# Sphere Packings, Lattices, and Infinite Dimensional Algebra

August 16 to August 20, 2004

at the

American Institute of Mathematics, Palo Alto, California

organized by

Lisa Carbone, Noam Elkies, and Jim Lepowsky

## Original Announcement

This workshop will focus on sphere packings and lattice packings, with particular attention to dimensions 8 and 24 and the connection with automorphic forms and Moonshine.

The problem of packing identical spheres as densely as possible in Euclidean space has a 400 year history, having been initiated by Johannes Kepler in 1611. Though the problem is unsolved in general today, attempts to solve it have led to the discovery of a wealth of mathematics.

Most of the densest known sphere packings are {\it lattice packings}. A lattice in ${\Bbb R}^{n}$ is an additive subgroup $L\subset {\Bbb R}^{n}$ which is generated by some basis for the real vector space ${\Bbb R}^{n}$. A {\it lattice packing} in ${\Bbb R}^{n}$ is then a sphere packing where the centers of spheres are placed at the points of a lattice $L\subset {\Bbb R}^{n}$, the radius of each sphere being half the length of the shortest non-zero vectors in $L$.

The study of sphere packings and lattice packings fits naturally into a broader class of packing problems, including error-correcting codes in information transmission. In fact the [24,12,8] extended binary code' of M. J. E. Golay led to the discovery by John Leech of one of the most remarkable lattices, the {\it Leech lattice} in ${\Bbb R}^{24}$.

The Leech lattice appears in several places in Moonshine' which is a term first coined by J. Conway and S. Norton in 1979 to describe the mysterious connections between finite sporadic simple groups and modular functions. The solution of the Monstrous moonshine conjectures by Frenkel, Lepowsky, Meurman and Borcherds expanded the Moonshine horizon to include interrelationships between lattices and hyperbolic reflection groups, generalized Kac-Moody Lie algebras, vertex (operator) algebras, automorphic forms and conformal field theory.

Finding optimal sphere and lattice packings is an active area of current research. Many of the intricate connections between lattices, packings groups, automorphic forms and problems in Moonshine are not fully understood and give rise to substantial open problems. The proposed workshop will bring together specialists from these diverse but related areas, providing a unique opportunity for them to interact.

Primarily, this workshop will consist of:

• An investigation of the techniques of Cohn and Elkies, and of Cohn and Kumar, which are conjectured to give rise to a proof that the $E_8$ root lattice and the Leech lattice give the densest sphere packings in ${\Bbb R}^{8}$ and ${\Bbb R}^{24}$ respectively.
• An introduction to the recent work of Frenkel, Lepowsky, and Meurman, and of Huang, which indicates deep connections between lattices, codes and conformal field theory.
In addition, the workshop will explore open problems relating to the Moonshine module, such as relationships between the Monster vertex algebra, its integral quadratic forms, generalized Kac-Moody algebras and hyperbolic reflection groups.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.