01 · The idea
In everyday language
Sphere packing asks how densely equal, nonoverlapping balls can fill space. In eight dimensions, the E8 lattice provides an exceptionally symmetric candidate. Viazovska did not compare it with every packing one by one. She constructed a special auxiliary function whose signs and Fourier-transform properties impose a universal upper bound on packing density. Modular forms supplied exactly the function needed to make the bound sharp. Because E8 attains that bound, no other packing can be denser. With collaborators, she soon adapted the approach to the Leech lattice in twenty-four dimensions, revealing a remarkable meeting of geometry, number theory and Fourier analysis.
Why it matters
The proof established exact optimal densities in exceptional dimensions and transformed the use of analytic bounds in packing problems. Related interpolation methods now connect optimal configurations with broad questions about energy minimization and Fourier analysis.
