Noncommutative Geometry Year 2000
| dc.creator | Connes, Alain | |
| dc.date | 2000-11-23 | |
| dc.date.accessioned | 2026-07-07T04:38:47Z | |
| dc.date.available | 2026-07-07T04:38:47Z | |
| dc.description | 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 $θ\to 1/θ$ for the NC torus $\Tb_θ^2$, unseen in perturbative expansions in $θ$ 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 $S_θ^4$ 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. | |
| dc.description | 67 pages | |
| dc.identifier | https://arxiv.org/abs/math/0011193 | |
| dc.identifier | http://arxiv.org/abs/math/0011193 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60420 | |
| dc.subject | Quantum Algebra | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | Number Theory | |
| dc.subject | Operator Algebras | |
| dc.title | Noncommutative Geometry Year 2000 | |
| dc.type | text |