Orbifolds are not commutative geometries

In this note we show that the crucial orientation condition for commutative geometries fails for the natural spectral triple of an orbifold M/G.Comment: 6 pages, Latex, no figure

Reconstruction of manifolds in noncommutative geometry

We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.Comment: 67 pages, no figures, Latex; major changes, a new Appendix

Transcendental obstructions to weak approximation on general K3 surfaces

We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between polarized K3 surfaces endowed with particular kinds of Brauer classes and cubic fourfolds.Comment: 24 pages, 3 figures, Magma scripts included at the end of the source file

Dixmier traces on noncompact isospectral deformations

We extend the isospectral deformations of Connes, Landi and Dubois-Violette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group $R^l$. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of $R^l$, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.Comment: 30 pages, no figures; several minor improvements, to appear in J. Funct. Ana

Fourier analysis on the affine group, quantization and noncompact Connes geometries

We find the Stratonovich-Weyl quantizer for the nonunimodular affine group of the line. A noncommutative product of functions on the half-plane, underlying a noncompact spectral triple in the sense of Connes, is obtained from it. The corresponding Wigner functions reproduce the time-frequency distributions of signal processing. The same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by Kirillov.Comment: 37 pages, Latex, uses TikZ package to draw 3 figures. Two new subsections, main results unchange