19,283 research outputs found
Noncommutative Geometry and Arithmetic
This is an overview of recent results aimed at developing a geometry of
noncommutative tori with real multiplication, with the purpose of providing a
parallel, for real quadratic fields, of the classical theory of elliptic curves
with complex multiplication for imaginary quadratic fields. This talk
concentrates on two main aspects: the relation of Stark numbers to the geometry
of noncommutative tori with real multiplication, and the shadows of modular
forms on the noncommutative boundary of modular curves, that is, the moduli
space of noncommutative tori. To appear in Proc. ICM 2010.Comment: 16 pages, LaTe
Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey
After a brief summary of the main properties of Poisson manifolds and Lie
algebroids in general, we survey recent work on the modular classes of Poisson
and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted
Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor
approach to the modular class concludes the paper.Comment: Dedicated to the memory of Thomas Branson. This is a contribution to
the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P.
Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods
and Applications) at http://www.emis.de/journals/SIGMA
Twenty-five years of two-dimensional rational conformal field theory
In this article we try to give a condensed panoramic view of the development
of two-dimensional rational conformal field theory in the last twenty-five
years.Comment: A review for the 50th anniversary of the Journal of Mathematical
Physics. Some references added, typos correcte
Modular Theory and Eyvind Wichmann's Contributions to modern Particle Physics Theory
Some of the consequences of Eyvind Wichmann's contributions to modular theory
and the QFT phase-space structure are presented. In order to show the power of
those ideas in contemporary problems, I selected the issue of algebraic
holography as well as a new nonperturbative constructive approach (based on the
modular structur of wedge-localized algebras and modular inclusions) and show
that these ideas are recent consequences of the pathbreaking work which
Wichmann together with his collaborator Bisognano initiated in the mid
70$^{ies}.Comment: A rogue address which entered chapter 2 has since been omitted. 21
pages, tcilatex, to be published in a special Festschrift volume, dedicated
to Prof. E. Wichmann on the occasion of his seventieth birthda
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