Understanding sphere packing in higher dimensions In a recent publication by the Ukrainian mathematician Maryna Viazovska the Kepler problem for dimension $8$ and $24$, namely the densest packing of spheres, was solved.
Admittedly it is very difficult to grasp the work for someone not involved in the field, but it would be incredibly valuable if, briefly, the methodology adopted, and the main breakthrough could be explained at a conceptual level. Such that one can have a rough idea how the solution came to be.
This is merely an attempt to get more input in order to understand some of the main ideas involved, as this is very exciting news given that the problem of densest packing of spheres in different dimensions has a long history.
 A: There are two things you need to understand.  The first is how to prove sphere packing bounds via harmonic analysis ("linear programming bounds").  My lecture notes from PCMI 2014 give an exposition of this theory, which covers the period up to, but not including, Viazovska's paper on eight dimensions.  This reduces the problem to finding radial functions in eight and twenty-four dimensions with certain roots and sign conditions, and the lecture notes discuss why everybody believed such functions exist.  The functions themselves are surprisingly subtle; see this paper with Steve Miller for numerical experiments and conjectures.
Then Viazovska's breakthrough lies in how to construct the right functions.  It's convenient to split them up into eigenfunctions of the Fourier transform, which have to have roots at certain locations.  One way to look at it is that she brute forces those roots by including a sine squared factor that produces the roots, and then she multiplies it by something that's essentially the Laplace transform of a (quasi)modular form.  To make this work, one has to answer three questions:


*

*What sort of modular forms play nice with the sine squared factor and produce radial eigenfunctions of the Fourier transform?

*Do there exist such modular forms that actually give the desired functions?

*How can we prove the inequalities these functions are supposed to satisfy?
Viazovska's methodology gives nice answers:


*

*This amounts to identifying the right level, weight, and depth (for quasimodular forms).

*Yes.  One can reverse engineer them based on the desired properties.

*In the best way one could hope for, namely at the level of the modular form itself (rather than having to worry about something subtle happening in the Laplace transform).
For details, see the eight and twenty-four dimensional papers.  As Peter Sarnak points out in Erica Klarreich's Quanta article, the actual construction is surprisingly simple.  If you know about linear programming bounds and basic facts about modular forms, the proof itself is very down to earth.
A: The image below depicts a hypersphere inscribed in a hypercube. The number of dimensions increases from left to right (2,3,4 and 5 dim).
As you can see, the higher the dimension, the higher is the number of "hypercorners" that are left uncovered by a single hypersphere sitting at the center, of diameter no greater than the side. These increase as the powers of two: 4, 8, 16 and so on.

It is easy to prove, by using the Gamma function, that the relative volume of the sphere to that of the hypercube (aka "density") decreases as the number of dimensions rises.
In dimension 8, you would have 2^8=256 hypercorners around the 8-dimensional sphere.
One first trivial attempt to pack non-overlapping spheres in a fractal manner is by means of an Apollonian (Leibniz) gasket - thus observe the second picture attached. As the number of dimensions increases, so does the amount of spheres that need to be fitted in this manner, in an exponential manner.

Of course, there are more efficient ways to pack spheres than this gasket. The simplest approximation is to think of unit hyperspheres: that is, of those having equal -unitary- volume. Or having equal surface area, in some other interpretations, as it is easy to translate from one to another (by using the pi formula).
Some more complex packing gaskets or schemes involve spheres of varying sizes, other than the classic Apollonian fractal, and their computation may involve optimization methods such as Linear Programming or meta/heuristics.
Viazovska's article does not deal with these, but only with unit balls, and benchmarks any packing method with the trivial classic lattice packing in 8 dimensions. Its conclusion is that there is none more efficient in density measures than the (E8) lattice.
A: Just for the sake of completeness, I'd like to add the links to the published versions and the reviews:
Viazovska, Maryna S., The sphere packing problem in dimension 8, Ann. Math. (2) 185, No. 3, 991-1015 (2017). Zbl 1373.52025.
Cohn, Henry; Kumar, Abhinav; Miller, Stephen D.; Radchenko, Danylo; Viazovska, Maryna, The sphere packing problem in dimension (24), Ann. Math. (2) 185, No. 3, 1017-1033 (2017). Zbl 1370.52037.
