Integrable Billiards in Cones

The talk is based on a joined work with Siyao Yin.
The classical Birkhoff conjecture states that if the plane billiards inside a closed smooth convex curve is integrable then the curve is an ellipse. In higher dimensions, all known integrable billiards are inside billiard tables consisting of pieces of quadrics. We study Birkhoff billiards in convex cones in ${\mathbb R}^n$ and prove that billiards in any $C^3$-smooth convex cone are integrable. This provides the first examples of integrable billiard tables in ${\mathbb R}^n$ not related to quadrics.