I will discuss some of the 50 year old history, and the first (very recent) rigorous results, on a basic question raised by Fermi, Pasta and Ulam in 1947 in a celebrated paper.

F, P and U wondered numerically about the long time behaviour of a long (say, infinite) atomic chain evolving under Hamiltonian dynamics, with nonlinear (say Lennard-Jones) nearest neighbour interaction.

Statistical mechanics reasoning suggests that the nonlinearity would promote asymptotically thermalized distribution of energy among the Fourier modes of the system. (For a linear chain, i.e. a quadratic nearest-neighbour potential, the Fourier spectrum would be time-independent). But the numerical experiments showed strong recurrence effects.

The main new theorem (joint with Robert Pego, Maryland) shows that on each sufficiently low energy surface, an open set of initial data has perpetually recurrent Fourier spectrum. In particular no long-time equidistribution occurs. The result confirms the numerical findings of F, P and U, and disproves statistical mechanics arguments, at low energy. At high energy the story remains open from a rigorous point of view.

Previous attempts (which will be discussed in the talk) tried to view FPU recurrences as Kolmogorov-Arnold-Moser type effecs, but didn't lead to rigorous results as errors could not be controlled as the number of particles (i.e. the system dimension) gets large. Our own work stays in infinite dimensions all the way. We start from the observation that exact solitary wave modes are present in the chain, and then show that these are stable globally in time in the sense of start close, stay close. The latter combines ideas from infinite-dimensional symplectic geometry (including the discovery of a natural symplectic form for the FPU chain) and a careful multiscale analysis which allows to approximate the (nonintegrable) chain by the (integrable) Korteweg-de Vries equation.