 
Summary: RESEARCH BLOG 4/6/04
SPHERE PACKING
There is an article today in the New York Times on Hales' proof of
the Kepler conjecture. The theory part of Hales' proof will be published
in the Annals of Math, whereas the computational aspects will be pub
lished in Discrete and Computational Geometry . Apparently, Annals
will not publish computer assisted proofs any more. This seems like a
reasonable compromise, especially if other journals are willing to pub
lish the computational parts of the proof, and the more mathematically
influential part of the proof is most likely the theoretical part. It still
would be quite interesting if someone could come up with a purely the
oretical argument, maybe along the lines of Cohn and Elkies' approach
to sphere packing estimates. I seem to recall from a talk of Cohn,
that this comes down to showing the existence of a certain function
which has zeroes (with correct multiplicities) at the distances between
vertices of the facecenteredcubic lattice (the optimal packing in R3
).
For certain packing problems in hyperbolic space, optimal bounds are
known. Optimal packings of spheres on hyperbolic surfaces are known,
and the regular packing by horospheres in H3
