Almost impossible \(E_8\) and Leech lattices
type: Item
instance of: scholarly article
title: Almost impossible \(E_8\) and Leech lattices
zbMATH Open document ID: 1479.11117
DOI: 10.4171/MAG-47
published in: European Mathematical Society Magazine
publication date: 2022-02-17T00:00:00Z
review text: Summary: We start this short note by introducing two remarkable mathematical objects: the \(E_8\) root lattice \(\Lambda_8\) in 8-dimensional Euclidean space and the Leech lattice \(\Lambda_{24}\) in 24-dimensional space. These two lattices stand out among their lattice sisters for several reasons. The first reason is that these both lattices are related to other unique and exceptional mathematical objects. The \(E_8\) lattice is the root lattice of the semisimple exceptional Lie algebra \(E_8\). The quotient of \(\Lambda_8\) by a suitable sublattice is isomorphic to the Hamming binary code of dimension 8 and minimum distance 4, which in its turn is an optimal error-correcting binary code with these parameters. The Leech lattice is famously connected to the exceptional finite simple groups, monstrous moonshine [\textit{J. H. Conway} and \textit{S. P. Norton}, Bull. Lond. Math. Soc. 11, 308--339 (1979; Zbl 0424.20010)] and the monster vertex algebra [\textit{R. E. Borcherds}, Invent. Math. 109, No. 2, 405--444 (1992; Zbl 0799.17014)]. Another reason is that \(\Lambda_8\) and \(\Lambda_{24}\) are solutions to a number of optimization problems. The \(E_8\) and Leech lattice provide optimal sphere packings in their respective dimensions [\textit{H. Cohn} et al., Ann. Math. (2) 185, No. 3, 1017--1033 (2017; Zbl 1370.52037); \textit{M. S. Viazovska}, ibid. 185, No. 3, 991--1015 (2017; Zbl 1373.52025)]. Also both lattices are universally optimal, which means that among all point configurations of the same density, the \(\Lambda_8\) and \(\Lambda_{24}\) have the smallest possible Gaussian energy [\textit{H. Cohn} et al., ``Universal optimality of the \(E_8\) and Leech lattices and interpolation formulas'', Preprint, \url{arXiv:1902.05438}]. The third reason for our interest in these lattices is less obvious. The optimality of the \(E_8\) and Leech lattices can be proven in a rather short way, while the solutions of analogous problems in other dimensions, even dimensions muc…
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11H31
Mathematics Subject Classification ID: 11H71
zbMATH DE Number: 7475672
author: Maryna Viazovska
MaRDI profile type: Publication
cites work: Monstrous moonshine and monstrous Lie superalgebras
cites work: New upper bounds on sphere packings. I
cites work: Universally optimal distribution of points on spheres
cites work: The sphere packing problem in dimension \(24\)
cites work: Monstrous Moonshine
cites work: What are all the best sphere packings in low dimensions?
cites work: Q4219187
cites work: Q4119093
cites work: Über die dichteste Kugellagerung
cites work: A proof of the Kepler conjecture
cites work: Notes on Sphere Packings
cites work: Q3876610
cites work: New bounds on the number of unit spheres that can touch a unit sphere in n dimensions
cites work: Q4826730
cites work: A Mathematical Theory of Communication
cites work: The sphere packing problem in dimension 8
description: scientific article; zbMATH DE number 7475672