The course is an introduction to the theory of classical and quantum
integrable systems. The classical part is concerned with constructing
solutions of (systems of) non-linear PDEs ; the quantum part is
concerned with the 'explicit' diagonalization of (family of) operators ;
in both case, their 'integrability' means that there are miracles
making these seemingly complicated problems solvable. These miracles are
closely related to the existence of many (hidden) symmetries. This
applies to a variety of models that are relevant in physics, including
examples of non-linear wave propagations, spin chains, free fermions,
many-body quantum systems, ... but also relevant in the geometry. We
will see various constructions of integrable systems from algebra and
geometry and general techniques to solve them, illustrated by important
examples such that the KdV equation, the KP hierarchy, the (classical
and quantum) Calogero-Moser system, the 6-vertex model, etc. Emphasis
will be put on explaining miracles.
The lectures are intended both for mathematicians and theoretically inclined physicists.
Prerequisite: a basic knowledge in multilinear algebra, complex analysis and differential geometry
- Course owner: Gaetan Borot