Advances in Random Matrix Theory, Zeta Functions, and Sphere Packing

Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: wh...

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Bibliographic Details
Published in:Proceedings of the National Academy of Sciences - PNAS 2000-11, Vol.97 (24), p.12963-12964
Main Authors: Hales, T. C., Sarnak, P., Pugh, M. C.
Format: Article
Language:English
Subjects:
Cannonballs
Eigenvalues
Family law
Fractions
From the Academy
From the Academy: Frontiers of Science Symposium
Mathematical functions
Mathematics
Matrix theory
Statistics
Symmetry
Zero
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Summary:Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues ("the energy levels") follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros 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 topics are distinct, so we present them separately with their own introductory remarks.
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.220396097