Seminar: A new lower bound for sphere packing

Speaker: Austin Eide

Title: A new lower bound for sphere packing

Abstract: I'll present a very recent result of Campos, Jenssen, Michelen, and Sahasrabudhe on sphere packing density. The sphere packing problem asks: what is the maximum proportion of $\mathbb{R}^{d}$ which can be covered by a disjoint collection of unit spheres? The exact value is known only for only a few (small) dimensions. One can also consider the asymptotic density as $d \to \infty$, which is the setting I'll discuss. Specifically, the result establishes that there exists a packing of density $(1-o(1))\frac{d \log d}{2^{d+1}}$. This is the first order-of-magnitude improvement on the problem since 1947. The proof proceeds via a reduction to a graph problem and makes abundant use of probabilistic tools (like the first moment method and Rödl Nibble, to name just a couple).