Emerging Applications of Number Theory

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.

With the advent of computers, 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. 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. 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.

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