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 the balls occupy. It was known exactly only for $d=1$ (trivial), $d=2$ (the hexagonal packing, A. Thue and L. Fejes Tóth), and $d=3$ (the face-centred cubic packing, T. Hales's 1998 proof of the Kepler conjecture). Cohn and Elkies's 2003 linear programming method produced numerical upper bounds for $\Delta_d$ that came astonishingly close to the densities of the $E_8$ lattice in $\mathbb{R}^8$ and the Leech lattice $\Lambda_{24}$ in $\mathbb{R}^{24}$ — within a factor of about $1.000001$ for $E_8$ — strongly suggesting, but not proving, that these two highly symmetric lattices were optimal.

In 2016 Maryna Viazovska proved $\Delta_8 = \pi^4/384$, the exact density of the $E_8$-lattice packing (Steps 2–5 below sketch her method); the very next week she, Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko adapted the same technique to prove $\Delta_{24} = \pi^{12}/12!$, the density of the Leech lattice (Step 6). These remain, together with dimensions $1$, $2$, and $3$, the only dimensions in which the sphere packing problem is fully solved.