Sphere packing in dimensions 8 and 24

Solved, 2016Geometry

Statement

In $8$ -dimensional Euclidean space $\mathbb{R}^8$ , no packing of congruent non-overlapping spheres has density greater than that of the $E_8$ lattice packing, $\Delta_8 = \frac{\pi^4}{384} \approx 0.25367$ ; in $24$ -dimensional Euclidean space $\mathbb{R}^{24}$ , no sphere packing has density greater than that of the Leech lattice $\Lambda_{24}$ packing, $\Delta_{24} = \frac{\pi^{12}}{12!} \approx 0.00192957$ .

On March 14, 2016, Maryna Viazovska posted a stunning 23-page preprint (`arXiv:1603.04246`, published in the Annals of Mathematics in 2017) solving the sphere packing problem in dimension $8$ by constructing the long-sought Cohn–Elkies 'magic function' via modular and quasimodular forms. Just one week later (`arXiv:1603.06518`), Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska adapted her modular-form construction to solve the $24$ -dimensional problem for the Leech lattice $\Lambda_{24}$ . Viazovska was awarded the 2022 Fields Medal for this work.

In 2019 (`arXiv:1905.03413`, published in the Annals of Mathematics in 2022), Cohn, Kumar, Miller, Radchenko, and Viazovska proved that $E_8$ and the Leech lattice $\Lambda_{24}$ are universally optimal: they minimize potential energy among all point configurations of fixed density for every completely monotonic potential function of squared distance (such as inverse power laws and Gaussians), and their proof established an exact Fourier interpolation formula reconstructing any radial Schwartz function from its values and derivatives at radii $\sqrt{2k}$ .

References

Maryna S. Viazovska (2017). The sphere packing problem in dimension 8 · DOI:10.4007/annals.2017.185.3.7 · arXiv:1603.04246
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska (2017). The sphere packing problem in dimension 24 · DOI:10.4007/annals.2017.185.3.8 · arXiv:1603.06518

History of the problem

1900David Hilbert includes the problem of the densest packing of spheres and congruent shapes in Euclidean space as part of Hilbert's 18th problem.David Hilbert
1934Hans Frederick Blichfeldt proves that the $E_8$ lattice achieves the maximum density among all lattice packings in $\mathbb{R}^8$ .
1965John Leech constructs the 24-dimensional Leech lattice $\Lambda_{24}$ from the binary Golay code, where each sphere touches $196,560$ neighbors.
2003Henry Cohn and Noam Elkies develop Fourier-analytic linear programming bounds for sphere packing and verify numerically to high precision that auxiliary 'magic functions' should yield the exact bounds in dimensions $8$ and $24$ .
2016Maryna Viazovska constructs the exact radial eigenfunctions in $\mathbb{R}^8$ using Laplace transforms of modular forms, and within a week joins Cohn, Kumar, Miller, and Radchenko to solve the $\mathbb{R}^{24}$ case.