Noncommutative Geometry and Number Theory: Where Arithmetic Meets Geometry and Physics - Aspects of Mathematics 2006 edition

Jacket Description/Back: In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local Lfactors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect. Table of Contents: The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local L-factors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect. Description for Sales People: Die Nichtkommutative Geometrie ist ein moderner Zweig der Mathematik. Sie stellt machtige Werkzeuge zur Verfugung, die es ermoglichen, "quantisierte'' Raume zu untersuchen. Anders als im Fall gewohnlicher Raume sind ihre Koordinatenalgebren nichtkommutativ und konnen daher Phanomene wie die Heisenbergsche Unscharferelation in der Quantenmechanik modellieren. Dieses Buch (in englischer Sprache) beschreibt, teilweise sogar auf einfuhrendem Niveau, den Zusammenhang von Nichtkommutativer Geometrie und klassischen Problemen der Zahlentheorie und die Verbindung mit Ideen aus der Physik. Es bringt die wichtigen Experten zusammen und gibt einen hervorragenden Uberblick uber "the state of the art" dieses Forschungsgebietes. "Marc Notes: A publication of the Max-Planck-Institute for Mathematics, Bonn.; The contributions to this volume partly reflect the two workshops Noncommutative Geometry and Number Theory that took place at the Max-Planck-Institut fur Mathematik in Bonn, in August 2003 and June 2004--Pref.; Includes bibliographical references. Biographical Note: Prof. Dr. Caterina Consani, Department of Mathematics, The Johns Hopkins University, Baltimore, USA Prof. Dr. Matilde Marcolli, Max-Planck Institute for Mathematics, Bonn, Germany