# Examples of exponentially many collisions in a hard ball system.****Cancelled due to campus closure****

## Meeting Details

Abstract: 20 years ago the topic of my talk at the ICM was a solution of a conjecture which goes back to Boltzmann and Ya. Sinai. It states that the number of collisions in a system of $n$ identical balls colliding elastically in empty space is uniformly bounded for all initial positions and velocities of the balls. The answer is affirmative and the proven upper bound is (poly) exponential in $n$. Little was known about many collisions can actually occur. On the line, $n(n-1)/2$ is a realizable maximum. The only non-trivial (and counter-intuitive) example in higher dimensions I am aware of is an observation by Thurston and Sandri who gave an example of 4 collisions between 3 balls in $R^2$. Recently, Sergei Ivanov and me proved that there are examples with exponentially many collisions between $n$ identical balls in $R^3$, even though the exponents in the lower and upper bounds do not match. The example is not explicit, we just prove its existence. A few related problems around entropy and other dynamical invariants will be discussed.