3/22/2004

Abhinav Kumar
Harvard University

I will give an overview of the proof that the Leech lattice gives the densest lattice packing in 24 dimensions (this is joint work with Henry Cohn). Furthermore, we show that no sphere packing in that dimension can exceed the Leech lattice's density by a factor of more than $1+1.65 \cdot 10^{-30}$. Our methods also give a new proof of the result of Blichfeldt and Vet\v{c}inkin that the $E_8$ lattice gives the densest lattice packing in 8 dimensions. The proof involves various combinatorial properties of the Leech and $E_8$ lattices, as well as computer verification of the properties of certain polynomials.