Colloquium Seminar: Recent progress in completely integrable PDE

I will survey recent work on low regularity conservationlaws, equicontinuity, and optimal well-posedness for several completely integrable systems. I will describe the method of commuting flows and the increasingly sophisticated techniques that have been required in order to achieve sharp results across a spectrum of integrable models. This is based on joint work with B. Harrop-Griffiths, R. Killip, T. Laurens, and M. Ntekoume.