Local strong Birkhoff conjecture of almost every ellipse

The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. We will consider a stronger notion of integrability, namely, integrability close to the boundary, and explain the proof of a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse.