Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska: Universal optimality of the E8 and Leech lattices and interpolation formulas Henry Cohn

A follow up paper on the tight bounds for sphere packings in eight and 24 dimensions. (Thanks, again, Steve, for letting me know.)

For the 2016 breakthroughs see this post, this post of John Baez, this article by Erica Klarreich on Quanta Magazine, and a Notices AMS article by  Henry Cohn   A conceptual breakthrough in sphere packing. See also, Henry Cohn’s 2010 paper Order and disorder in energy minimization, and Maryna Viazovska’s ICM 2018 videotaped lecture.

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska: Universal optimality of the E8 and Leech lattices and interpolation formulas

Abstract: We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions 8 and 24, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions.

The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\hat f$ at the radii √2π for integers n ≥ 1 in $R^8$ and n ≥ 2 in $R^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska’s work on sphere packing and placing it in the context of a more conceptual theory.

1. Gil Kalai says: