What's the most space-efficient way to stack spheres in three-dimensional space? Although the answer is "obvious", a rigorous proof has eluded mathematicians for centuries, having only recently been found by Hales in 1998, an immense computerized proof-by-exhaustion which is impossible to verify by hand. The more general problem of packing (n-1)-spheres into n-dimensional Euclidean (or some other) space is still only poorly understood, in spite of many striking connections linking this problem to other areas of mathematics. In this talk, we shall explore these connections, discuss some of the methods used to prove sphere packing density bounds, and review some results concerning sphere packing densities in high dimensions.
Lecture notes for this talk can be found here.