These talks will be self-contained presentations of mathematical concepts and theorems. They are intended to be accessible to all mathematics students.

Nonlinear lattice dynamics, where the acceleration of a particle is dependent on its nearest neighbours, provide simple models of molecular chains studied in physics, chemistry and biology, and are the starting point for the numerical analysis of partial differential equations. Toy models of such chains can be easily constructed and used to demonstrate their dynamics. I will discuss three nonlinear lattices: the famous Fermi-Pasta-Ulam model, the Toda lattice and anharmonic lattices. The various types of nonlinear waves that can occur in these lattices will be demonstrated, together with the analogies and differences with some continuous systems.