Noncommutative Geometry and Number Theory

Prof. Dr. Caterina Consani, Department of Mathematics, The Johns Hopkins University, Baltimore, USA
Prof. Dr. Matilde Marcolli, Max-Planck Institute for Mathematics, Bonn, Germany

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.

Recent interactions between the fields of Noncommutative Geometry and Number Theory

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.

The Hecke algebra of a reductive p-adic group: a geometric conjecture.- Hilbert modular forms and the Ramanujan conjecture.- Farey fractions and two-dimensional tori.- Transgressions of the Godbillon-Vey Class and Rademacher functions.- Archimedean cohomology revisited.- A twisted Burnside theorem for countable groups and Reidemeister numbers.- to Hopf-Cyclic Cohomology.- The non-abelian (or non-linear) method of Chabauty.- The residues of quantum field theory - numbers we should know.- Phase transitions with spontaneous symmetry breaking on Hecke C*-algebras from number fields.- On harmonic maps in noncommutative geometry.- Towards the fractional quantum Hall effect: a noncommutative geometry perspective.- Homological algebra for Schwartz algebras of reductive p-adic groups.- A non-commutative geometry approach to the representation theory of reductive p-adic groups: Homology of Hecke algebras, a survey and some new results.- Three examples of non-commutative boundaries of Shimura varieties.- Holomorphic bundles on 2-dimensional noncommutative toric orbifolds.- A New short proof of the local index formula of Atiyah-Singer.