# Mathematical Sciences Research Institute

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# Seminar

Special Seminar: On polynomially integrable billiards on surfaces of constant curvature September 11, 2018 (04:00 PM PDT - 05:00 PM PDT)
Parent Program: Hamiltonian systems, from topology to applications through analysis MSRI: Simons Auditorium
Speaker(s) Alexey Glutsyuk (École Normale Supérieure de Lyon)
Description No Description
Video
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.
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.