A Conceptual Breakthrough in Sphere Packing

On March 14, 2016, the world of mathematics received an extraordinary Pi Day surprise when Maryna Viazovska posted to the arXiv a solution of the sphere packing problem in eight dimensions [15]. Her proof shows that the ????8 root lattice is the densest sphere packing in eight dimensions, via a beautiful and conceptually simple argument. Sphere packing is notorious for complicated proofs of intuitively obvious facts, as well as hopelessly difficult unsolved problems, so it’s wonderful to see a relatively simple proof of a deep theorem in sphere packing. No proof of optimality had been known for any dimension above three, and Viazovska’s paper does not even address four through seven dimensions. Instead, it relies on remarkable properties of the ????8 lattice. Her proof is thus a notable contribution to the story of ????8, and more generally the story of exceptional structures in mathematics.