How can states of matter be classified? Beyond the usual considerations based on geometric symmetries, the application of concepts derived from topology is currently leading to some fascinating developments. Initially proposed to tackle certain solid-state problems such as the quantum Hall effect, these concepts are now being applied to the fields of atomic physics and optics.
Topological materials are characterized by specific robust properties, which persist when the parameters of the system are (moderately) modified or disorder is added. Topological materials can, for example, exhibit quantized conductance or lossless propagating edge states, providing the basis for novel devices, particularly in photonics.
This year's lecture set out to establish the foundations of this new field of research for the case of particles or waves moving in a spatially periodic environment. Starting with the notion of geometric phase (or Berry phase), we successively addressed problems in one, then two dimensions. We have also illustrated our results with a series of recent experiments involving cold atomic gases and photons.