Going back to a booklet of Kepler in 1611, the problem of packing
identical spheres as densely as possible in Euclidean space has a
history of almost 400 years. Nevertheless, the problem in arbitrary
dimensions is far from being solved. Attempts to solve it have led
to the discovery of a wealth of mathematics, and the theory of sphere
packings today is an active field of research providing many
challenging open problems. It has connections to various mathematical
areas, including number theory, coding theory, group theory, and
harmonic analysis. The aim of this seminar is to introduce the
participants to some of these mathematical flowers and to leave them
with many loose ends for further study.
Topics will include: Exceptional structures, lattices, codes, groups,
designs, two-point homogeneous spaces, linear programming bounds,
spherical codes, hyperbolic sphere packings, positive definite quadratic
forms, reduction theories.