Emerging Applications of Number Theory

Dennis A. Hejhal, Joel Friedman, Martin C. Gutzwiller, Andrew M. Odlyzko

688 pages, parution le 01/01/1999

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Résumé

Most people tend to view number theory as the very paradigm of pure mathematics. With the advent of computers, however, number theory has been finding an increasing number of applications in practical settings, such as in cryptography, random number generation, coding theory, and even concert hall acoustics. Yet other applications are still emerging - providing number theorists with some major new areas of opportunity.

The 1996 IMA summer program on Emerging Applications of Number Theory was aimed at stimulating further work with some of these newest (and most attractive) applications. Concentration was on number theory's recent links with: (a) wave phenomena in quantum mechanics (more specifically, quantum chaos); and (b) graph theory (especially expander graphs and related spectral theory).

This volume contains the contributed papers from that meeting and will be of interest to anyone intrigued by novel applications of modern number-theoretical techniques.

Contents

Foreword
Preface

Trace formula for quantum integrable systems, lattice-point problem, and small divisors 1
Theta-lifts of Maass waveforms 39
The transfer operator approach to Selberg's zeta function and modular and Maass wave forms for PSL (2,Z) 73
Chaos and deviation from uniform distribution: eigenfunction computation; applied modular arithmetic 143
Logarithmic Sobolev techniques for random walks on graphs 175
Eigenvalue statistics in quantum ideal gases 187
Multifractal spectrum and Laplace spectrum 201
Number theory and atomic densities 205
Explicit formulas and oscillations 219
Energy fluctuation analysis in integrable billiards in hyperbolic geometry 269
On eigenfunctions of the Laplacian for Hecke triangle groups 291
Eigenvalue spacings for regular graphs 317
Classical limits of eigenfunctions for some completely integrable systems 329
Does a quantum particle know the time? 355
Level spacings for Cayley graphs 373
Eigenvalues of Ramanujan graphs 387
Theta sums, Eisenstein series, and the semiclassical dynamics of a precessing spin 405
Random walks on generalized Euclidean graphs 451
Two proofs of Ihara's theorem 469
Playing billiards with microwaves - quantum manifestations of classical chaos 479
Characters of the symmetric groups: formulas, estimates and applications 525
Number theory and formal languages 547
Expander graphs and amenable quotients 571
Ramanujan hypergraphs and Ramanujan geometries 583
Constructing error-correcting codes from expander graphs 591
Multipath zeta functions of graphs 601
Eigenvalues of the Laplacian for Bianchi groups 617
A survey of discrete trace formulas 643

List of Participants 683
Program Schedule 685
Afterword

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Dennis A. Hejhal, Joel Friedman, Martin C. Gutzwiller, Andrew M. Odlyzko
Parution	01/01/1999
Nb. de pages	688
Format	16 x 24
Couverture	Relié
Poids	1089g
Intérieur	Noir et Blanc
EAN13	9780387988245
ISBN13	978-0-387-98824-5