The classical sphere packing problem asks: what is the densest possible arrangement of identical, non-overlapping spheres in ℝ^d? Over the past century, this question has been intensely studied by mathematicians and physicists alike. In this talk I will discuss some different perspectives on this problem along with some recent progress. In particular, I will sketch a proof that there exists a sphere packing in ℝ^d with density at least
(1+o(1)) 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 factor to Rogers’ bound from 1947. This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.