Title: Billiard Trajectories inside Cones

Abstract: The report discusses billiard trajectories inside an $n$-dimensional cone over a strictly convex closed manifold $M$. It is shown that if $M$ is a  $C^3$-smooth manifold, then every trajectory has a finite number of reflections and, in this case, the billiard admits first integrals whose values uniquely determine all billiard trajectories. At the same time, there exists a $C^2$-smooth manifold $M$ and a billiard trajectory in the cone such that this trajectory has infinitely many reflections in finite time.