Worked solution: Viazovska's modular form solution to sphere packing in dimensions 8 and 24

Frequently asked questions

Is this proof computer-assisted like the Kepler conjecture proof?

Only lightly: computers were used to search numerically for good candidate auxiliary functions and to verify some of the modular-form identities and coefficient computations along the way, but the final published proof is a short, fully human-checkable analytic argument (about 23 pages for $d=8$ ), unlike Flyspeck's reliance on a proof assistant to verify thousands of pages of case analysis.

Source

Maryna Viazovska, with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko, 2016

The sphere packing problem in dimension 8

arXiv:1603.04246v1

Condensed summaryComputer-assisted

Step 9 of 9: A bridge between number theory and geometry, recognised with the Fields Medal

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In plain words

Before 2016, nobody had connected the dots between the abstract, century-old theory of modular forms and the very concrete, physical question of stacking balls; Viazovska's proof revealed that this connection was there all along, waiting for the right function to be built. The mathematical community's reaction was closer to delight than disbelief — one leading number theorist called the argument 'stunningly simple, as all great things are' — and it earned her the 2022 Fields Medal, mathematics' highest honour, making her only the second woman to receive it.

\text{Fields Medal, ICM 2022}

Detailed analysis

Maryna Viazovska received the Fields Medal at the International Congress of Mathematicians in July 2022, principally for the $d=8$ sphere packing proof together with the follow-up $d=24$ work; the citation highlighted the surprising bridge her method built between the analytic theory of modular forms and extremal problems in Euclidean geometry, a bridge that had not been anticipated even by specialists in either separate field. Beyond resolving two specific dimensions, the construction opened a new research direction, universal optimality: Cohn, Kumar, Miller, Radchenko, and Viazovska later proved (2019–2022) that $E_8$ and the Leech lattice are not merely optimal for the packing problem but simultaneously optimal for an entire family of related 'energy minimisation' problems, using the same interpolation-of-modular-forms toolkit built here.

The uniqueness statement in this proof (Steps 6–7) also carries physical weight: it certifies that the crystal-like structures long used in coding theory and in the study of exceptional Lie groups are not merely good, but provably the best possible way to pack space in these two special dimensions.