Proof — E8 lattice sphere packing
Statement
Optimal sphere packing in 8 and 24 dimensions. Proved in ZFC.
Proof sketch
Viazovska (2016) constructs an explicit radial Schwartz function on R^8 using modular forms. Its sign and Fourier-transform constraints match the Cohn-Elkies linear programming framework exactly. Poisson summation on the E8 lattice shows the upper bound equals E8 density, proving optimality in dimension 8. A related extension with Cohn, Kumar, Miller, and Radchenko established optimality in dimension 24 (Leech lattice).
Source
Fields Medal 2022. Optimal sphere packing in 8 and 24 dimensions. Proved in ZFC.
Domains
mathematics