The sphere packing problem in dimension N asks for an arrangement of non-overlapping spheres of equal radius which occupies the largest possible proportion of the corresponding Euclidean space. This problem has a long and fascinating history. In 1611 Johannes Kepler conjectured that the best possible packing in dimension 3 is obtained by a face centered cubic and hexagonal arrangements of spheres. A proof of this legendary conjecture has finally been published in 2005 by Thomas Hales. The analogous problem in dimension 2 has been solved by Laszlo Fejes Toth in 1940, and this really is the extent of our current knowledge. If, however, one only considers lattice packings, i.e. arrangements of spheres with centers at points of a lattice, more is known. In this talk, I will introduce the sphere packing problem, briefly surveying its history and known results. I will then restrict to lattice packings, describing a connection between the problem of finding an optimal lattice packing in a given dimension and minimization problem for Epstein zeta function on the space of unimodular lattices in this dimension. I will also introduce some important classes of lattices which are expected to solve these related problems, and will demostrate these concepts on the well understood 2-dimensional case. If time allows, I will conclude with a certain approximation lemma which shows how good of a packing density one can expect from lattices with rational bases coming from one of these important classes of lattices in dimension 2. Lecture given at the Claremont Colloquium, November 2007.
© 2007 Lenny Fukshansky
Fukshansky, Lenny. "Sphere packing, lattices, and Epstein zeta function." Claremont Colloquium, Claremont, California, November 2007.