Noncommutative Geometry and Number Theory
Where Arithmetic meets Geometry and Physics
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Beschreibung
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.
In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. Research across these ?elds has now reached an imp- tant turning point, as shows the increasing interest with which the mathematical community approaches these topics. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new c- nections between the ?elds of number theory, algebraic geometry and noncom- tative geometry. Thecontributionstothisvolumepartlyre?ectthetwoworkshops“Noncom- tative Geometry and Number Theory” that took place at the Max–Planck–Institut f¨ ur Mathematik in Bonn, in August 2003 and June 2004. The two workshops were the ?rst activity entirely dedicated to the interplay between these two ?elds of mathematics. An important part of the activities, which is also re?ected in this volume, came from the hindsight of physics which often provides new perspectives onnumber theoretic problems that make it possible to employ the tools of nonc- mutative geometry, well designed to describe the quantum world.
Aktuelles aus der mathematischen Forschung
Die Nichtkommutative Geometrie ist ein moderner Zweig der Mathematik. Sie stellt mächtige Werkzeuge zur Verfügung, die es ermöglichen, "quantisierte'' Räume zu untersuchen. Anders als im Fall gewöhnlicher Räume sind ihre Koordinatenalgebren nichtkommutativ und können daher Phänomene wie die Heisenbergsche Unschärferelation in der Quantenmechanik modellieren. Dieses Buch (in englischer Sprache) beschreibt, teilweise sogar auf einführendem 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 Überblick über "the state of the art" dieses Forschungsgebietes.
Autor*in
Caterina Consani
Themen in »Noncommutative Geometry and Number Theory«
Archimedean cohomology Farey fractions Formula of Atiyah-Singer Fractional quantum Hall effect Hecke algebra Hopf cyclic cohomology Method of Chabauty Rademacher functions Ramanujan conjecture Reidemeister numbers Schwartz algebra number theory