A dynamical system is called integrable if the equations describing it admit enough conserved quantities. By Noether's theorem, conserved quantities correspond to symmetries, and this fact suggests that a fruitful way to analyze an integrable system is  to use methods from geometry and representation theory, rather than analytical or numerical ones.

We will focus on KdV equation (and some generalisations of it), which is an infinite-dimensional integrable systems, hence requires infinite-dimensional lie algebras. We will see that the lie algebra describing KdV has two important irreducible representations, and a surprising isomorphism between them (boson-fermion correspondence) allows us to simplify the KdV equation.

If time permits, we can briefly discuss connections with related topics such as geometry, algebraic curves, (pseudo) differential operators, and others.