A Breakthrough by Maryna Viazovska Leading to the Long Awaited Solutions for the Densest Packing Problem in Dimensions 8 and 24

Maryna Viazovska

The news

Maryna Viazovska has solved the densest packing problem in dimension eight! Subsequently, Maryna Viazovska with Henry Cohn, Steve Miller, Abhinav Kumar, and Danilo Radchenko solved the densest packing problem in 24 dimensions!

Here are the links to the papers:

Maryna Viazovska, The sphere packing problem in dimension 8

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska,
The sphere packing problem in dimension 24

(I thank Steve Miller and Peter Sarnak for telling me about it.)

Additional sources:  An article by Frank Morgan in The Huffington Post; A blog post by John Baez on the n-Category Cafe; An article by Erica Klarreich on Quanta Magazine; A blog post in Mathbya girl;

Update: The February 2017 issue of the Notices of the AMS has a beautiful paper by Henry Cohn  entitled A conceptual breakthrough in sphere packing about the developments described in the post.

Some Background

Kepler, Gauss, Hales, Cohn and Kumar. A central mathematical problem is to find the densest sphere packing in . The case is known as the Kepler conjecture. Gauss solved it for lattice packings, and Thomas Hales proved it for general packing using a massive use of computations. Cohn and Kumar settled the lattice case for dimensions 8 and 24.

Conway and Sloane. The “bible” regarding sphere packing is the classic book by two major player of the theory John Conway and Neil Sloane.

Hales and Fejes Toth.  Announced in 1998 and published a few years later, Hales’ proof  relies on some early work of Laszlo Fejes Toth. Since a full verification would require developing much of  the whole project from scratch, Hales himself led a team of researchers to find a formal proof which was published in 2015.

Lie and  Leech.   Lower bounds for higher dimensions. For some dimensions, special lattices of Lie type give surprisingly dense lattice packings. The Leech lattice gives a remarkably dense packing in dimension 24.

Minkowski,…, Ball, Vance and Venkatesh,  For asymptotically large dimensions a probabilistic method by Minkowski gives the best known lower bound up to small (but exciting) improvements. It gives a packing of density $2 \cdot 2^{-n}$.  Here is a slide from a lecture by Henry Cohn on the state of the art for the asymptotic question. (And here is the link to the slides of the full lecture.)

Delsartes, Kabatiansky and Levenshtein. Upper bounds via linear programming. Delsartes’ linear programming method (that can be seen as a Fourier/spectral attack with special features,) had led to important results towards general upper bounds by Kabatiansky and Levenshtein.

Cohn and Elkies developed related spectral methods applying directly to sphere packing, which allow to improve the upper bounds in dimensions 4–31 and give strikingly good results in dimensions 8 and 24.  Cohn and Kumar used these linear programming methods to settle the densest lattice problem in dimensions 8 and 24 and to give extremely good numerical upper bounds for the non-lattice case.

This is the starting point for Viazovska’s breakthrough.

Related problems/issues  to keep in mind: The densest packing problem in other dimensions and when the dimension tends to infinity; Kissing numbers and spherical codes; Upper bounds for error correcting codes; packing in other symmetric spaces; packing covering and tiling in combinatorics and geometry.

Viazovska’s breakthrough

The little I can tell you is that for a solution one needs to identify certain functions to plug in to the spectral machine. And Maryna’s starting point was some familiar extraordinary elliptic functions and modular forms.  More details on the comment section are most welcome.  (Update:) John Baez wrote on the n-Category Cafe some elementary comments on the proofs: E8 is the best.

The 24-dimensional case

A key ingredient for the result in dimension 24 is the earlier numerical rationality conjectures by Cohn and Miller.  Those now appear in the preprint: Henry Cohn, Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24 .

Congratulations to Maryna, Abhinav,  Danilo,  Henry, and Steve!

I remember a decade ago that Steve Miller explained to me some developments, ideas, and dreams  regarding two problems. One was the sphere packing problem in dimensions 8 and 24 that he now took part in solving, and the other was the irrationality questions regarding zeta functions at odd integers (and maybe also the Euler constant.)  Time to move to the second problem, Steve 🙂

(And a trivia question: name a player in both these stories. As usual if you answer in the comment section please give a zero-knowledge answer demonstrating that you know the solution without revealing it.)