\textbf{counting collisions.}20 years ago the topic of my talk at the icm was a solution of a problem which goes back to boltzmann and 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, there can be $n(n-1)/2$ collisions, and this is he 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. \noindent \textbf{a survival guide for a feeble fish and homogenization of the g-equation.} how fish can get from a to b in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water? this is related to g-equation and has applications to its homogenization. g-equation which is believed to govern many combustion processes. based on a joint work with s. ivanov and a. novikov.