy transport from macroscopic to microscopic modes should occur) has now been settled in the regime of low overall energy. (The answer is No. Joint work with Robert Pego, using solitary wave theory, renormalization group ideas and symplectic geometry ideas.)