|Location:||MSRI: Simons Auditorium|
Abstract: The famous Birkhoff Conjecture deals with convex bounded planar billiards with smooth boundary. Recall that a caustic of a planar billiard is a curve $\gamma$ such that each its tangent line is reflected from the boundary of the billiard to a line again tangent to $\gamma$. A billiard is Birkhoff caustic integrable, if an interior neighborhood of its boundary admits a foliation by closed caustics such that the billiard boundary is also its leaf. The Birkhoff Conjecture states that every Birkhoff caustic integrable planar billiard is an ellipse. Recently V.Kaloshin and A.Sorrentino proved its local version: every Birkhoff integrable deformation of an ellipse is an ellipse.
Birkhoff integrability is equivalent to the Liouville integrability of the billiard flow: existence of a first integral independent with the module of the speed. The algebraic version of the Birkhoff Conjecture, which was first studied by Sergey Bolotin, concerns polynomially integrable billiards, where the billiard flow admits a first integral polynomial in the speed that is non-constant on the unit level hypersurface of the module of the speed.
In this talk we present a brief survey of Birkhoff Conjecture and a complete solution of its algebraic version. We prove that each polynomially integrable planar billiard with $C^2$-smooth non-linear connected boundary is an ellipse. We classify polynomially integrable billiards with piecewise smooth boundaries on all surfaces of constant curvature: plane, sphere, hyperbolic plane. These are joint results with Mikhail Bialy and Andrey Mironov.
The talk is based on the preprint https://arxiv.org/abs/1706.04030