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Description: 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 |
