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

## Department of Mathematics Colloquium

## Meeting Details

For more information about this meeting, contact Kristin Berrigan, Stephanie Geyer, Wen-Ching W. Li.

**Speaker:** Dima Burago, Penn State University

**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.

## Room Reservation Information

**Room Number:** 114 McAllister

**Date:** 01/24/2019

**Time:** 3:30pm - 4:30pm