# Almost impossible E₈ and Leech lattices

### Maryna Viazovska

École Polytechnique Fédérale de Lausanne, Switzerland

## Abstract

We start this short note by introducing two remarkable mathematical
objects: the $E_{8}$ root lattice $Lambda8$ in 8-dimensional Euclidean
space and the Leech lattice $Lambda24$ 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 $Lambda8$ 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 [7] and the monster
vertex algebra [1].

Another reason is that $Lambda8$ and $Lambda24$ are solutions to a number
of optimization problems. The $E_{8}$ and Leech lattice provide optimal
sphere packings in their respective dimensions [5, 23]. Also
both lattices are universally optimal, which means that among all
point configurations of the same density, the $Lambda8$ and $Lambda24$ have the
smallest possible Gaussian energy [6].

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 much smaller than 8 and 24, is still
wide open. Finally, this last property seems to be inherited by other
geometric objects obtained from $Lambda8$ and $Lambda24$, such as Hamming
code, Golay code and the sets of shortest vectors of both lattices.

## Cite this article

Maryna Viazovska, Almost impossible $E_{8}$ and Leech lattices. Eur. Math. Soc. Mag. 121 (2021), pp. 4–8

DOI 10.4171/MAG/47