The classical sphere packing problem asks: what is the densest possible arrangement of identical, non-overlapping spheres in $\mathbb{R}^d$?
I will discuss a recent proof that there exists a sphere packing with density at least
\[
(1-o(1))\frac{d \log d}{2^{d+1}}.
\]
This improves upon previous bounds by a factor of order $\log d$ and is the first improvement by more than a constant to Rogers' bound from 1947.
This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.