Отправлено: 21.11.09 10:57. Заголовок: Advances in random matrix theory, zeta functions, and sphere packing

Advances in random matrix theory, zeta functions, and sphere packing
T. C. Hales, P. Sarnak, and M. C. Pugh*
+ Author Affiliations

Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395
Abstract
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.

Footnotes
↵ * To whom reprint requests should be addressed. E-mail: mpugh@math.upenn.edu.
This paper is a summary of a session presented at the 11th annual symposium on Frontiers of Science, held November 11–13, 1999, at the Arnold and Mabel Beckman Center of the National Academies of Sciences and Engineering in Irvine, CA.
Article published online before print: Proc. Natl. Acad. Sci. USA, 10.1073/pnas.220396097.
Article and publication date are at www.pnas.org/cgi/doi/10.1073/pnas.220396097
Copyright © 2000, The National Academy of Sciences Advances in random matrix theory, zeta functions, and sphere packing
T. C. Hales, P. Sarnak, and M. C. Pugh*
+ Author Affiliations

Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395
Abstract
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.

Footnotes
↵ * To whom reprint requests should be addressed. E-mail: mpugh@math.upenn.edu.
This paper is a summary of a session presented at the 11th annual symposium on Frontiers of Science, held November 11–13, 1999, at the Arnold and Mabel Beckman Center of the National Academies of Sciences and Engineering in Irvine, CA.
Article published online before print: Proc. Natl. Acad. Sci. USA, 10.1073/pnas.220396097.
Article and publication date are at www.pnas.org/cgi/doi/10.1073/pnas.220396097
Copyright © 2000, The National Academy of Sciences T. C. Hales, P. Sarnak, and M. C. Pugh*
+ Author Affiliations

Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395
Abstract
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.

This paper is a summary of a session presented at the 11th annual symposium on Frontiers of Science, held November 11–13, 1999, at the Arnold and Mabel Beckman Center of the National Academies of Sciences and Engineering in Irvine, CA.
Article published online before print: Proc. Natl. Acad. Sci. USA, 10.1073/pnas.220396097.
Article and publication date are at www.pnas.org/cgi/doi/10.1073/pnas.220396097
Copyright © 2000, The National Academy of Sciences