$\vec{w}h\alpha\mathfrak{t}\;\; \forall\mathbb{R}\varepsilon\ldots$

sphere packing lower and upper bounds?

Ji Hoon Chun (TU Berlin)
2024/05/03, 13:00 s.t.
Before the MATH+ Friday colloquium talk by Julian Sahasrabudhe (U Cambridge)
FU Berlin, Arnimallee 3, Room SR 119
About what?

In Euclidean space, the densest sphere packings and their densities are only known in dimensions 1, 2 (Thue, Fejes Tóth), 3 (Hales), 8 (Viazovska), and 24 (Cohn et al.). However, several nontrivial lower and upper bounds for the density δ(d) of the densest packing in dimension d have been established. A simple "folklore" result states that δ(d) ≥ 1/2^d. In this talk we present the intuition and details of three other lower and upper bounds for δ(d): the Minkowski–Hlawka theorem for a lower bound, Blichfeldt's upper bound, and Rogers's upper bound. These results, among others, place δ(d) within a narrow strip of possible densities.