Noncommutative Geometry Year 2000
Abstract
We describe basic concepts of noncommutative geometry and a general construction extending the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. Basic tools of the theory, K-theory, Cyclic cohomology, Morita equivalence, Operator theoretic index theorems, Hopf algebra symmetry are reviewed. They cover the global aspects of noncommutative spaces, such as the transformation $\theta \to 1/\theta$ for the NC torus $\Tb_{\theta}^2$, unseen in perturbative expansions in $\theta$ such as star or Moyal products. We discuss the foundational problem of "what is a manifold in NCG" and explain the role of Poincare duality in K-homology which is the basic reason for the spectral point of view. When specializing to 4-geometries this leads to the universal "Instanton algebra". We describe our work with G. Landi which gives NC-spheres $S_{\theta}^4$ from representations of the Instanton algebra. We show that any compact Riemannian spin manifold whose isometry group has rank $r \geq 2$ admits isospectral deformations to noncommutative geometries. We give a survey of our work with H. Moscovici on the transverse geometry of foliations which yields a diffeomorphism invariant geometry on the bundle of metrics on a manifold and a natural extension of cyclic cohomology to Hopf algebras. Then, our work with D. Kreimer on renormalization and the Riemann-Hilbert problem. Finally we describe the spectral realization of zeros of zeta and L-functions from the noncommutative space of Adele classes on a global field and its relation with the Arthur-Selberg trace formula in the Langlands program. We end with a tentalizing connection between the renormalization group and the missing Galois theory at Archimedian places.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- November 2000
- DOI:
- 10.48550/arXiv.math/0011193
- arXiv:
- arXiv:math/0011193
- Bibcode:
- 2000math.....11193C
- Keywords:
-
- Quantum Algebra;
- Operator Algebras;
- Number Theory
- E-Print:
- 67 pages