01 · A life in mathematics
The person behind the work
Maryna Viazovska was born in Kyiv and became a number theorist whose work solved an exceptionally geometric problem. At her 2022 Fields Medal she held the Chair of Number Theory at EPFL in Lausanne. She proved that the E8 lattice gives the densest packing of equal spheres in eight dimensions. Soon afterward, with four collaborators, she established the analogous optimality of the Leech lattice in twenty-four dimensions. Her construction of special functions drew on modular forms and Fourier analysis. It converted a seemingly impossible comparison with every conceivable packing into a sharp bound attained by a highly symmetric arrangement.
The contribution, explained
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.