The problem
Determine the maximum packing density of congruent spheres in \mathbb{R}⁸ (answer: the E₈ lattice, density π⁴/2⁸) and \mathbb{R}^{24} (the Leech lattice, π^{12}/12!), and prove optimality.
Optimal Sphere Packing in Dimensions 8 and 24
historicReferences
History & significance
Kepler settled the question's three-dimensional ancestor (see our historic shelf); Cohn and Elkies (2003) developed linear-programming bounds that came tantalisingly close in dimensions 8 and 24 — missing optimality by a whisker for thirteen years.
The resolution (human proof)
Proven March-April 2016 by Maryna Viazovska (EPFL, then Humboldt/Berlin): in a single week she constructed the interpolating 'magic function' — a modular form whose Fourier transform matches it at the right nodes — fixing the LP bound exactly for E₈; within days she and Cohn, Kumar, Miller and Radchenko did the same for the Leech lattice. Published in Annals the same year; the 2022 Fields Medal followed. Mathematics rarely moves this fast, and rarely this beautifully.