The topic of my four lectures are integrable PDEs, a class of evolution equations widely used in applications in physics and the engineering sciences, but also within mathematics such as in geometry and probability theory. I plan to discuss the method of normal forms, being one of the principal tools to analyze them, and present three recent results in the field, each of them representing a different application of this method: (1) a KAM theorem for semi-linear Hamiltonian perturbations of the defocusingNLS equation on the circle; (2) new results on the wellposedness / illposednessof the KdV, KdV2, and mKdV equations in spaces of functions of low regularityand spaces of distributions; (3) a study of the Toda lattice, describing in alimiting regime, the asymptotics when the number of particles tends to infinity. 

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