Advances in random matrix theory, zeta functions, and sphere packing

Over four hundred years ago, Sir Waiter Raleigh asked his mathematical assi stant to find formulas for the number of cannonballs in regularly stacked p iles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: wh y is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part o f the 282-page proof relies on long computer verifications. Random matrix t heory was developed by physicists to describe the spectra of complex nuclei . In particular, the statistical fluctuations of the eigenvalues ("the ener gy levels") follow certain universal laws based on symmetry types. We descr ibe these and then discuss the remarkable appearance of these laws for zero s of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These t opics are distinct, 50 we present them separately with their own introducto ry remarks.