G.D.Birkhoff introduced a mathematical billiard inside of a convex domain as the motion
of a massless particle with elastic reflection at the boundary. A theorem of Poncelet says
that the billiard inside an ellipse is integrable, in the sense that the neighborhood of the
boundary is foliated by smooth closed curves and each billiard orbit near the boundary
is tangent to one and only one such curve (in this particular case, a confocal ellipse).
A famous conjecture by Birkhoff claims that ellipses are the only domains with this
property. We show a local version of this conjecture - namely, that a small perturbation
of an ellipse has this property only if it is itself an ellipse. It turns out that the method
of proof gives an insight into deformational spectral rigidity of planar axis symmetric
domains and gives a partial answer to a question of P. Sarnak.
This is based on several papers with Avila, De Simoi, G.Huang, Sorrentino, Q. Wei.
The talk will be accessible to a general audience.