Sphere Packings, Kissing Numbers, and Lattices
Stub. The sphere-packing problem and kissing numbers as the geometric counterpart of the coding bounds of the previous note; the basic theory of lattices (definitions, examples, a catalogue of named lattices) as the natural structure for constructing dense packings; the E8 and Leech lattices as the two most celebrated examples; and Viazovska's proof that E8 (and later, with coauthors, the Leech lattice) achieves the provably optimal sphere-packing density in dimensions 8 and 24.
Corresponds to Lecture topic 4 and the lattice-theory reading materials (lecture notes, examples of lattices, the lattice catalogue, Viazovska’s paper). A sphere packing arranges non-overlapping spheres to maximize the fraction of space covered; the kissing number of a packing is how many spheres can simultaneously touch one sphere — both questions dual, in spirit, to the coding-theory bounds of linear-codes-bounds-and-macwilliams-identities (a code with large minimum distance is exactly a packing of Hamming balls). Lattices — discrete subgroups of closed under addition — are the natural way to generate highly symmetric, often optimal, packings, and the course surveys a catalogue of named examples. The E8 lattice (in 8 dimensions) and the Leech lattice (in 24 dimensions) are the two standout cases, both linked to exceptional coding-theoretic and group-theoretic structures; Viazovska’s breakthrough (2016 Fields Medal–winning work) proved E8 achieves the provably optimal packing density in dimension 8, via modular forms, with the 24-dimensional Leech case resolved shortly after by Viazovska and coauthors using the same technique.
Outline (TODO — flesh out each)
- Sphere packings and kissing numbers
- Lattice theory — definitions and basic examples
- A catalogue of lattices
- The E8 lattice
- The Leech lattice
- Viazovska’s optimality proof (E8, and the Leech lattice)