Frontiers in Number Theory, Physics, and Geometry I (PDF)

The relation between mathematics and physics has a long history, in which the role of number theory and of other more abstract parts of mathematics has recently become more prominent.
More than ten years after a first meeting in 1989 between number theorists and physicists at the Centre de Physique des Houches, a second 2-week event focused on the broader interface of number theory, geometry, and physics.
This book is the result of that exciting meeting, and collects, in 2 volumes, extended versions of the lecture courses, followed by shorter texts on special topics, of eminent mathematicians and physicists.
The present volume has three parts: Random matrices, Zeta functions, Dynamical systems.
The companion volume is subtitled: On Conformal Field Theories, Discrete Groups and Renormalization and will be published in 2006 (Springer, 3-540-30307-3). TOC:Part I: Quantum and Arithmetical Chaos.- Notes on L-functions and Random Matrix Theory.- Energy Level Statistics, Lattice Point Problems, and Almost Modular Functions.- Arithmetic Quantum Chaos of Maass Waveforms.- Large N Expansion for Normal and Complex Matrix Ensembles.- Symmetries Arising from Free Probability Theory.- Universality and Randomness for the Graphs and Metric Spaces. Part II: From Physics to Number Theory Via Noncommutative Geometry.- More Zeta Functions for the Riemann Zeros.- Hilbert Spaces of Entire Functions and L-Functions.- Dynamical Zeta functions and Closed Orbits for Geodesic and Hyperbolic Flows. Part III: Continued Fraction Algorithms for Interval Exchange Maps: an Introduction.- Brjuno Numbers and Dynamical Systems.- Some Properties of Real and Complex Brjuno Functions. Part IV: Appendices.