Georgia Institute of Technology College of Sciences

Dynamical billiards constitute a very natural class of Hamiltonian systems: in 1927 George Birkhoff conjectured that, among all billiards inside smooth planar convex domains, only billiards in ellipses are integrable. In this talk we will prove a version of this conjecture for convex domains that are sufficiently close to an ellipse of small eccentricity. We will also describe some remarkable relation with inverse spectral theory and spectral rigidity of planar convex domains. Our techniques can in fact be fruitfully adapted to prove spectral rigidity among generic (finitely) smooth axially symmetric domains which are sufficiently close to a circle. This gives a partial answer to a question by P. Sarnak.