As suggested by Gaurish4Math, in todays post I’ll be discussing the recent breakthrough in the sphere packing problem.
What is the Sphere Packing Problem?
A sphere packing is an arrangement of spheres that do not overlap within a containing space. The problem is to find the best packing of spheres in dimension n. The densest packing of spheres was only known in dimensions 0, 1, 2 and 3, before this breakthrough.
The E8 lattice can be characterised as the “unique, positive-definite, even, uni-modular lattice of rank 8”. It is formed by taking all the sums of the vectors in the root system. This root system contains all roots (a1,a2,a3,a4,a5,a6,a7,a8) where all ai are integers or all ai are integers plus 1/2, the sum is an even integer, and sum of the squares is 2.
It can be expressed as follows:
Maryna Viazovska published a paper proving that if you centre spheres at the points of the E8 lattice, you get the densest packing of spheres in 8 dimensions. Following this, Viazovska joined Cohn, Kumar, Miller and Radchenko to prove that the Leech lattice gives the densest packing of spheres in 24 dimensions.
The starting point for Viazovska’s breakthrough was a method developed by Cohn and Elkies in 2001 that improved the upper bounds to the density in dimensions 4-31, giving extremely good results for dimensions 8 and 24; they showed that the densest packing in 8 dimensions could be no more than 1.000001 times as dense as that coming from the E8 lattice (Viazovska then reduced this to 1). They showed that, provided a function satisfies a number of conditions, it will give the upper bound on the density. The problem was to find this function.
However, finding the right function proves enigmatic and to find it Maryna had find Fourier transforms of some modular forms and prove certain estimates about them.
Congratulations to Viazovska, Cohn, Kumar, Miller and Radchenko for this fantastic achievement!
Apologies that this post is quite short – the mathematics involved is quite complex and hard to resume in a post. Hope you enjoyed it! Mx