  In 1963 Ya. G. Sinai formulated a modern version of Boltzmann's ergodic
hypothesis, what we now call the ``Boltzmann-Sinai Ergodic Hypothesis'': The
billiard system of $N$ ($N\ge 2$) hard balls of unit mass moving on the flat
torus $\mathbb{T}^\nu=\mathbb{R}^\nu/\mathbb{Z}^\nu$ ($\nu\ge 2$) is ergodic
after we make the standard reductions by fixing the values of trivial invariant
quantities. It took fifty years and the efforts of several people, including
Sinai himself, until this conjecture was finally proved. In this short survey
we provide a quick review of the closing part of this process, by showing how
Sinai's original ideas developed further between 2000 and 2013, eventually
leading the proof of the conjecture.
