Emerging Applications of Number TheoryDennis A. Hejhal, Joel Friedman, Martin C. Gutzwiller, Andrew M. Odlyzko 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
| 1 | |
Thetalifts of Maass waveforms | 39 |
The transfer operator approach to Selbergs zeta function | 73 |
Logarithmic Sobolev techniques for random walks | 175 |
Multifractal spectrum and Laplace spectrum | 201 |
Explicit formulas and oscillations | 219 |
Energy fluctuation analysis in integrable billiards | 269 |
On eigenfunctions of the Laplacian for Hecke | 291 |
Theta sums Eisenstein series and the semiclassical | 405 |
Random walks on generalized Euclidean graphs | 451 |
Two proofs of Iharas theorem | 469 |
formulas estimates | 525 |
Number theory and formal languages | 547 |
Expander graphs and amenable quotients | 571 |
Ramanujan hypergraphs and Ramanujan geometries | 583 |
Multipath zeta functions of graphs | 601 |
Eigenvalue spacings for regular graphs | 317 |
Does a quantum particle know the time? | 355 |
Level spacings for Cayley graphs | 373 |
Eigenvalues of Ramanujan graphs | 387 |
Eigenvalues of the Laplacian for Bianchi groups | 617 |
A survey of discrete trace formulas | 643 |
List of Participants | 683 |
Other editions - View all
Emerging Applications of Number Theory Dennis A Hejhal,Joel Friedman,Martin C Gutzwiller No preview available - 1999 |
Emerging Applications of Number Theory Dennis A. Hejhal,Joel Friedman,Martin C. Gutzwiller,Andrew M. Odlyzko No preview available - 2012 |
Common terms and phrases
algebra algorithm arithmetic asymptotic automorphic billiard bound Cayley graphs chaotic classical closed geodesics coefficients computations conjecture conjugacy class constant corresponding cusp forms defined denote density discrete edges eigenfunctions eigenvalues Eisenstein series equation ergodic example expander expander graphs finite Fourier geodesic flow Hecke operators Hence horocycle hyperbolic integral k-regular Laplacian Lemma level spacing limit linear log log Math Mathematics matrix measure modular multiplication normalized number theory orbits Phys Poisson polynomial prime problem proof properties Proposition prove quadratic quantum quantum chaos quaternion Ramanujan graphs random walk regular regular graph representation Riemann Riemann zeta function Sarnak Selberg semiclassical sequence SL2 Fp spacing distribution spectral spectrum ẞk statistics subgroup symmetric Theorem theta trace formula transform vertex vertices zeros zeta function