Collisions of hard balls and geometry of non-positive curvature

Professor Dmitri Burago (Pennsylvania State University)

Consider a system of several hard balls moving in empty space and colliding elastically. It is intuitively clear that the number of collisions can not be arbitrarily big, while the number of balls, their masses and radii are fixed. Surprisingly, this problem remained open for over twenty years. One also expects the existence of an analogous bound on the number of collisions in unitary time for hard ball gas in a box. (This bound then may have a certain thermodynamical meaning.) Proofs of these statements (obtained jointly with S. Ferleger and A. Kononenko) are based on purely geometrical considerations, which become very transparent from the viewpoint of geometry of non-positively-curved spaces. The same approach also yields other dynamical information, i.e. estimations on topological entropy, and raises new problems of both geometrical, dynamical and combinatorial character.