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Frontiers in Number Theory, Physics, and Geometry I: On Random Matrices, Zeta Functions, and Dynamical Systemsby Pierre Vanhove
Synopses & ReviewsPublisher Comments: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). Synopsis:This book presents pedagogical contributions on selected topics relating Number Theory, Theoretical Physics and Geometry. The parts are composed of long self-contained pedagogical lectures followed by shorter contributions on specific subjects organized by theme. Most courses and short contributions go up to the recent developments in the fields; some of them follow their author's original viewpoints. There are contributions on Random Matrix Theory, Quantum Chaos, Non-commutative Geometry, Zeta functions, and Dynamical Systems. The chapters of this book are extended versions of lectures given at a meeting entitled Number Theory, Physics and Geometry, held at Les Houches in March 2003, which gathered mathematicians and physicists. Table of ContentsPart 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.
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