#
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.