On the periodicity conjecture for Y-systems

Y-systems are a certain family of algebraic recurrence equations, which emerged in the early nineties in the study of the Thermodynamic Bethe Ansatz. They are naturally associated to arbitrary pairs of Dynkin diagrams, and the periodicity conjecture asserts that all solutions to those systems are periodic with period equal to twice the sum of the respective Coxeter numbers. Although this conjecture has by now been largely proved, there remain open questions. In this talk, I will discuss several approaches to the problem and present some recent developments.