Ebook: 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.

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Content:
Front Matter....Pages i-xiii
Trace Formula for Quantum Integrable Systems, Lattice-Point Problem, and Small Divisors....Pages 1-38
Theta-Lifts of Maass Waveforms....Pages 39-72
The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ?)....Pages 73-141
Chaos and Deviation from Uniform Distribution: Eigenfunction Computation; Applied Modular Arithmetic....Pages 143-174
Logarithmic Sobolev Techniques for Random Walks on Graphs....Pages 175-186
Eigenvalue Statistics in Quantum Ideal Gases....Pages 187-199
Multifractal Spectrum and Laplace Spectrum....Pages 201-204
Number Theory and Atomic Densities....Pages 205-218
Explicit Formulas and Oscillations....Pages 219-267
Energy Fluctuation Analysis in Integrable Billiards in Hyperbolic Geometry....Pages 269-289
On Eigenfunctions of the Laplacian for Hecke Triangle Groups....Pages 291-315
Eigenvalue Spacings for Regular Graphs....Pages 317-327
Classical Limits of Eigenfunctions for Some Completely Integrable Systems....Pages 329-354
Does a Quantum Particle Know the Time?....Pages 355-371
Level Spacings for Cayley Graphs....Pages 373-386
Eigenvalues of Ramanujan Graphs....Pages 387-403
Theta Sums, Eisenstein Series, and the Semiclassical Dynamics of a Precessing Spin....Pages 405-450
Random Walks on Generalized Euclidean Graphs....Pages 451-468
Two Proofs of Ihara’s Theorem....Pages 469-478
Playing Billiards with Microwaves — Quantum Manifestations of Classical Chaos....Pages 479-523
Characters of the Symmetric Groups: Formulas, Estimates and Applications....Pages 525-545
Number Theory and Formal Languages....Pages 547-570
Expander Graphs and Amenable Quotients....Pages 571-581
Ramanujan Hypergraphs and Ramanujan Geometries....Pages 583-590
Constructing Error-Correcting Codes from Expander Graphs....Pages 591-600
Multipath Zeta Functions of Graphs....Pages 601-615
Eigenvalues of the Laplacian for Bianchi Groups....Pages 617-641
A Survey of Discrete Trace Formulas....Pages 643-681
Back Matter....Pages 683-697

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Content:
Front Matter....Pages i-xiii
Trace Formula for Quantum Integrable Systems, Lattice-Point Problem, and Small Divisors....Pages 1-38
Theta-Lifts of Maass Waveforms....Pages 39-72
The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ?)....Pages 73-141
Chaos and Deviation from Uniform Distribution: Eigenfunction Computation; Applied Modular Arithmetic....Pages 143-174
Logarithmic Sobolev Techniques for Random Walks on Graphs....Pages 175-186
Eigenvalue Statistics in Quantum Ideal Gases....Pages 187-199
Multifractal Spectrum and Laplace Spectrum....Pages 201-204
Number Theory and Atomic Densities....Pages 205-218
Explicit Formulas and Oscillations....Pages 219-267
Energy Fluctuation Analysis in Integrable Billiards in Hyperbolic Geometry....Pages 269-289
On Eigenfunctions of the Laplacian for Hecke Triangle Groups....Pages 291-315
Eigenvalue Spacings for Regular Graphs....Pages 317-327
Classical Limits of Eigenfunctions for Some Completely Integrable Systems....Pages 329-354
Does a Quantum Particle Know the Time?....Pages 355-371
Level Spacings for Cayley Graphs....Pages 373-386
Eigenvalues of Ramanujan Graphs....Pages 387-403
Theta Sums, Eisenstein Series, and the Semiclassical Dynamics of a Precessing Spin....Pages 405-450
Random Walks on Generalized Euclidean Graphs....Pages 451-468
Two Proofs of Ihara’s Theorem....Pages 469-478
Playing Billiards with Microwaves — Quantum Manifestations of Classical Chaos....Pages 479-523
Characters of the Symmetric Groups: Formulas, Estimates and Applications....Pages 525-545
Number Theory and Formal Languages....Pages 547-570
Expander Graphs and Amenable Quotients....Pages 571-581
Ramanujan Hypergraphs and Ramanujan Geometries....Pages 583-590
Constructing Error-Correcting Codes from Expander Graphs....Pages 591-600
Multipath Zeta Functions of Graphs....Pages 601-615
Eigenvalues of the Laplacian for Bianchi Groups....Pages 617-641
A Survey of Discrete Trace Formulas....Pages 643-681
Back Matter....Pages 683-697
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