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Spectral truncations in noncommutative geometry and operator systems
Alain Connes, Walter D. van Suijlekom
In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in…
Noncommutative Geometry, the spectral standpoint
Alain Connes
We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for…
On an idea of Michael Atiyah
Alain Connes
In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit-Thompson on the solvability of finite groups of odd order, the iterat…
The term a_4 in the heat kernel expansion of noncommutative tori
Alain Connes, Farzad Fathizadeh
We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a_4 t…
Modular Curvature for Noncommutative Two-Tori
Alain Connes, Henri Moscovici
In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of s…
On the arithmetic of the BC-system
Alain Connes, Caterina Consani
For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the…
The Witt construction in characteristic one and Quantization
Alain Connes
We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs of a perfect semi-ring of characteristic one and an element strictly large…
The Gauss-Bonnet Theorem for the noncommutative two torus
Alain Connes, Paula Tretkoff
In this paper we show that the value at zero of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent…
A walk in the noncommutative garden
Alain Connes, Matilde Marcolli
This text is written for the volume of the school/conference "Noncommutative Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry,…
Noncommutative geometry and motives: the thermodynamics of endomotives
Alain Connes, Caterina Consani, Matilde Marcolli
We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spe…
Non commutative finite dimensional manifolds II. Moduli space and structure of non commutative 3-spheres
Alain Connes, Michel Dubois-Violette
This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic…
Transgressions of the Godbillon-Vey class and Rademacher functions
Alain Connes, Henri Moscovici
We construct, out of modular symbols, 1-traces that are invariant with respect to the actions of the Hopf algebra $\Hc_{1}$ on the crossed product $\Ac_{Q}$ of the algebra of modul…
Background independent geometry and Hopf cyclic cohomology
Alain Connes, Henri Moscovici
This is primarily a survey of the way in which Hopf cyclic cohomology has emerged and evolved, in close relationship with the application of the noncommutative local index formula…
Rankin-Cohen Brackets and the Hopf Algebra of Transverse Geometry
Alain Connes, Henri Moscovici
We settle in this paper a question left open in our paper ``Modular Hecke algebras and their Hopf symmetry'', by showing how to extend the Rankin-Cohen brackets from modular forms…
Modular Hecke Algebras and their Hopf Symmetry
Alain Connes, Henri Moscovici
We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hec…
Cyclic Cohomology, Quantum group Symmetries and the Local Index Formula for SUq(2)
Alain Connes
We analyse the noncommutative space underlying the quantum group SUq(2) from the spectral point of view which is the basis of noncommutative geometry, and show how the general theo…
Yang-Mills algebra
Alain Connes, Michel Dubois-Violette
Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang-Mills connection over the (s+1)-dimensional pseudo Euclidean space are poin…
Noncommutative Geometry Year 2000
Alain Connes
We describe basic concepts of noncommutative geometry and a general construction extending the familiar duality between ordinary spaces and commutative algebras to a duality betwee…
Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
Alain Connes, Giovanni Landi
We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of $\Rb^n$. They arise naturally from basic considerations of noncom…
Cyclic Cohomology, Hopf Algebras and the Modular Theory
Alain Connes, Henri Moscovici
We associate canonically a cyclic module to any Hopf algebra endowed with a modular pair, consisting of a group-like element and a character, in involution. This provides the key c…
Cyclic cohomology and Hopf algebras
Alain Connes, Henri Moscovici
We show by a direct computation that, for any Hopf algebra with a modulus-like character, the formulas first introduced in [CM] in the context of characteristic classes for actions…