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The Uncanny Precision of the Spectral Action
Ali H. Chamseddine, Alain Connes
Noncommutative geometry has been slowly emerging as a new paradigm of geometry which starts from quantum mechanics. One of its key features is that the new geometry is spectral in…
A unitary invariant in Riemannian geometry
Alain Connes
We introduce an invariant of Riemannian geometry which measures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of th…
On the spectral characterization of manifolds
Alain Connes
We show that the first five of the axioms we had formulated on spectral triples suffice (in a slightly stronger form) to characterize the spectral triples associated to smooth comp…
On the notion of geometry over $\F_1$
Alain Connes, Caterina Consani
We refine the notion of variety over the "field with one element" developed by C. Soulé by introducing a grading in the associated functor to the category of sets, and show that th…
Fun with $\F_1$
Alain Connes, Caterina Consani, Matilde Marcolli
We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost--Connes naturally arises by extension of scalars from the "field with one element"…