Joan Verdera - The regularity of the boundary of a vortex patch and commutators of singular integrals

I will introduce briefly the vorticity form of the Euler equation in the plane and show how singular integrals appear immediately. Then I will introduce vortex patches and the problem of regularity of the boundary. I will describe some elements of a short proof I have found recently, which also solves the regularity problem for other transport equations. Commutators of singular integrals play a key role, as it is well-known.