Recall that the classical sphere packing problem is the following: What is the maximum proportion of \(\mathbb{R}^d\) that can be covered by a collection of non-overlapping spheres of volume \(1\)?
The exact value of this maximum is only known in dimensions 1, 2, 3, 8 and 24; and the cases 8 and 24 were only resolved very recently (resulting in the Fields Medal for Maryna Viazovska in 2022). A new paper from December on the arXiv (arXiv:2312.10026) now provides an improvement of the known lower bound for the sphere packing problem: It is at least \[(1-o(1))\frac{d \log(d)}{2^{d+1}}.\] This is the first serious improvement of the lower bound since 1947.