Sphere Packing Theorems in Dimensions 8 and 24

In 2016, Maryna Viazovska proved that the E8 lattice gives the densest possible sphere packing in eight-dimensional space, whether the packing is required to be regular or not, using a method built from the Laplace transform of a carefully chosen modular function together with Fourier analysis and the Poisson summation formula. Shortly afterward, Viazovska and collaborators extended the same technique to twenty-four dimensions, proving that the Leech lattice gives the densest possible sphere packing there as well. Both proofs settled century-old open questions and were praised for their surprising simplicity, with the mathematician Peter Sarnak calling the eight-dimensional proof stunningly simple.