In this talk, I discuss the energy-critical half-wave maps equation (HWM). It has been known for quite some time that (HWM) is completely integrable with a Lax pair structure. However, the question about global-in-time existence of solutions has been completely open so far — even for smooth and sufficiently small initial data. I will present very recent results that prove global well-posedness for rational initial data (with no size restriction) along with a general soliton resolution result in the large-time limit. The proofs strongly exploit the Lax structure of (HWM) in combination with an explicit flow formula. This is joint work with Patrick Gérard (Paris-Saclay).[-]

In this talk I will report on some of the progress made by the author and collaborators on the topic of nonlinear diffusion equations involving long distance interactions in the form of fractional Laplacian operators. The nonlinearities are of the following types: porous medium, fast diffusion or p-Laplacian. Results cover well-posedness, regularity, free bouncadaries, asymptotics, extinction, and others. Differences with standard diffusion have been specially examined.[-]