Collisions of hard balls and Alexandrov spaces of curvature bounded above

Consider a system of several hard balls moving freely and colliding elastically in Euclidean space. Does there exist an upper bound on the number of collisions which depends only on the number of balls and their masses? One may ask the same question for a system of balls in a box, but now it makes sense to ask for such an estimation in a fixed time interval. This problem (which had been posed by Ya. Sinai in 1978) has been solved by means of geometric considerations in a “generalized unfolding space”, which happens to be a non-positively curved length space. The same approach helped to solve several other problems concerning dynamical properties of semi-dispersing billiard systems and led to a new circle of problems in the border between geometry, algebra and combinatorics. The talk is based on a joint work with S. Ferleger, A. Kononenko and partially B. Kleiner.