Particular attention will be paid to the utility of the “boundaries at infinity” that arise from these geometric conditions.

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2. Just 20 years ago the topic of my talk at the ICM was a solution of a problem which goes back to Boltzmann and has been formulated mathematically by Ya. Sinai. The conjecture of Boltzmann-Sinai 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 exponential in $n$. The question is how many collisions can actually occur. On the line, one sees that there can be $n(n-1)/2$ collisions, and this is the maximum. Since the line embeds in any Euclidean space, the same example works in all dimensions. The only non-trivial (and counter-intuitive) example 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. ]]>

Join Zoom Meeting (Meeting ID: 740 238 195) https://vanderbilt.zoom.us/j/740238195

If you open the Zoom app and type in the meeting ID, you should be able to join. You may want to arrive a couple of minutes early, so we are all in the virtual seminar room by the time Ben starts. The plan is to follow the talk with a virtual beer & pizza session

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