Birkhoff Conjecture for convex planar billiards and spectral rigidity>
Vadim Kaloshin

The classical Birkhoff conjecture states that the only integrable
convex planar domains are circles and ellipses. In a joint work with A.
Avila and J. De Simoi we show that this conjecture is true for
perturbations of ellipses of small eccentricity. It turns out that the
method of proof shed some light on deformational spectral rigidity of
symmetric convex domains.