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

The sphere packing constant $\Delta_d$ is the supremum, over all packings of non-overlapping unit balls in $\mathbb{R}^d$, of the fraction of space covered. It was known exactly only for $d=1$ (trivially), $d=2$ (the hexagonal packing, Thue and Fejes Tóth), and $d=3$ (the face-centred cubic packing, Hales's 1998 proof of the Kepler conjecture — one of the most complex proofs ever completed). In every higher dimension only bounds existed.

The suspects in dimensions 8 and 24 were long identified: the $E_8$ root lattice gives density $\pi^4/384\approx0.2537$ in $\mathbb{R}^8$, and the Leech lattice $\Lambda_{24}$ gives $\pi^{12}/12!\approx0.00193$ in $\mathbb{R}^{24}$. Cohn and Elkies's 2003 linear programming method produced numerical upper bounds within a factor of about $1.000001$ of these densities — strong evidence of optimality, but a bound is not a theorem.

In 2016 Viazovska proved $\Delta_8=\pi^4/384$ by constructing the 'magic function' whose existence Cohn and Elkies had conjectured (Steps 1–5); one week later she, Cohn, Kumar, Miller, and Radchenko adapted it to prove $\Delta_{24}=\pi^{12}/12!$ (Step 6). Dimensions 1, 2, 3, 8, 24 remain the only solved cases — a strikingly sparse list.