This course will give a pedagogical introduction to (quantum) integrability, a topic in mathematical physics with applications ranging from experiments in condensed-matter physics to high-energy theory. The aim is to show some highlights of the field, with a glimpse of the underlying algebraic structures, while keeping technicalities to a minimum.

The provisional plan of the course is as follows; details can be adjusted to suit the audience.

Part I will cover the (standard) basics of integrability, more or less following my lecture notes arXiv:1501.06805:

• the Heisenberg spin chain,
• the six-vertex model,
• the exact characterisation of their spectrum by Bethe ansatz,
• an application to alternating-sign matrices (ASM).

Part II is about the (less standard) basics of long-range integrability:

• the Haldane–Shastry spin chain,
• the quantum Calogero–Sutherland system,
• the exact and explicit characterisation of their spectrum.