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The Birkhoff conjecture says that the boundary of a strictly convexintegrable billiard table is necessarily an ellipse. In this article, weconsider a stronger notion of integrability, namely, integrability close to theboundary, and prove a local version of this conjecture: a small perturbation ofalmost every ellipse that preserves integrability near the boundary, is itselfan ellipse. We apply this result to study local spectral rigidity of ellipsesusing the connection between the wave trace of the Laplacian and the dynamicsnear the boundary and establish rigidity for almost all of them.

Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse