269. | Spectral Triples and Zeta-CyclesForthcoming Alain Connes, Caterina Consani Enseignement Mathématique, Forthcoming. Journal ArticleAbstract @article{Connes2021bb, title = {Spectral Triples and Zeta-Cycles}, author = {Alain Connes and Caterina Consani}, editor = {European Math. Society}, url = {https://alainconnes.org/wp-content/uploads/zeta-cycles-3.pdf}, year = {2021}, date = {2021-09-15}, journal = {Enseignement Mathématique}, abstract = { We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S. We show both numerically and conceptually that the associated eigenvectors are obtained by a simple arithmetic operation of finite sum using prolate spheroidal wave functions associated to the scale S. Then we use these functions to condition the canonical spectral triple of the circle of length L=2 Log(S) in such a way that they belong to the kernel of the perturbed Dirac operator. We give numerical evidence that, when one varies L, the low lying spectrum of the perturbed spectral triple resembles the low lying zeros of the Riemann zeta function. We justify conceptually this result and show that, for each eigenvalue, the coincidence is perfect for the special values of the length L of the circle for which the two natural ways of realizing the perturbation give the same eigenvalue. This fact is tested numerically by reproducing the first thirty one zeros of the Riemann zeta function from our spectral side, and estimate the probability of having obtained this agreement at random, as a very small number whose first fifty decimal places are all zero. The theoretical concept which emerges is that of zeta cycle and our main result establishes its relation with the critical zeros of the Riemann zeta function and with the spectral realization of these zeros obtained by the first author.}, keywords = {}, pubstate = {forthcoming}, tppubtype = {article} } We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S. We show both numerically and conceptually that the associated eigenvectors are obtained by a simple arithmetic operation of finite sum using prolate spheroidal wave functions associated to the scale S. Then we use these functions to condition the canonical spectral triple of the circle of length L=2 Log(S) in such a way that they belong to the kernel of the perturbed Dirac operator. We give numerical evidence that, when one varies L, the low lying spectrum of the perturbed spectral triple resembles the low lying zeros of the Riemann zeta function. We justify conceptually this result and show that, for each eigenvalue, the coincidence is perfect for the special values of the length L of the circle for which the two natural ways of realizing the perturbation give the same eigenvalue. This fact is tested numerically by reproducing the first thirty one zeros of the Riemann zeta function from our spectral side, and estimate the probability of having obtained this agreement at random, as a very small number whose first fifty decimal places are all zero. The theoretical concept which emerges is that of zeta cycle and our main result establishes its relation with the critical zeros of the Riemann zeta function and with the spectral realization of these zeros obtained by the first author. |

268. | Quasi-inner functions and local factors Alain Connes, Caterina Consani Journal of Number Theory, 226 , pp. 139-167, 2021. Journal ArticleAbstract @article{connes-consani, title = {Quasi-inner functions and local factors}, author = {Alain Connes and Caterina Consani}, editor = {Elsevier}, url = {/wp-content/uploads/quasi-inner.pdf}, year = {2021}, date = {2021-09-06}, journal = {Journal of Number Theory}, volume = {226}, pages = {139-167}, abstract = {We introduce the notion of {it quasi-inner} function and show that the product u=ρ∞∏ρv of m+1 ratios of local {L-}factors {ρv(z)=γv(z)/γv(1−z)} over a finite set F of places of the field of rational numbers {inclusive of} the archimedean place is {quasi-inner} on the left of the critical line R(z)=12 in the following sense. The off diagonal part u21 of the matrix of the multiplication by u in the orthogonal decomposition of the Hilbert space L2 of square integrable functions on the critical line into the Hardy space H2 and its orthogonal complement is a compact operator. When interpreted on the unit disk, the quasi-inner condition means that the associated Haenkel matrix is compact. We show that none of the individual non-archimedean ratios ρv is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio ρ∞ into a product of m quasi-inner functions whose product with each ρv retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part u22 for the quasi-inner function u=ρ∞, and when u(F)=∏v∈Fρv the kernels of the u(F)22 form an inductive system of infinite dimensional spaces which are the semi-local analogues of (classical) Sonin's spaces.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We introduce the notion of {it quasi-inner} function and show that the product u=ρ∞∏ρv of m+1 ratios of local {L-}factors {ρv(z)=γv(z)/γv(1−z)} over a finite set F of places of the field of rational numbers {inclusive of} the archimedean place is {quasi-inner} on the left of the critical line R(z)=12 in the following sense. The off diagonal part u21 of the matrix of the multiplication by u in the orthogonal decomposition of the Hilbert space L2 of square integrable functions on the critical line into the Hardy space H2 and its orthogonal complement is a compact operator. When interpreted on the unit disk, the quasi-inner condition means that the associated Haenkel matrix is compact. We show that none of the individual non-archimedean ratios ρv is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio ρ∞ into a product of m quasi-inner functions whose product with each ρv retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part u22 for the quasi-inner function u=ρ∞, and when u(F)=∏v∈Fρv the kernels of the u(F)22 form an inductive system of infinite dimensional spaces which are the semi-local analogues of (classical) Sonin's spaces. |

267. | On absolute algebraic geometry, the affine case Forthcoming Alain Connes, Caterina Consani Advances in Mathematics, pp. 1–45, Forthcoming. Journal ArticleAbstract @article{Connes2021bb, title = {On absolute algebraic geometry, the affine case }, author = {Alain Connes and Caterina Consani}, editor = {Elsevier}, url = {https://alainconnes.org/wp-content/uploads/JAG2020-1.pdf}, year = {2021}, date = {2021-07-20}, journal = {Advances in Mathematics}, pages = {1--45}, abstract = {We develop algebraic geometry for general Segal’s Γ-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under SpecZ).}, keywords = {}, pubstate = {forthcoming}, tppubtype = {article} } We develop algebraic geometry for general Segal’s Γ-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under SpecZ). |

266. | Weil positivity and trace formula, the archimedean placeForthcoming Alain Connes, Caterina Consani Selecta Mathematica, Forthcoming. Journal ArticleAbstract @article{Connes2021bb, title = {Weil positivity and trace formula, the archimedean place}, author = {Alain Connes and Caterina Consani}, editor = {Birkhauser}, url = {https://alainconnes.org/wp-content/uploads/Selecta.pdf}, year = {2021}, date = {2021-07-20}, journal = {Selecta Mathematica}, abstract = {We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula. We explore in great details the simplest case of the single archimedean place. The root of this result is the positivity of the trace of the scaling action compressed onto the orthogonal complement of the range of the cutoff projections associated to the cutoff in phase space, for Lambda=1. We express the difference between the Weil distribution and the trace associated to the above compression of the scaling action, in terms of prolate spheroidal wave functions, and use, as a key device, the theory of hermitian Toeplitz matrices to control that difference. All the concepts and tools used in this paper make sense in the general semi-local case, where Weil positivity implies RH.}, keywords = {}, pubstate = {forthcoming}, tppubtype = {article} } We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula. We explore in great details the simplest case of the single archimedean place. The root of this result is the positivity of the trace of the scaling action compressed onto the orthogonal complement of the range of the cutoff projections associated to the cutoff in phase space, for Lambda=1. We express the difference between the Weil distribution and the trace associated to the above compression of the scaling action, in terms of prolate spheroidal wave functions, and use, as a key device, the theory of hermitian Toeplitz matrices to control that difference. All the concepts and tools used in this paper make sense in the general semi-local case, where Weil positivity implies RH. |

265. | Noncommutative Geometry, the Spectral Standpoint Alain Connes Cambridge University Press (Ed.): Cambridge University Press, 2021. Book ChapterAbstract @inbook{ncg-spectral, title = {Noncommutative Geometry, the Spectral Standpoint}, author = {Alain Connes }, editor = {Cambridge University Press}, url = {/wp-content/uploads/NCGspectral.pdf}, year = {2021}, date = {2021-03-18}, publisher = {Cambridge University Press}, abstract = {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 noncommutative tori, 2) Advances on the Baum-Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the pseudo-differential calculi using smooth groupoids, 4) The development of Hopf cyclic cohomology, 5) The increasing role of topological cyclic homology in number theory, and of the lambda operations in archimedean cohomology, 6) The understanding of the renormalization group as a motivic Galois group, 7) The development of quantum field theory on noncommutative spaces, 8) The discovery of a simple equation whose irreducible representations correspond to 4-dimensional spin geometries with quantized volume and give an explanation of the Lagrangian of the standard model coupled to gravity, 9) The discovery that very natural toposes such as the scaling site provide the missing algebro-geometric structure on the noncommutative adele class space underlying the spectral realization of zeros of L-functions.}, keywords = {}, pubstate = {published}, tppubtype = {inbook} } 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 noncommutative tori, 2) Advances on the Baum-Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the pseudo-differential calculi using smooth groupoids, 4) The development of Hopf cyclic cohomology, 5) The increasing role of topological cyclic homology in number theory, and of the lambda operations in archimedean cohomology, 6) The understanding of the renormalization group as a motivic Galois group, 7) The development of quantum field theory on noncommutative spaces, 8) The discovery of a simple equation whose irreducible representations correspond to 4-dimensional spin geometries with quantized volume and give an explanation of the Lagrangian of the standard model coupled to gravity, 9) The discovery that very natural toposes such as the scaling site provide the missing algebro-geometric structure on the noncommutative adele class space underlying the spectral realization of zeros of L-functions. |

264. | Spectral truncation in noncommutative geometry and operator systems Alain Connes, Walter D. van Suijlekom Communication in Mathematical Physics, 383 , pp. 2021–2067, 2021. Journal ArticleAbstract @article{operatorsystems, title = {Spectral truncation in noncommutative geometry and operator systems}, author = {Alain Connes and Walter D. van Suijlekom }, editor = {Springer}, url = {https://alainconnes.org/wp-content/uploads/withwalter.pdf}, year = {2021}, date = {2021-02-02}, journal = {Communication in Mathematical Physics}, volume = {383}, pages = {2021--2067}, abstract = {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 momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional role played by C∗-algebras is taken over by operator systems. As part of the techniques we treat C∗-envelopes, dual operator systems and stable equivalence. We define a propagation number for operator systems, which we show to be an invariant under stable equivalence and use to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle. These are operator systems of finite-dimensional Toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval. It turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i.e. the distance on the state space associated to the Dirac operator. The main property of the spectral truncation is that it keeps the isometry group intact. In contrast, if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete, even though in this case the operator system is a C∗-algebra. We analyze this in the context of the finite Fourier transform. The extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation d(x,y)<ε between two points, or more generally a tolerance relation which naturally gives rise to an operator system.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional role played by C∗-algebras is taken over by operator systems. As part of the techniques we treat C∗-envelopes, dual operator systems and stable equivalence. We define a propagation number for operator systems, which we show to be an invariant under stable equivalence and use to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle. These are operator systems of finite-dimensional Toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval. It turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i.e. the distance on the state space associated to the Dirac operator. The main property of the spectral truncation is that it keeps the isometry group intact. In contrast, if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete, even though in this case the operator system is a C∗-algebra. We analyze this in the context of the finite Fourier transform. The extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation d(x,y)<ε between two points, or more generally a tolerance relation which naturally gives rise to an operator system. |

263. | Segal's Gamma Rings and Universal Arithmetic Alain Connes, Caterina Consani Quarterly Journal of Mathematics, pp. 1-29, 2020. Journal ArticleAbstract @article{Segals, title = {Segal's Gamma Rings and Universal Arithmetic}, author = {Alain Connes and Caterina Consani }, url = {/wp-content/uploads/segals-GammaRings.pdf}, year = {2020}, date = {2020-12-02}, journal = {Quarterly Journal of Mathematics}, pages = {1-29}, abstract = {Segal’s Γ-rings provide a natural framework for absolute algebraic geometry. We use Almkvist’s global Witt construction to explore the relation with J. Borger F1-geometry and compute the Witt functor-ring W0(S) of the simplest Γ-ring S. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between λ-rings and the Arithmetic site. Then, we concentrate on the Arakelov compactification Spec Z which acquires a structure sheaf of S-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D on SpecZ, we show how to associate to D a Γ-space that encodes, in homotopical terms, the Riemann-Roch problem for D.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Segal’s Γ-rings provide a natural framework for absolute algebraic geometry. We use Almkvist’s global Witt construction to explore the relation with J. Borger F1-geometry and compute the Witt functor-ring W0(S) of the simplest Γ-ring S. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between λ-rings and the Arithmetic site. Then, we concentrate on the Arakelov compactification Spec Z which acquires a structure sheaf of S-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D on SpecZ, we show how to associate to D a Γ-space that encodes, in homotopical terms, the Riemann-Roch problem for D. |

262. | Alain Connes Journal of Mathematics and Music, 15 (1), pp. 1–16, 2020, ISSN: 1745-9737. Journal ArticleAbstract @article{Connes2021, title = {Motivic Rhythms}, author = {Alain Connes}, url = {/wp-content/uploads/motivic-rythms.pdf}, doi = {10.1080/17459737.2020.1817587}, issn = {1745-9737}, year = {2020}, date = {2020-08-28}, journal = {Journal of Mathematics and Music}, volume = {15}, number = {1}, pages = {1--16}, abstract = {In this article on mathematics and music, we explain how one can "listen to motives" as rhythmic interpreters. In the simplest instance which is the one we shall consider, the motive is simply the H1 of the reduction modulo a prime p of an hyperelliptic curve (defined over Q). The corresponding time onsets are given by the arguments of the complex eigenvalues of the Frobenius. We find a surprising relation between mathematical properties of the motives and the ideas on rhythms developed by the composer Olivier Messiaen.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this article on mathematics and music, we explain how one can "listen to motives" as rhythmic interpreters. In the simplest instance which is the one we shall consider, the motive is simply the H1 of the reduction modulo a prime p of an hyperelliptic curve (defined over Q). The corresponding time onsets are given by the arguments of the complex eigenvalues of the Frobenius. We find a surprising relation between mathematical properties of the motives and the ideas on rhythms developed by the composer Olivier Messiaen. |

261. | Alain Connes, Caterina Consani J. Operator Theory, 85 (1), pp. 257–276, 2019, ISSN: 0379-4024. Journal ArticleAbstract @article{Connes2021b, title = {The scaling Hamiltonian}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/scalingH.pdf}, doi = {10.7900/jot}, issn = {0379-4024}, year = {2019}, date = {2019-10-31}, journal = {J. Operator Theory}, volume = {85}, number = {1}, pages = {257--276}, abstract = {We first explain the link between the Berry-Keating Hamiltonian and the spectral realization of zeros of the Riemann zeta function ζ of [10], and why there is no conflict at the semi-classical level between the “absorption" picture of [10] and the semiclassical “emission" computations of [1, 2], while the minus sign manifests itself in the Maslov phases. We then use the quantized calculus to analyse the recent attempt of X.-J. Li at proving Weil’s positivity, and understand its limit. We then propose an operator theoretic semi-local framework directly related to the Riemann Hypothesis.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We first explain the link between the Berry-Keating Hamiltonian and the spectral realization of zeros of the Riemann zeta function ζ of [10], and why there is no conflict at the semi-classical level between the “absorption" picture of [10] and the semiclassical “emission" computations of [1, 2], while the minus sign manifests itself in the Maslov phases. We then use the quantized calculus to analyse the recent attempt of X.-J. Li at proving Weil’s positivity, and understand its limit. We then propose an operator theoretic semi-local framework directly related to the Riemann Hypothesis. |

260. | Sir Michael Atiyah, a Knight Mathematician Alain Connes, Joseph Kouneiher Notices Amer. Math. Soc., 66 (10), pp. 1660–1671, 2019, ISSN: 0002-9920. Journal ArticleAbstract @article{Connes2019, title = {Sir Michael Atiyah, a Knight Mathematician}, author = {Alain Connes and Joseph Kouneiher }, url = {/wp-content/uploads/Atiyah-Kouneiher-1.pdf}, issn = {0002-9920}, year = {2019}, date = {2019-10-17}, journal = {Notices Amer. Math. Soc.}, volume = {66}, number = {10}, pages = {1660--1671}, abstract = {Sir Michael Atiyah was considered one of the world’s foremost mathematicians. He is best known for his work in algebraic topology and the codevelopment of a branch of mathematics called topological K-theory together with the Atiyah-Singer index theorem for which he received Fields Medal (1966). He also received the Abel Prize (2004) along with Isadore M. Singer for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and for their outstanding role in building new bridges between mathematics and theoretical physics. Indeed, his work has helped theoretical physicists to advance their understanding of quantum field theory and general relativity.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Sir Michael Atiyah was considered one of the world’s foremost mathematicians. He is best known for his work in algebraic topology and the codevelopment of a branch of mathematics called topological K-theory together with the Atiyah-Singer index theorem for which he received Fields Medal (1966). He also received the Abel Prize (2004) along with Isadore M. Singer for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and for their outstanding role in building new bridges between mathematics and theoretical physics. Indeed, his work has helped theoretical physicists to advance their understanding of quantum field theory and general relativity. |

259. | Alain Connes CNRS Éditions, Paris, 2019, ISBN: 978-2-271-12712-9. BookAbstract @book{MR3970991, title = {La géométrie et le quantique}, author = {Alain Connes}, isbn = {978-2-271-12712-9}, year = {2019}, date = {2019-09-26}, pages = {73}, publisher = {CNRS Éditions}, address = {Paris}, abstract = {En 1637, Descartes révolutionne la manière que l'on a de faire de la géométrie : en associant à chaque point de l'espace trois coordonnées, il pose les bases de la géométrie algébrique. Cette géométrie est dite "commutative" : le produit de deux quantités ne dépend pas de l'ordre des termes, et A × B = B × A. Cette propriété est fondamentale, l'ensemble de l'édifice mathématique en dépend. Mais au début du XXe siècle, la découverte du monde quantique vient tout bouleverser. L'espace géométrique des états d'un système microscopique, un atome par exemple, s'enrichit de nouvelles propriétés, qui ne commutent plus. Il faut donc adapter l'ensemble des outils mathématiques. Cette nouvelle géométrie, dite "non commutative", devenue essentielle à la recherche en physique, a été développée par Alain Connes. En un texte court, vif et fascinant, ce grand mathématicien nous introduit à la poésie de sa discipline.}, keywords = {}, pubstate = {published}, tppubtype = {book} } En 1637, Descartes révolutionne la manière que l'on a de faire de la géométrie : en associant à chaque point de l'espace trois coordonnées, il pose les bases de la géométrie algébrique. Cette géométrie est dite "commutative" : le produit de deux quantités ne dépend pas de l'ordre des termes, et A × B = B × A. Cette propriété est fondamentale, l'ensemble de l'édifice mathématique en dépend. Mais au début du XXe siècle, la découverte du monde quantique vient tout bouleverser. L'espace géométrique des états d'un système microscopique, un atome par exemple, s'enrichit de nouvelles propriétés, qui ne commutent plus. Il faut donc adapter l'ensemble des outils mathématiques. Cette nouvelle géométrie, dite "non commutative", devenue essentielle à la recherche en physique, a été développée par Alain Connes. En un texte court, vif et fascinant, ce grand mathématicien nous introduit à la poésie de sa discipline. |

258. | Alain Connes, Caterina Consani Theory Appl. Categ., 35 (6), pp. 155–178, 2019. Journal ArticleAbstract @article{Connes2020, title = {Sp̅e̅c̅ ℤ and the Gromov norm}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/1905.03310-2-1.pdf}, year = {2019}, date = {2019-05-08}, journal = {Theory Appl. Categ.}, volume = {35}, number = {6}, pages = {155--178}, abstract = {We define the homology of a simplicial set with coefficients in a Segal’s Γ-set (s-module). We show the relevance of this new homology with values in s-modules by proving that taking as coefficients the s-modules at the archimedean place over the structure sheaf on SpecZ as in [2], one obtains on the singular homology with real coefficients of a topological space X, a norm equivalent to the Gromov norm. Moreover, we prove that the two norms agree when X is an oriented compact Riemann surface.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We define the homology of a simplicial set with coefficients in a Segal’s Γ-set (s-module). We show the relevance of this new homology with values in s-modules by proving that taking as coefficients the s-modules at the archimedean place over the structure sheaf on SpecZ as in [2], one obtains on the singular homology with real coefficients of a topological space X, a norm equivalent to the Gromov norm. Moreover, we prove that the two norms agree when X is an oriented compact Riemann surface. |

257. | The term ?₄ in the heat kernel expansion of noncommutative tori Alain Connes, Farzad Fathizadeh Münster J. Math., 12 (2), pp. 239–410, 2019, ISSN: 1867-5778. Journal ArticleAbstract @article{MR4030920, title = {The term ?₄ in the heat kernel expansion of noncommutative tori}, author = {Alain Connes and Farzad Fathizadeh}, url = {/wp-content/uploads/terma4.pdf}, doi = {10.17879/53149724705}, issn = {1867-5778}, year = {2019}, date = {2019-01-01}, journal = {Münster J. Math.}, volume = {12}, number = {2}, pages = {239--410}, abstract = {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 a4 that appears in its corresponding small-time heat kernel expansion. The final formula involves one variable functions and lengthy two, three and four variable functions of the modular automorphism of the state that encodes the conformal perturbation of the flat metric. We confirm the validity of the calculated expressions by showing that they satisfy a family of conceptually predicted functional relations. By studying these functional rela- tions abstractly, we derive a partial differential system which involves a natural action of cyclic groups of order two, three and four and a flow in parameter space. We discover symmetries of the calculated expressions with respect to the action of the cyclic groups. In passing, we show that the main ingredients of our calculations, which come from a rearrangement lemma and relations between the derivatives up to order four of the conformal factor and those of its logarithm, can be derived by finite differences from the generating function of the Bernoulli numbers and its multiplicative inverse. We then shed light on the significance of exponential polynomials and their smooth fractions in understanding the gen- eral structure of the noncommutative geometric invariants appearing in the heat kernel expansion. As an application of our results we obtain the a4 term for non- commutative four tori which split as products of two tori. These four tori are not conformally flat and the a4 term gives a first hint of the Riemann curvature and the higher dimensional modular structure.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 a4 that appears in its corresponding small-time heat kernel expansion. The final formula involves one variable functions and lengthy two, three and four variable functions of the modular automorphism of the state that encodes the conformal perturbation of the flat metric. We confirm the validity of the calculated expressions by showing that they satisfy a family of conceptually predicted functional relations. By studying these functional rela- tions abstractly, we derive a partial differential system which involves a natural action of cyclic groups of order two, three and four and a flow in parameter space. We discover symmetries of the calculated expressions with respect to the action of the cyclic groups. In passing, we show that the main ingredients of our calculations, which come from a rearrangement lemma and relations between the derivatives up to order four of the conformal factor and those of its logarithm, can be derived by finite differences from the generating function of the Bernoulli numbers and its multiplicative inverse. We then shed light on the significance of exponential polynomials and their smooth fractions in understanding the gen- eral structure of the noncommutative geometric invariants appearing in the heat kernel expansion. As an application of our results we obtain the a4 term for non- commutative four tori which split as products of two tori. These four tori are not conformally flat and the a4 term gives a first hint of the Riemann curvature and the higher dimensional modular structure. |

256. | Conformal trace theorem for Julia sets of quadratic polynomials Alain Connes, Edward McDonald, Fedor A. Sukochev, Dimitriy V. Zanin Ergodic Theory Dynam. Systems, 39 (9), pp. 2481–2506, 2019, ISSN: 0143-3857. Journal ArticleAbstract @article{MR3989126, title = {Conformal trace theorem for Julia sets of quadratic polynomials}, author = {Alain Connes and Edward McDonald and Fedor A. Sukochev and Dimitriy V. Zanin}, url = {/wp-content/uploads/ConnesMSZ_conformal_trace_th_for_julia_sets_published.pdf}, doi = {10.1017/etds.2017.124}, issn = {0143-3857}, year = {2019}, date = {2019-01-01}, journal = {Ergodic Theory Dynam. Systems}, volume = {39}, number = {9}, pages = {2481--2506}, abstract = {If c is in the main cardioid of the Mandelbrot set, then the Julia set J of the map φc : z ?→ z2 + c is a Jordan curve of Hausdorff dimension p ∈ [1, 2). We provide a full proof of a formula for the Hausdorff measure on J first announced by the first named author in 1996.}, keywords = {}, pubstate = {published}, tppubtype = {article} } If c is in the main cardioid of the Mandelbrot set, then the Julia set J of the map φc : z ?→ z2 + c is a Jordan curve of Hausdorff dimension p ∈ [1, 2). We provide a full proof of a formula for the Hausdorff measure on J first announced by the first named author in 1996. |

255. | Noncommutative geometry for symmetric non-self-adjoint operators Alain Connes, Galina Levitina, Edward McDonald, Fedor A. Sukochev, Dimitriy V. Zanin J. Funct. Anal., 277 (3), pp. 889–936, 2019, ISSN: 0022-1236. Journal ArticleAbstract @article{Connes2019b, title = {Noncommutative geometry for symmetric non-self-adjoint operators}, author = {Alain Connes and Galina Levitina and Edward McDonald and Fedor A. Sukochev and Dimitriy V. Zanin}, doi = {10.1016/j.jfa.2018.12.012}, issn = {0022-1236}, year = {2019}, date = {2019-01-01}, journal = {J. Funct. Anal.}, volume = {277}, number = {3}, pages = {889--936}, abstract = {We introduce the notion of a pre-spectral triple, which is a generalisa- tion of a spectral triple (A, H, D) where D is no longer required to be self- adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative ge- ometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on domains in Rd, d ≥ 2.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We introduce the notion of a pre-spectral triple, which is a generalisa- tion of a spectral triple (A, H, D) where D is no longer required to be self- adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative ge- ometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on domains in Rd, d ≥ 2. |

254. | Homological algebra in characteristic one Alain Connes, Caterina Consani High. Struct., 3 (1), pp. 155–247, 2019. Journal ArticleAbstract @article{MR3939048, title = {Homological algebra in characteristic one}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/Homological.pdf}, year = {2019}, date = {2019-01-01}, journal = {High. Struct.}, volume = {3}, number = {1}, pages = {155--247}, abstract = {This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category Bmod of modules over the Boolean semifield B := {0, 1} is the replacement for the category of abelian groups. We show that the semi-additive category Bmod fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on Bmod we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category Bmods is simply Bmod in the topos of sets endowed with an involution and as such it shares with Bmod most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category Bmods is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between Bmods and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of Bmods the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category Bmod of modules over the Boolean semifield B := {0, 1} is the replacement for the category of abelian groups. We show that the semi-additive category Bmod fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on Bmod we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category Bmods is simply Bmod in the topos of sets endowed with an involution and as such it shares with Bmod most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category Bmods is a semi-exact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between Bmods and the category of operators in Hilbert space, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of Bmods the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square. |

253. | Iteration of the exterior power on representation rings Alain Connes J. Geom. Phys., 141 , pp. 1–10, 2019, ISSN: 0393-0440. Journal ArticleAbstract @article{MR3926806, title = {Iteration of the exterior power on representation rings}, author = {Alain Connes}, url = {/wp-content/uploads/Michael-odd-1.pdf}, doi = {10.1016/j.geomphys.2019.02.013}, issn = {0393-0440}, year = {2019}, date = {2019-01-01}, journal = {J. Geom. Phys.}, volume = {141}, pages = {1--10}, abstract = {The Feit–Thompson theorem on the solvability of finite groups of odd order was very much on Michael Atiyah’s mind during his participation in the 2017 Shanghai conference on noncom- mutative geometry. Michael’s lively presence there, and his inexhaustible enthusiasm for all mathematics –old, new and yet to be created– were highlights of the meeting. The idea of Michael’s that we shall examine in this note was conceived by him during his flight home from Shanghai. It is a new strategy for FT, based on the iterative process sketched by him. His idea is to use this process to construct a non-trivial character of the finite group G. Taken too literally this cannot work because it would apply to the group of permutations of G which commute with the involution g ?→ g^-1 and this group only has characters of even order. The goal of the present paper, as a tribute to a luminous mathematical imagination that never dimmed, is to take seriously his proposal and to show that, understanding it in a broader sense, one arrives at a very interesting idea.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The Feit–Thompson theorem on the solvability of finite groups of odd order was very much on Michael Atiyah’s mind during his participation in the 2017 Shanghai conference on noncom- mutative geometry. Michael’s lively presence there, and his inexhaustible enthusiasm for all mathematics –old, new and yet to be created– were highlights of the meeting. The idea of Michael’s that we shall examine in this note was conceived by him during his flight home from Shanghai. It is a new strategy for FT, based on the iterative process sketched by him. His idea is to use this process to construct a non-trivial character of the finite group G. Taken too literally this cannot work because it would apply to the group of permutations of G which commute with the involution g ?→ g^-1 and this group only has characters of even order. The goal of the present paper, as a tribute to a luminous mathematical imagination that never dimmed, is to take seriously his proposal and to show that, understanding it in a broader sense, one arrives at a very interesting idea. |

252. | Alain Connes J. Number Theory, 194 , pp. 1–7, 2018, ISSN: 0022-314X. Journal ArticleAbstract @article{MR3860465, title = {Around Wilson's theorem}, author = {Alain Connes}, url = {/wp-content/uploads/wilson-1.pdf}, doi = {10.1016/j.jnt.2018.07.014}, issn = {0022-314X}, year = {2018}, date = {2018-09-11}, journal = {J. Number Theory}, volume = {194}, pages = {1--7}, abstract = {We study the series s(n,x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson’s theorem we show that the integer part of s(n,x) for x = Pi/2 is the number of primes less or equal to n and we get a similar formula for x a rational multiple of Pi. We show that for almost all x in the Lebesgue measure s(n,x) is equivalent to n/2 when n tends to infinity, while for almost all x in the Baire sense, 1/2 is a limit point of the ratio of s(n,x) to the number of primes less or equal to n.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We study the series s(n,x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson’s theorem we show that the integer part of s(n,x) for x = Pi/2 is the number of primes less or equal to n and we get a similar formula for x a rational multiple of Pi. We show that for almost all x in the Lebesgue measure s(n,x) is equivalent to n/2 when n tends to infinity, while for almost all x in the Baire sense, 1/2 is a limit point of the ratio of s(n,x) to the number of primes less or equal to n. |

251. | Entropy and the spectral action Ali H. Chamseddine, Alain Connes, Walter D. van Suijlekom Comm. Math. Phys., 373 (2), pp. 457–471, 2018, ISSN: 0010-3616. Journal ArticleAbstract @article{Chamseddine2020, title = {Entropy and the spectral action}, author = {Ali H. Chamseddine and Alain Connes and Walter D. van Suijlekom}, url = {/wp-content/uploads/1809.02944-3.pdf}, doi = {10.1007/s00220-019-03297-8}, issn = {0010-3616}, year = {2018}, date = {2018-09-09}, journal = {Comm. Math. Phys.}, volume = {373}, number = {2}, pages = {457--471}, abstract = {We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple. We show that this entropy is given by the spectral action of the spectral triple for a specific universal function. The main result of our paper is the surprising relation between this function and the Riemann zeta function. It manifests itself in particular by the values of the coefficients c(d) by which it multiplies the d dimensional terms in the heat expansion of the spectral triple. We find that c(d) is the product of the Riemann xi function evaluated at −d by an elementary expression. In particular c(4) is a rational multiple of ζ(5) and c(2) a rational multiple of ζ(3). The functional equation gives a duality between the coefficients in positive dimension, which govern the high energy expansion, and the coefficients in negative dimension, exchanging even dimension with odd dimension.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple. We show that this entropy is given by the spectral action of the spectral triple for a specific universal function. The main result of our paper is the surprising relation between this function and the Riemann zeta function. It manifests itself in particular by the values of the coefficients c(d) by which it multiplies the d dimensional terms in the heat expansion of the spectral triple. We find that c(d) is the product of the Riemann xi function evaluated at −d by an elementary expression. In particular c(4) is a rational multiple of ζ(5) and c(2) a rational multiple of ζ(3). The functional equation gives a duality between the coefficients in positive dimension, which govern the high energy expansion, and the coefficients in negative dimension, exchanging even dimension with odd dimension. |

250. | Alain Connes, Danye Chéreau, Jacques Dixmier 2018, ISBN: 9782738141361. BookAbstract @book{Connes2018b, title = {Le Spectre d'Atacama}, author = {Alain Connes and Danye Chéreau and Jacques Dixmier}, isbn = {9782738141361}, year = {2018}, date = {2018-01-31}, abstract = {Ce roman nous invite à partager la quête de la vérité dans la peau d’un scientifique, d’une physicienne rescapée d’un séjour quantique et d’un as de l’informatique. Chemin faisant, le lien entre espace et musique se révèle, avec Messiaen et son Quatuor pour la fin du Temps, avec les spectres mystérieux captés par l’observatoire d’Alma au Chili. À l’aléa du quantique vont se mêler celui, rebelle, des nombres premiers et celui des topos du grand mathématicien Alexandre Grothendieck, dont on écoutera les motifs. Le diable aussi jouera son rôle. Il sévit dans le machine learning qui l’emporte au jeu sans que l’on puisse comprendre pourquoi. Qui gagnera la partie ? Alain Connes est mathématicien, professeur au Collège de France, titulaire de la chaire d’Analyse et Géométrie, membre de l’Académie des sciences et de plusieurs académies étrangères, dont la National Academy of Sciences des États-Unis. Il a obtenu la médaille Fields en 1982. Danye Chéreau a une formation littéraire. Les surprises de la vie lui ont permis de découvrir le milieu scientifique et le monde des chercheurs sur lesquels elle porte un regard toujours curieux, amusé et attendri. Jacques Dixmier a professé aux universités de Toulouse, de Dijon et de Paris. Il est mathématicien « pur », mais certains de ses domaines de recherche (algèbres d’opérateurs, représentations des groupes, algèbres enveloppantes) sont utiles en mécanique quantique. Il a aussi publié des nouvelles de science-fiction.}, keywords = {}, pubstate = {published}, tppubtype = {book} } Ce roman nous invite à partager la quête de la vérité dans la peau d’un scientifique, d’une physicienne rescapée d’un séjour quantique et d’un as de l’informatique. Chemin faisant, le lien entre espace et musique se révèle, avec Messiaen et son Quatuor pour la fin du Temps, avec les spectres mystérieux captés par l’observatoire d’Alma au Chili. À l’aléa du quantique vont se mêler celui, rebelle, des nombres premiers et celui des topos du grand mathématicien Alexandre Grothendieck, dont on écoutera les motifs. Le diable aussi jouera son rôle. Il sévit dans le machine learning qui l’emporte au jeu sans que l’on puisse comprendre pourquoi. Qui gagnera la partie ? Alain Connes est mathématicien, professeur au Collège de France, titulaire de la chaire d’Analyse et Géométrie, membre de l’Académie des sciences et de plusieurs académies étrangères, dont la National Academy of Sciences des États-Unis. Il a obtenu la médaille Fields en 1982. Danye Chéreau a une formation littéraire. Les surprises de la vie lui ont permis de découvrir le milieu scientifique et le monde des chercheurs sur lesquels elle porte un regard toujours curieux, amusé et attendri. Jacques Dixmier a professé aux universités de Toulouse, de Dijon et de Paris. Il est mathématicien « pur », mais certains de ses domaines de recherche (algèbres d’opérateurs, représentations des groupes, algèbres enveloppantes) sont utiles en mécanique quantique. Il a aussi publié des nouvelles de science-fiction. |

249. | Alain Connes Foundations of mathematics and physics one century after Ħilbert, pp. 159–196, Springer, Cham, 2018. IncollectionAbstract @incollection{Connes2018, title = {Geometry and the quantum}, author = {Alain Connes}, url = {/wp-content/uploads/geometryquantum2kounehier.pdf}, year = {2018}, date = {2018-01-01}, booktitle = {Foundations of mathematics and physics one century after Ħilbert}, pages = {159--196}, publisher = {Springer, Cham}, abstract = {The ideas of noncommutative geometry are deeply rooted in both physics, with the predominant influence of the discovery of Quantum Mechanics, and in mathematics where it emerged from the great variety of examples of “noncommutative spaces” i.e. of geometric spaces which are best encoded algebraically by a noncommutative algebra. It is an honor to present an overview of the state of the art of the interplay of noncommutative geometry with physics on the occasion of the celebration of the centenary of Hilbert’s work on the foundations of physics. Indeed, the ideas which I will explain, those of noncommutative geometry (NCG) in relation to our model of space-time, owe a lot to Hilbert and this is so in two respects. First of course by the fundamental role of Hilbert space in the formalism of Quantum Mechanics as formalized by von Neumann, see §1.1. But also because, as explained in details in [32,38], one can consider Hilbert to be the first person to have speculated about a unified theory of electromagnetism and gravitation, we come to this point soon in §1.2.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } The ideas of noncommutative geometry are deeply rooted in both physics, with the predominant influence of the discovery of Quantum Mechanics, and in mathematics where it emerged from the great variety of examples of “noncommutative spaces” i.e. of geometric spaces which are best encoded algebraically by a noncommutative algebra. It is an honor to present an overview of the state of the art of the interplay of noncommutative geometry with physics on the occasion of the celebration of the centenary of Hilbert’s work on the foundations of physics. Indeed, the ideas which I will explain, those of noncommutative geometry (NCG) in relation to our model of space-time, owe a lot to Hilbert and this is so in two respects. First of course by the fundamental role of Hilbert space in the formalism of Quantum Mechanics as formalized by von Neumann, see §1.1. But also because, as explained in details in [32,38], one can consider Hilbert to be the first person to have speculated about a unified theory of electromagnetism and gravitation, we come to this point soon in §1.2. |

248. | Trace theorem for quasi-Fuchsian groups Alain Connes, Fedor A. Sukochev, Dimitriy V. Zanin Mat. Sb., 208 (10), pp. 59–90, 2017, ISSN: 0368-8666. Journal ArticleAbstract @article{MR3706885, title = {Trace theorem for quasi-Fuchsian groups}, author = {Alain Connes and Fedor A. Sukochev and Dimitriy V. Zanin}, url = {/wp-content/uploads/1703.05447.pdf}, doi = {10.4213/sm8794}, issn = {0368-8666}, year = {2017}, date = {2017-01-01}, journal = {Mat. Sb.}, volume = {208}, number = {10}, pages = {59--90}, abstract = {We complete the proof of the Trace Theorem in the quantized calculus for quasi-Fuchsian group which was stated and sketched, but not fully proved, on pp. 322-325 in the book “Noncommutative Geometry”of the first author.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We complete the proof of the Trace Theorem in the quantized calculus for quasi-Fuchsian group which was stated and sketched, but not fully proved, on pp. 322-325 in the book “Noncommutative Geometry”of the first author. |

247. | Alain Connes, Caterina Consani Selecta Math. (N.S.), 23 (3), pp. 1803–1850, 2017, ISSN: 1022-1824. Journal ArticleAbstract @article{MR3663595, title = {Geometry of the scaling site}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/scalingsite.pdf}, doi = {10.1007/s00029-017-0313-y}, issn = {1022-1824}, year = {2017}, date = {2017-01-01}, journal = {Selecta Math. (N.S.)}, volume = {23}, number = {3}, pages = {1803--1850}, abstract = {In this paper we construct the scaling site S by implementing the extension of scalars on the arithmetic site A , from the smallest Boolean semifield B to the tropical semifield R+max . The obtained semiringed topos is the Grothendieck topos [0, ∞)x ? N*, semi-direct product of the Euclidean half-line and the monoid N* of positive integers acting by multiplication, endowed with the structure sheaf of piecewise affine, convex functions with integral slopes. We show that pointwise [0, ∞) x N* coincides with the adele class space of Q and that this latter space inherits the geometric structure of a tropical curve. We restrict this construction to the periodic orbit of the scaling flow associated to each prime p and obtain a quasi-tropical structure which turns this orbit into a variant Cp = R∗+/pZ of the classical Jacobi description C∗/qZ of an elliptic curve. On Cp, we develop the theory of Cartier divisors, determine the structure of the quotient Div(Cp)/P of the abelian group of divisors by the subgroup of principal divisors, develop the theory of theta functions, and prove the Riemann-Roch formula which involves real valued dimensions, as in the type II index theory. We show that one would have been led to the same definition of S by analyzing the well known results on the localization of zeros of analytic functions involving Newton polygons in the non-archimedean case and the Jensen’s formula in the complex case.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper we construct the scaling site S by implementing the extension of scalars on the arithmetic site A , from the smallest Boolean semifield B to the tropical semifield R+max . The obtained semiringed topos is the Grothendieck topos [0, ∞)x ? N*, semi-direct product of the Euclidean half-line and the monoid N* of positive integers acting by multiplication, endowed with the structure sheaf of piecewise affine, convex functions with integral slopes. We show that pointwise [0, ∞) x N* coincides with the adele class space of Q and that this latter space inherits the geometric structure of a tropical curve. We restrict this construction to the periodic orbit of the scaling flow associated to each prime p and obtain a quasi-tropical structure which turns this orbit into a variant Cp = R∗+/pZ of the classical Jacobi description C∗/qZ of an elliptic curve. On Cp, we develop the theory of Cartier divisors, determine the structure of the quotient Div(Cp)/P of the abelian group of divisors by the subgroup of principal divisors, develop the theory of theta functions, and prove the Riemann-Roch formula which involves real valued dimensions, as in the type II index theory. We show that one would have been led to the same definition of S by analyzing the well known results on the localization of zeros of analytic functions involving Newton polygons in the non-archimedean case and the Jensen’s formula in the complex case. |

246. | An essay on the Riemann hypothesis Alain Connes Open problems in mathematics, pp. 225–257, Springer, [Cham], 2016. IncollectionAbstract @incollection{MR3526936, title = {An essay on the Riemann hypothesis}, author = {Alain Connes}, url = {/wp-content/uploads/EssayRH.pdf}, year = {2016}, date = {2016-01-01}, booktitle = {Open problems in mathematics}, pages = {225--257}, publisher = {Springer, [Cham]}, abstract = {The Riemann hypothesis is, and will hopefully remain for a long time, a great moti- vation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is “analytic" and is based on Riemannian spaces and Selberg’s work on the trace formula and its comparison with the explicit formulas. The second is based on algebraic geometry and the Riemann-Roch theorem. We establish a framework in which one can transpose many of the ingredients of the Weil proof as reformulated by Mattuck, Tate and Grothendieck. This framework is elaborate and involves noncommutative geometry, Grothendieck toposes and tropical geometry. We point out the remaining difficulties and show that RH gives a strong moti- vation to develop algebraic geometry in the emerging world of characteristic one. Finally we briefly discuss a third strategy based on the development of a suitable “Weil coho- mology", the role of Segal’s Γ-rings and of topological cyclic homology as a model for “absolute algebra" and as a cohomological tool.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } The Riemann hypothesis is, and will hopefully remain for a long time, a great moti- vation to uncover and explore new parts of the mathematical world. After reviewing its impact on the development of algebraic geometry we discuss three strategies, working concretely at the level of the explicit formulas. The first strategy is “analytic" and is based on Riemannian spaces and Selberg’s work on the trace formula and its comparison with the explicit formulas. The second is based on algebraic geometry and the Riemann-Roch theorem. We establish a framework in which one can transpose many of the ingredients of the Weil proof as reformulated by Mattuck, Tate and Grothendieck. This framework is elaborate and involves noncommutative geometry, Grothendieck toposes and tropical geometry. We point out the remaining difficulties and show that RH gives a strong moti- vation to develop algebraic geometry in the emerging world of characteristic one. Finally we briefly discuss a third strategy based on the development of a suitable “Weil coho- mology", the role of Segal’s Γ-rings and of topological cyclic homology as a model for “absolute algebra" and as a cohomological tool. |

245. | Geometry of the arithmetic site Alain Connes, Caterina Consani Adv. Math., 291 , pp. 274–329, 2016, ISSN: 0001-8708. Journal ArticleAbstract @article{MR3459019, title = {Geometry of the arithmetic site}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/arithmeticsite.pdf}, doi = {10.1016/j.aim.2015.11.045}, issn = {0001-8708}, year = {2016}, date = {2016-01-01}, journal = {Adv. Math.}, volume = {291}, pages = {274--329}, abstract = {We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the ad`ele class space of the field of rational numbers by the maximal compact subgroup of the id`ele class group, which we had previously shown to yield the correct counting function to obtain the complete Riemann zeta function as Hasse-Weil zeta function, is the set of geometric points of the arithmetic site over the semifield of tropical real numbers. The action of the multiplicative group of positive real numbers on the ad`ele class space corresponds to the action of the Frobenius automorphisms on the above geometric points. The underlying topological space of the arithmetic site is the topos of functors from the multiplicative semigroup of non-zero natural numbers to the category of sets. The structure sheaf is made by semirings of characteristic one and is given globally by the semifield of tropical integers. In spite of the countable combinatorial nature of the arithmetic site, this space admits a one parameter semigroup of Frobenius correspondences obtained as sub-varieties of the square of the site. This square is a semi-ringed topos whose structure sheaf involves Newton polygons. Finally, we show that the arithmetic site is intimately related to the structure of the (absolute) point in non-commutative geometry.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the ad`ele class space of the field of rational numbers by the maximal compact subgroup of the id`ele class group, which we had previously shown to yield the correct counting function to obtain the complete Riemann zeta function as Hasse-Weil zeta function, is the set of geometric points of the arithmetic site over the semifield of tropical real numbers. The action of the multiplicative group of positive real numbers on the ad`ele class space corresponds to the action of the Frobenius automorphisms on the above geometric points. The underlying topological space of the arithmetic site is the topos of functors from the multiplicative semigroup of non-zero natural numbers to the category of sets. The structure sheaf is made by semirings of characteristic one and is given globally by the semifield of tropical integers. In spite of the countable combinatorial nature of the arithmetic site, this space admits a one parameter semigroup of Frobenius correspondences obtained as sub-varieties of the square of the site. This square is a semi-ringed topos whose structure sheaf involves Newton polygons. Finally, we show that the arithmetic site is intimately related to the structure of the (absolute) point in non-commutative geometry. |

244. | Absolute algebra and Segal's Γ-rings Alain Connes, Caterina Consani J. Number Theory, 162 , pp. 518–551, 2016, ISSN: 0022-314X. Journal ArticleAbstract @article{MR3448278, title = {Absolute algebra and Segal's Γ-rings}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/Segalgammarings.pdf}, doi = {10.1016/j.jnt.2015.12.002}, issn = {0022-314X}, year = {2016}, date = {2016-01-01}, journal = {J. Number Theory}, volume = {162}, pages = {518--551}, abstract = {We show that the basic categorical concept of an S-algebra as derived from the theory of Segal’s Γ -sets provides a unifying description of several construc- tions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoïds, semirings and hyperrings as well as the development by means of monads and generalized rings in Arakelov geometry. The assembly map determines a functorial way to associate an S-algebra to a monad on pointed sets. The notion of an S-algebra is very familiar in algebraic topology where it also provides a suitable groundwork to the definition of topological cyclic homol- ogy. The main contribution of this paper is to point out its relevance and unifying role in arithmetic, in relation with the development of an algebraic geometry over symmetric closed monoidal categories.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We show that the basic categorical concept of an S-algebra as derived from the theory of Segal’s Γ -sets provides a unifying description of several construc- tions attempting to model an algebraic geometry over the absolute point. It merges, in particular, the approaches using monoïds, semirings and hyperrings as well as the development by means of monads and generalized rings in Arakelov geometry. The assembly map determines a functorial way to associate an S-algebra to a monad on pointed sets. The notion of an S-algebra is very familiar in algebraic topology where it also provides a suitable groundwork to the definition of topological cyclic homol- ogy. The main contribution of this paper is to point out its relevance and unifying role in arithmetic, in relation with the development of an algebraic geometry over symmetric closed monoidal categories. |

243. | Alain Connes, Caterina Consani C. R. Math. Acad. Sci. Paris, 354 (1), pp. 1–6, 2016, ISSN: 1631-073X. Journal ArticleAbstract @article{Connes2016, title = {The scaling site}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/The-scaling-site-2016.pdf}, doi = {10.1016/j.crma.2015.09.027}, issn = {1631-073X}, year = {2016}, date = {2016-01-01}, journal = {C. R. Math. Acad. Sci. Paris}, volume = {354}, number = {1}, pages = {1--6}, abstract = {We investigate the semi-ringed topos obtained from the arithmetic site A of [3,4], by max extension of scalars from the smallest Boolean semifield B to the tropical semifield R+ . The obtained site [0, ∞) ? N× is the semi-direct product of the Euclidean half-line and the monoid N× of positive integers acting by multiplication. Its points are the same as the points A (Rmax) of A over Rmax and form the quotient of the adele class space of Q by ++ the action of the maximal compact subgroup Zˆ∗ of the idèle class group. The structure sheaf of the scaling topos endows it with a natural structure of tropical curve over the topos N?×. The restriction of this structure to the periodic orbits of the scaling flow gives, for each prime p, an analogue C p of an elliptic curve whose Jacobian is Z/(p − 1)Z. The Riemann–Roch formula holds on Cp and involves real-valued dimensions and real degrees for divisors.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We investigate the semi-ringed topos obtained from the arithmetic site A of [3,4], by max extension of scalars from the smallest Boolean semifield B to the tropical semifield R+ . The obtained site [0, ∞) ? N× is the semi-direct product of the Euclidean half-line and the monoid N× of positive integers acting by multiplication. Its points are the same as the points A (Rmax) of A over Rmax and form the quotient of the adele class space of Q by ++ the action of the maximal compact subgroup Zˆ∗ of the idèle class group. The structure sheaf of the scaling topos endows it with a natural structure of tropical curve over the topos N?×. The restriction of this structure to the periodic orbits of the scaling flow gives, for each prime p, an analogue C p of an elliptic curve whose Jacobian is Z/(p − 1)Z. The Riemann–Roch formula holds on Cp and involves real-valued dimensions and real degrees for divisors. |

242. | Alain Connes, Vaughan Jones, Magdalena Musat, Mikael Rørdam Notices Amer. Math. Soc., 63 (1), pp. 48–49, 2016, ISSN: 0002-9920. Journal ArticleAbstract @article{MR3410853, title = {Uffe Haagerup—In memoriam}, author = {Alain Connes and Vaughan Jones and Magdalena Musat and Mikael Rørdam}, url = {/wp-content/uploads/UffeH.pdf}, doi = {10.1090/noti1303}, issn = {0002-9920}, year = {2016}, date = {2016-01-01}, journal = {Notices Amer. Math. Soc.}, volume = {63}, number = {1}, pages = {48--49}, abstract = {Uffe Haagerup, a world-renowned analyst and leading figure in operator algebras, passed away on July 5, 2015, in a tragic drowning accident near his summer house in Faaborg, Denmark. He spent most of his career as a professor at the University of Southern Denmark (Odense), and during 2010– 2014 he was a professor at the University of Copenhagen while holding an Advanced European Research Council (ERC) grant. Uffe was a uniquely gifted mathematician of incredible analytic power and insight, which he generously shared with his many collaborators. His kindness and warm personality were greatly valued by his many friends and colleagues in Denmark and abroad. Uffe was born December 19, 1949, in the town of Kolding, but grew up in Faaborg, the younger of two brothers. At age ten Uffe started helping the local surveyor and solved difficult trigonometric problems in the process. His problem-solving skills earned him national acclaim when a few years later his proposed development plan for a large summer house area was chosen over the one of a Copenhagen contractor. }, keywords = {}, pubstate = {published}, tppubtype = {article} } Uffe Haagerup, a world-renowned analyst and leading figure in operator algebras, passed away on July 5, 2015, in a tragic drowning accident near his summer house in Faaborg, Denmark. He spent most of his career as a professor at the University of Southern Denmark (Odense), and during 2010– 2014 he was a professor at the University of Copenhagen while holding an Advanced European Research Council (ERC) grant. Uffe was a uniquely gifted mathematician of incredible analytic power and insight, which he generously shared with his many collaborators. His kindness and warm personality were greatly valued by his many friends and colleagues in Denmark and abroad. Uffe was born December 19, 1949, in the town of Kolding, but grew up in Faaborg, the younger of two brothers. At age ten Uffe started helping the local surveyor and solved difficult trigonometric problems in the process. His problem-solving skills earned him national acclaim when a few years later his proposed development plan for a large summer house area was chosen over the one of a Copenhagen contractor. |

241. | Grand unification in the spectral Pati–Salam model Ali H. Chamseddine, Alain Connes, Walter D. van Suijlekom J. High Energy Phys., (11), pp. 011, front matter+12, 2015, ISSN: 1126-6708. Journal ArticleAbstract @article{MR3455566, title = {Grand unification in the spectral Pati–Salam model}, author = {Ali H. Chamseddine and Alain Connes and Walter D. van Suijlekom}, url = {/wp-content/uploads/GUpatisalam.pdf}, doi = {10.1007/JHEP11(2015)011}, issn = {1126-6708}, year = {2015}, date = {2015-01-01}, journal = {J. High Energy Phys.}, number = {11}, pages = {011, front matter+12}, abstract = {We analyze the running at one-loop of the gauge couplings in the spectral Pati–Salam model that was derived in the framework of noncommutative geometry. There are a few different scenario’s for the scalar particle content which are determined by the precise form of the Dirac operator for the finite noncommutative space. We consider these different scenarios and establish for all of them unification of the Pati–Salam gauge couplings. The boundary conditions are set by the usual RG flow for the Standard Model couplings at an intermediate mass scale at which the Pati–Salam symmetry is broken.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We analyze the running at one-loop of the gauge couplings in the spectral Pati–Salam model that was derived in the framework of noncommutative geometry. There are a few different scenario’s for the scalar particle content which are determined by the precise form of the Dirac operator for the finite noncommutative space. We consider these different scenarios and establish for all of them unification of the Pati–Salam gauge couplings. The boundary conditions are set by the usual RG flow for the Standard Model couplings at an intermediate mass scale at which the Pati–Salam symmetry is broken. |

240. | Quanta of geometry: noncommutative aspects Ali H. Chamseddine, Alain Connes, Viatcheslav Mukhanov Phys. Rev. Lett., 114 (9), pp. 091302, 5, 2015, ISSN: 0031-9007. Journal ArticleAbstract @article{MR3437516, title = {Quanta of geometry: noncommutative aspects}, author = {Ali H. Chamseddine and Alain Connes and Viatcheslav Mukhanov}, url = {/wp-content/uploads/quantaofgeomCCM.pdf}, doi = {10.1103/PhysRevLett.114.091302}, issn = {0031-9007}, year = {2015}, date = {2015-01-01}, journal = {Phys. Rev. Lett.}, volume = {114}, number = {9}, pages = {091302, 5}, abstract = {In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin-manifolds with large quantized volume are then obtained as solutions. The two algebras M2 (H) and M4 (C) are obtained which are the exact constituents of the Standard Model. Using the two maps from M4 to S4 the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter and area quantization of black holes.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin-manifolds with large quantized volume are then obtained as solutions. The two algebras M2 (H) and M4 (C) are obtained which are the exact constituents of the Standard Model. Using the two maps from M4 to S4 the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter and area quantization of black holes. |

239. | The cyclic and epicyclic sites Alain Connes, Caterina Consani Rend. Semin. Mat. Univ. Padova, 134 , pp. 197–237, 2015, ISSN: 0041-8994. Journal ArticleAbstract @article{MR3428418, title = {The cyclic and epicyclic sites}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/The-cyclic-and-epicyclic-sites-2015.pdf}, doi = {10.4171/RSMUP/134-5}, issn = {0041-8994}, year = {2015}, date = {2015-01-01}, journal = {Rend. Semin. Mat. Univ. Padova}, volume = {134}, pages = {197--237}, abstract = {We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of “max-plus integers” Zmax. An object of this category is a pair (E,K) of a semimodule E over an algebraic extension K of Zmax. The morphisms are projective classes of semilinear maps between semimodules. The epicyclic topos sits over the arithmetic topos N̂× of [6] and the fibers of the associated geometric morphism correspond to the cyclic site. In two appendices we review the role of the cyclic and epicyclic toposes as the geometric structures supporting cyclic homology and the lambda operations.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of “max-plus integers” Zmax. An object of this category is a pair (E,K) of a semimodule E over an algebraic extension K of Zmax. The morphisms are projective classes of semilinear maps between semimodules. The epicyclic topos sits over the arithmetic topos N̂× of [6] and the fibers of the associated geometric morphism correspond to the cyclic site. In two appendices we review the role of the cyclic and epicyclic toposes as the geometric structures supporting cyclic homology and the lambda operations. |

238. | Universal thickening of the field of real numbers Alain Connes, Caterina Consani Advances in the theory of numbers, 77 , pp. 11–74, Fields Inst. Res. Math. Sci., Toronto, ON, 2015. IncollectionAbstract @incollection{MR3409323, title = {Universal thickening of the field of real numbers}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/Universal-thickening-of-the-field-of-real-numbers-2015.pdf}, doi = {10.1007/978-1-4939-3201-6_2}, year = {2015}, date = {2015-01-01}, booktitle = {Advances in the theory of numbers}, volume = {77}, pages = {11--74}, publisher = {Fields Inst. Res. Math. Sci., Toronto, ON}, series = {Fields Inst. Commun.}, abstract = {This paper establishes an analogue of the construction of the rings of periods of p-adic Hodge theory (cf. e.g. [12, 13, 14]) when a p-adic field is replaced by the field R of real numbers. We show that the original ideas of M. Krasner, which were motivated by the correspondence he first unveiled between Galois theories in unequal characteristics [18], reappear unavoidably when the above analogy is developed. The interest in pursuing this construction is enhanced by our recent dis- covery of the Arithmetic Site [7] with its structure sheaf of semirings of characteristic 1, whose geometric points involve in a crucial manner the tropical semifield max R+ . The encounter of a structure of characteristic 1 which is deeply related to the non-commutative geometric approach to the Riemann Hypothesis has motivated our search for the replacement of the p-adic constructions at the real archimedean place.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } This paper establishes an analogue of the construction of the rings of periods of p-adic Hodge theory (cf. e.g. [12, 13, 14]) when a p-adic field is replaced by the field R of real numbers. We show that the original ideas of M. Krasner, which were motivated by the correspondence he first unveiled between Galois theories in unequal characteristics [18], reappear unavoidably when the above analogy is developed. The interest in pursuing this construction is enhanced by our recent dis- covery of the Arithmetic Site [7] with its structure sheaf of semirings of characteristic 1, whose geometric points involve in a crucial manner the tropical semifield max R+ . The encounter of a structure of characteristic 1 which is deeply related to the non-commutative geometric approach to the Riemann Hypothesis has motivated our search for the replacement of the p-adic constructions at the real archimedean place. |

237. | Projective geometry in characteristic one and the epicyclic category Alain Connes, Caterina Consani Nagoya Math. J., 217 , pp. 95–132, 2015, ISSN: 0027-7630. Journal ArticleAbstract @article{MR3343840, title = {Projective geometry in characteristic one and the epicyclic category}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/Projective-geometry-in-characteristic-one-and-the-epicyclic-site-2015.pdf}, doi = {10.1215/00277630-2887960}, issn = {0027-7630}, year = {2015}, date = {2015-01-01}, journal = {Nagoya Math. J.}, volume = {217}, pages = {95--132}, abstract = {We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield of “max-plus integers” Zmax. Finite dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of Zmax. The associated projective spaces are finite and provide a mathematically consistent interpretation of J. Tits’ original idea of a geometry over the absolute point. The self- duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield of “max-plus integers” Zmax. Finite dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of Zmax. The associated projective spaces are finite and provide a mathematically consistent interpretation of J. Tits’ original idea of a geometry over the absolute point. The self- duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning. |

236. | Cyclic structures and the topos of simplicial sets Alain Connes, Caterina Consani J. Pure Appl. Algebra, 219 (4), pp. 1211–1235, 2015, ISSN: 0022-4049. Journal ArticleAbstract @article{MR3282133, title = {Cyclic structures and the topos of simplicial sets}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/Cyclic-structures-and-the-topos-2015.pdf}, doi = {10.1016/j.jpaa.2014.06.002}, issn = {0022-4049}, year = {2015}, date = {2015-01-01}, journal = {J. Pure Appl. Algebra}, volume = {219}, number = {4}, pages = {1211--1235}, abstract = {Given a point p of the topos Δ of simplicial sets and the corresponding flat covariant functor F : Δ −→ Sets, we determine the extensions of F to the cyclic category ˆΛ ⊃ Δ. We show that to each such cyclic structure on a point p of Δ corresponds a group Gp, that such groups can be noncommutative and that each Gp is described as the quotient of a left-ordered group by the subgroup generated by a central element. Moreover for any cyclic set X the fiber (or geometric realization) of the underlying simplicial set of X at p inherits canonically the structure of a Gp-space. This gives a far reaching generalization of the well-known circle action on the geometric realization of cyclic sets.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Given a point p of the topos Δ of simplicial sets and the corresponding flat covariant functor F : Δ −→ Sets, we determine the extensions of F to the cyclic category ˆΛ ⊃ Δ. We show that to each such cyclic structure on a point p of Δ corresponds a group Gp, that such groups can be noncommutative and that each Gp is described as the quotient of a left-ordered group by the subgroup generated by a central element. Moreover for any cyclic set X the fiber (or geometric realization) of the underlying simplicial set of X at p inherits canonically the structure of a Gp-space. This gives a far reaching generalization of the well-known circle action on the geometric realization of cyclic sets. |

235. | Geometry and the quantum: basics Ali H. Chamseddine, Alain Connes, Viatcheslav Mukhanov J. High Energy Phys., (12), pp. 098, front matter+24, 2014, ISSN: 1126-6708. Journal ArticleAbstract @article{Chamseddine2014, title = {Geometry and the quantum: basics}, author = {Ali H. Chamseddine and Alain Connes and Viatcheslav Mukhanov}, url = {/wp-content/uploads/geometryquantumCCM.pdf}, doi = {10.1007/JHEP12(2014)098}, issn = {1126-6708}, year = {2014}, date = {2014-01-01}, journal = {J. High Energy Phys.}, number = {12}, pages = {098, front matter+24}, abstract = {Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will rep- resent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M2(H) and M4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non- commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume > 4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the “particle picture” for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will rep- resent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M2(H) and M4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non- commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume > 4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the “particle picture” for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication. |

234. | Alain Connes, Caterina Consani C. R. Math. Acad. Sci. Paris, 352 (12), pp. 971–975, 2014, ISSN: 1631-073X. Journal ArticleAbstract @article{MR3276804, title = {The arithmetic site}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/The-arithmetic-site-2014.pdf}, doi = {10.1016/j.crma.2014.07.009}, issn = {1631-073X}, year = {2014}, date = {2014-01-01}, journal = {C. R. Math. Acad. Sci. Paris}, volume = {352}, number = {12}, pages = {971--975}, abstract = {We show that the non-commutative geometric approach to the Riemann zeta function has an algebraic geometric incarnation: the “Arithmetic Site”. This site involves the tropical ̄ ?× semiring N viewed as a sheaf on the topos N dual to the multiplicative semigroup of positive integers. We realize the Frobenius correspondences in the square of the “Arithmetic Site”.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We show that the non-commutative geometric approach to the Riemann zeta function has an algebraic geometric incarnation: the “Arithmetic Site”. This site involves the tropical ̄ ?× semiring N viewed as a sheaf on the topos N dual to the multiplicative semigroup of positive integers. We realize the Frobenius correspondences in the square of the “Arithmetic Site”. |

233. | On the arithmetic of the BC-system Alain Connes, Caterina Consani J. Noncommut. Geom., 8 (3), pp. 873–945, 2014, ISSN: 1661-6952. Journal ArticleAbstract @article{MR3261604, title = {On the arithmetic of the BC-system}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/On-the-arithmetic-of-the-BC-2014.pdf}, doi = {10.4171/JNCG/173}, issn = {1661-6952}, year = {2014}, date = {2014-01-01}, journal = {J. Noncommut. Geom.}, volume = {8}, number = {3}, pages = {873--945}, abstract = {For each prime p and each embedding σ of the multiplicative group of an algebraic closure of Fp as complex roots of unity, we construct a p-adic indecomposable representation πσ of the integral BC-system as additive endomorphisms of the big Witt ̄ring of the algebraic closure of Fp. The obtained representations are the p-adic analogues of the complex extremal KMS∞ states of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over C is replaced, in the p-adic case, by the p-adic L-functions and the polylogarithms whose values at roots of unity encode the KMS states. We use Iwasawa theory to extend the KMS theory to a covering of the completion Cp of an algebraic closure of Qp. We show that our previous work on the hyperring structure of the ad`ele class space, combines with p-adic analysis to refine the space of valuations on the cyclotomic extension of Q as a noncommutative space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the “arithmetic site”. }, keywords = {}, pubstate = {published}, tppubtype = {article} } For each prime p and each embedding σ of the multiplicative group of an algebraic closure of Fp as complex roots of unity, we construct a p-adic indecomposable representation πσ of the integral BC-system as additive endomorphisms of the big Witt ̄ring of the algebraic closure of Fp. The obtained representations are the p-adic analogues of the complex extremal KMS∞ states of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over C is replaced, in the p-adic case, by the p-adic L-functions and the polylogarithms whose values at roots of unity encode the KMS states. We use Iwasawa theory to extend the KMS theory to a covering of the completion Cp of an algebraic closure of Qp. We show that our previous work on the hyperring structure of the ad`ele class space, combines with p-adic analysis to refine the space of valuations on the cyclotomic extension of Q as a noncommutative space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the “arithmetic site”. |

232. | Cyclic homology, Serre's local factors and λ-operations Alain Connes, Caterina Consani J. K-Theory, 14 (1), pp. 1–45, 2014, ISSN: 1865-2433. Journal ArticleAbstract @article{MR3238256, title = {Cyclic homology, Serre's local factors and λ-operations}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/Cyclic-homology-Serre-local-2014.pdf}, doi = {10.1017/is014006012jkt270}, issn = {1865-2433}, year = {2014}, date = {2014-01-01}, journal = {J. K-Theory}, volume = {14}, number = {1}, pages = {1--45}, abstract = {We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring A1 D Q?j1K?, provides the right theory to obtain, using ?-operations, Serre’s archimedean local factors of the complex L-function of X as regularized determinants.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring A1 D Q?j1K?, provides the right theory to obtain, using ?-operations, Serre’s archimedean local factors of the complex L-function of X as regularized determinants. |

231. | Modular curvature for noncommutative two-tori Alain Connes, Henri Moscovici J. Amer. Math. Soc., 27 (3), pp. 639–684, 2014, ISSN: 0894-0347. Journal ArticleAbstract @article{MR3194491, title = {Modular curvature for noncommutative two-tori}, author = {Alain Connes and Henri Moscovici}, url = {/wp-content/uploads/modularcurvature.pdf}, doi = {10.1090/S0894-0347-2014-00793-1}, issn = {0894-0347}, year = {2014}, date = {2014-01-01}, journal = {J. Amer. Math. Soc.}, volume = {27}, number = {3}, pages = {639--684}, abstract = { In this paper we investigate the curvature of conformal deforma- tions by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two functions in the modular operator corresponding to the non-tracial weight defined by the conformal factor, applied to expressions involving the derivatives of the same factor. The first is a generating function for the Bernoulli numbers and is applied to the noncommutative Laplacian of the conformal factor, while the second is a two-variable function and is applied to a quadratic form in the first derivatives of the factor. Further outcomes of the paper include a variational proof of the Gauss-Bonnet theorem for non- commutative 2-tori, the modular analogue of Polyakov’s conformal anomaly formula for regularized determinants of Laplacians, a conceptual understand- ing of the modular curvature as gradient of the Ray-Singer analytic torsion, and the proof using operator positivity that the scale invariant version of the latter assumes its extreme value only at the flat metric.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper we investigate the curvature of conformal deforma- tions by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two functions in the modular operator corresponding to the non-tracial weight defined by the conformal factor, applied to expressions involving the derivatives of the same factor. The first is a generating function for the Bernoulli numbers and is applied to the noncommutative Laplacian of the conformal factor, while the second is a two-variable function and is applied to a quadratic form in the first derivatives of the factor. Further outcomes of the paper include a variational proof of the Gauss-Bonnet theorem for non- commutative 2-tori, the modular analogue of Polyakov’s conformal anomaly formula for regularized determinants of Laplacians, a conceptual understand- ing of the modular curvature as gradient of the Ray-Singer analytic torsion, and the proof using operator positivity that the scale invariant version of the latter assumes its extreme value only at the flat metric. |

230. | Alain Connes, Danye Chéreau, Jacques Dixmier 2013, ISBN: 9782738129833. BookAbstract @book{Connes2013, title = {Le théâtre quantique}, author = {Alain Connes and Danye Chéreau and Jacques Dixmier}, isbn = {9782738129833}, year = {2013}, date = {2013-05-07}, abstract = {Ce livre vous propose de penetrer au coeur du theatre quantique, en vous offrant une voie d'acces rapide a l'univers magique de la mecanique quantique, decouverte essentielle du XXe siecle qui defie notre intuition. Vous vous prendrez a rever de ces photons capricieux, de leurs etats intriques, de ces experiences troublantes aux conclusions inattendues. Voici une fantaisie initiatique qui aborde de maniere innovante le probleme du temps, les paradoxes de la mecanique quantique et les interrogations sur la simulation des fonctions cerebrales, a travers une intrigue policiere originale et les aventures d'une physicienne attachante, passionnee et prete a tout. Laissez-vous entrainer dans un monde enchante ou l'alea quantique est le tic-tac de l'horloge divine. Alain Connes est mathematicien, professeur au College de France, titulaire de la chaire d'Analyse et Geometrie, membre de l'Academie des sciences et de plusieurs academies etrangeres dont la National Academy of Sciences des Etats-Unis. Il a obtenu la medaille Fields en 1982. Danye Chereau a une formation litteraire. Refractaire aux mathematiques dans sa jeunesse, les surprises de la vie lui ont permis de decouvrir le milieu scientifique et le monde des chercheurs sur lesquels elle porte un regard toujours curieux, amuse et attendri. Jacques Dixmier a professe aux universites de Toulouse, Dijon, Paris. Il est mathematicien pur, mais certains de ses domaines de recherche (algebres d'operateurs, representations des groupes, algebres enveloppantes) sont utiles en mecanique quantique. Il a aussi publie des nouvelles de science-fiction.}, keywords = {}, pubstate = {published}, tppubtype = {book} } Ce livre vous propose de penetrer au coeur du theatre quantique, en vous offrant une voie d'acces rapide a l'univers magique de la mecanique quantique, decouverte essentielle du XXe siecle qui defie notre intuition. Vous vous prendrez a rever de ces photons capricieux, de leurs etats intriques, de ces experiences troublantes aux conclusions inattendues. Voici une fantaisie initiatique qui aborde de maniere innovante le probleme du temps, les paradoxes de la mecanique quantique et les interrogations sur la simulation des fonctions cerebrales, a travers une intrigue policiere originale et les aventures d'une physicienne attachante, passionnee et prete a tout. Laissez-vous entrainer dans un monde enchante ou l'alea quantique est le tic-tac de l'horloge divine. Alain Connes est mathematicien, professeur au College de France, titulaire de la chaire d'Analyse et Geometrie, membre de l'Academie des sciences et de plusieurs academies etrangeres dont la National Academy of Sciences des Etats-Unis. Il a obtenu la medaille Fields en 1982. Danye Chereau a une formation litteraire. Refractaire aux mathematiques dans sa jeunesse, les surprises de la vie lui ont permis de decouvrir le milieu scientifique et le monde des chercheurs sur lesquels elle porte un regard toujours curieux, amuse et attendri. Jacques Dixmier a professe aux universites de Toulouse, Dijon, Paris. Il est mathematicien pur, mais certains de ses domaines de recherche (algebres d'operateurs, representations des groupes, algebres enveloppantes) sont utiles en mecanique quantique. Il a aussi publie des nouvelles de science-fiction. |

229. | Inner fluctuations in noncommutative geometry without the first order condition Ali H. Chamseddine, Alain Connes, Walter D. van Suijlekom J. Geom. Phys., 73 , pp. 222–234, 2013, ISSN: 0393-0440. Journal ArticleAbstract @article{MR3090113, title = {Inner fluctuations in noncommutative geometry without the first order condition}, author = {Ali H. Chamseddine and Alain Connes and Walter D. van Suijlekom}, url = {/wp-content/uploads/innerfluctuations.pdf}, doi = {10.1016/j.geomphys.2013.06.006}, issn = {0393-0440}, year = {2013}, date = {2013-01-01}, journal = {J. Geom. Phys.}, volume = {73}, pages = {222--234}, abstract = {We extend inner fluctuations to spectral triples that do not fulfill the first-order condition. This involves the addition of a quadratic term to the usual linear terms. We find a semi-group of inner fluctuations, which only depends on the involutive algebra A and which extends the unitary group of A. This has a key application in noncommutative spectral models beyond the Standard Model, of which we consider here a toy model.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We extend inner fluctuations to spectral triples that do not fulfill the first-order condition. This involves the addition of a quadratic term to the usual linear terms. We find a semi-group of inner fluctuations, which only depends on the involutive algebra A and which extends the unitary group of A. This has a key application in noncommutative spectral models beyond the Standard Model, of which we consider here a toy model. |

228. | On the spectral characterization of manifolds Alain Connes J. Noncommut. Geom., 7 (1), pp. 1–82, 2013, ISSN: 1661-6952. Journal ArticleAbstract @article{MR3032810, title = {On the spectral characterization of manifolds}, author = {Alain Connes}, url = {/wp-content/uploads/spectramaniflods.pdf}, doi = {10.4171/JNCG/108}, issn = {1661-6952}, year = {2013}, date = {2013-01-01}, journal = {J. Noncommut. Geom.}, volume = {7}, number = {1}, pages = {1--82}, abstract = {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 compact manifolds. The algebra, which is assumed to be commutative, is shown to be isomorphic to the algebra of all smooth functions on a unique smooth oriented compact manifold, while the operator is shown to be of Dirac type and the metric to be Riemannian.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 compact manifolds. The algebra, which is assumed to be commutative, is shown to be isomorphic to the algebra of all smooth functions on a unique smooth oriented compact manifold, while the operator is shown to be of Dirac type and the metric to be Riemannian. |

227. | Resilience of the spectral standard model Ali H. Chamseddine, Alain Connes J. High Energy Phys., (9), pp. 104, front matter+10, 2012, ISSN: 1126-6708. Journal ArticleAbstract @article{MR3044924, title = {Resilience of the spectral standard model}, author = {Ali H. Chamseddine and Alain Connes}, url = {/wp-content/uploads/Resilience-2.pdf}, doi = {10.1007/JHEP09(2012)104}, issn = {1126-6708}, year = {2012}, date = {2012-01-01}, journal = {J. High Energy Phys.}, number = {9}, pages = {104, front matter+10}, abstract = {We show that the inconsistency between the spectral Standard Model and the experimental value of the Higgs mass is resolved by the presence of a real scalar field strongly coupled to the Higgs field. This scalar field was already present in the spectral model and we wrongly neglected it in our previous computations. It was shown recently by several authors, independently of the spectral approach, that such a strongly coupled scalar field stabilizes the Standard Model up to unification scale in spite of the low value of the Higgs mass. In this letter we show that the noncommutative neutral singlet modifies substantially the RG analysis, invalidates our previous prediction of Higgs mass in the range 160–180 Gev, and restores the consistency of the noncommutative geometric model with the low Higgs mass.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We show that the inconsistency between the spectral Standard Model and the experimental value of the Higgs mass is resolved by the presence of a real scalar field strongly coupled to the Higgs field. This scalar field was already present in the spectral model and we wrongly neglected it in our previous computations. It was shown recently by several authors, independently of the spectral approach, that such a strongly coupled scalar field stabilizes the Standard Model up to unification scale in spite of the low value of the Higgs mass. In this letter we show that the noncommutative neutral singlet modifies substantially the RG analysis, invalidates our previous prediction of Higgs mass in the range 160–180 Gev, and restores the consistency of the noncommutative geometric model with the low Higgs mass. |

226. | Spectral action for Robertson-Walker metrics Ali H. Chamseddine, Alain Connes J. High Energy Phys., (10), pp. 101, front matter + 29, 2012, ISSN: 1126-6708. Journal ArticleAbstract @article{MR3033848, title = {Spectral action for Robertson-Walker metrics}, author = {Ali H. Chamseddine and Alain Connes}, url = {/wp-content/uploads/Robertsonwalker.pdf}, doi = {10.1007/JHEP10(2012)101}, issn = {1126-6708}, year = {2012}, date = {2012-01-01}, journal = {J. High Energy Phys.}, number = {10}, pages = {101, front matter + 29}, abstract = {We use the Euler–Maclaurin formula and the Feynman–Kac formula to extend our previous method of computation of the spectral action based on the Poisson summation formula. We show how to compute directly the spectral action for the general case of Robertson–Walker metrics. We check the terms of the expansion up to a6 against the known universal formulas of Gilkey and compute the expansion up to a10 using our direct method.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We use the Euler–Maclaurin formula and the Feynman–Kac formula to extend our previous method of computation of the spectral action based on the Poisson summation formula. We show how to compute directly the spectral action for the general case of Robertson–Walker metrics. We check the terms of the expansion up to a6 against the known universal formulas of Gilkey and compute the expansion up to a10 using our direct method. |

225. | The Gauss-Bonnet theorem for the noncommutative two torus Alain Connes, Paula Tretkoff Noncommutative geometry, arithmetic, and related topics, pp. 141–158, Johns Hopkins Univ. Press, Baltimore, MD, 2011. IncollectionAbstract @incollection{MR2907006, title = {The Gauss-Bonnet theorem for the noncommutative two torus}, author = {Alain Connes and Paula Tretkoff}, url = {/wp-content/uploads/gaussbonnet.pdf}, year = {2011}, date = {2011-01-01}, booktitle = {Noncommutative geometry, arithmetic, and related topics}, pages = {141--158}, publisher = {Johns Hopkins Univ. Press, Baltimore, MD}, abstract = {In this paper we shall show that the value at the origin, ζ(0), of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent of the choice of the volume element (Weyl factor) given by a (non-unimodular) state. We had obtained, in the late eighties, in an unpublished computation, a general formula for ζ(0) involving modified logarithms of the modular operator of the state. We give here the detailed computation and prove that the result is independent of the Weyl factor as in the classical case, thus proving the analogue of the Gauss-Bonnet theorem for the noncommutative two torus.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } In this paper we shall show that the value at the origin, ζ(0), of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent of the choice of the volume element (Weyl factor) given by a (non-unimodular) state. We had obtained, in the late eighties, in an unpublished computation, a general formula for ζ(0) involving modified logarithms of the modular operator of the state. We give here the detailed computation and prove that the result is independent of the Weyl factor as in the classical case, thus proving the analogue of the Gauss-Bonnet theorem for the noncommutative two torus. |

224. | Characteristic 1, entropy and the absolute point Alain Connes, Caterina Consani Noncommutative geometry, arithmetic, and related topics, pp. 75–139, Johns Hopkins Univ. Press, Baltimore, MD, 2011. IncollectionAbstract @incollection{MR2907005, title = {Characteristic 1, entropy and the absolute point}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/char1absolutepoint.pdf}, year = {2011}, date = {2011-01-01}, booktitle = {Noncommutative geometry, arithmetic, and related topics}, pages = {75--139}, publisher = {Johns Hopkins Univ. Press, Baltimore, MD}, abstract = {We show that the mathematical meaning of working in character- istic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point SpecF1. After introducing the notion of “perfect” semi-ring of characteristic one, we explain how to adapt the construction of the Witt ring in characteris- tic p > 1 to the limit case of characteristic one. This construction also unveils an interesting connection with entropy and thermodynamics, while shedding a new light on the classical Witt construction itself. We simplify our earlier construction of the geometric realization of an F1-scheme and extend our ear- lier computations of the zeta function to cover the case of F1-schemes with torsion. Then, we show that the study of the additive structures on monoids provides a natural map M ?→ A(M) from monoids to sets which comes close to fulfill the requirements for the hypothetical curve Spec Z over the absolute point SpecF1. Finally, we test the computation of the zeta function on elliptic curves over the rational numbers.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We show that the mathematical meaning of working in character- istic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point SpecF1. After introducing the notion of “perfect” semi-ring of characteristic one, we explain how to adapt the construction of the Witt ring in characteris- tic p > 1 to the limit case of characteristic one. This construction also unveils an interesting connection with entropy and thermodynamics, while shedding a new light on the classical Witt construction itself. We simplify our earlier construction of the geometric realization of an F1-scheme and extend our ear- lier computations of the zeta function to cover the case of F1-schemes with torsion. Then, we show that the study of the additive structures on monoids provides a natural map M ?→ A(M) from monoids to sets which comes close to fulfill the requirements for the hypothetical curve Spec Z over the absolute point SpecF1. Finally, we test the computation of the zeta function on elliptic curves over the rational numbers. |

223. | Alain Connes Jpn. J. Math., 6 (1), pp. 1–44, 2011, ISSN: 0289-2316. Journal ArticleAbstract @article{Connes2011, title = {The BC-system and ?-functions}, author = {Alain Connes}, url = {/wp-content/uploads/TheBC-systemAndL-functions.pdf}, doi = {10.1007/s11537-011-1035-0}, issn = {0289-2316}, year = {2011}, date = {2011-01-01}, journal = {Jpn. J. Math.}, volume = {6}, number = {1}, pages = {1--44}, abstract = {In these lectures we survey some relations between L-functions and the BC-system, including new results obtained in collaboration with C. Consani. For each prime p and embedding? of the multiplicative group of an algebraic closure of Fp as complex roots of unity, we construct a p-adic indecomposable representation ?? of the integral BC-system. This construction is done using the identification of the big Witt ring of Fp and by implementing the Artin–Hasse exponentials. The obtained representations are the p-adic analogues of the complex, extremal KMS1 states of the BC-system. We use the theory of p-adic L-functions to determine the par- tition function. Together with the analogue of the Witt construction in characteristic one, these results provide further evidence towards the construction of an analogue, for the global field of rational numbers, of the curve which provides the geometric support for the arithmetic of function fields.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In these lectures we survey some relations between L-functions and the BC-system, including new results obtained in collaboration with C. Consani. For each prime p and embedding? of the multiplicative group of an algebraic closure of Fp as complex roots of unity, we construct a p-adic indecomposable representation ?? of the integral BC-system. This construction is done using the identification of the big Witt ring of Fp and by implementing the Artin–Hasse exponentials. The obtained representations are the p-adic analogues of the complex, extremal KMS1 states of the BC-system. We use the theory of p-adic L-functions to determine the par- tition function. Together with the analogue of the Witt construction in characteristic one, these results provide further evidence towards the construction of an analogue, for the global field of rational numbers, of the curve which provides the geometric support for the arithmetic of function fields. |

222. | The Witt construction in characteristic one and quantization Alain Connes Noncommutative geometry and global analysis, 546 , pp. 83–113, Amer. Math. Soc., Providence, RI, 2011. IncollectionAbstract @incollection{MR2815131, title = {The Witt construction in characteristic one and quantization}, author = {Alain Connes}, url = {/wp-content/uploads/Wittcar1.pdf}, doi = {10.1090/conm/546/10785}, year = {2011}, date = {2011-01-01}, booktitle = {Noncommutative geometry and global analysis}, volume = {546}, pages = {83--113}, publisher = {Amer. Math. Soc., Providence, RI}, series = {Contemp. Math.}, abstract = {We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs (R,ρ) of a perfect semi-ring R of characteristic one and an element ρ > 1 of R to real Banach algebras. We find that the entropy function occurs uniquely as the analogue of the Teichmu ̈ller polynomials in characteristic one. We then apply the construction to the semi-field Rmax which plays a central role in idempotent analysis and tropical + geometry. Our construction gives the inverse process of the “dequantization” and provides a first hint towards an extension Run of the field of real numbers relevant both in number theory and quantum physics.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs (R,ρ) of a perfect semi-ring R of characteristic one and an element ρ > 1 of R to real Banach algebras. We find that the entropy function occurs uniquely as the analogue of the Teichmu ̈ller polynomials in characteristic one. We then apply the construction to the semi-field Rmax which plays a central role in idempotent analysis and tropical + geometry. Our construction gives the inverse process of the “dequantization” and provides a first hint towards an extension Run of the field of real numbers relevant both in number theory and quantum physics. |

221. | On the notion of geometry over ?₁ Alain Connes, Caterina Consani J. Algebraic Geom., 20 (3), pp. 525–557, 2011, ISSN: 1056-3911. Journal ArticleAbstract @article{MR2786665, title = {On the notion of geometry over ?₁}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/GeomoverF1.pdf}, doi = {10.1090/S1056-3911-2010-00535-8}, issn = {1056-3911}, year = {2011}, date = {2011-01-01}, journal = {J. Algebraic Geom.}, volume = {20}, number = {3}, pages = {525--557}, abstract = {We refine the notion of variety over the “field with one element” developed by C. Soul ́e by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soul ́e by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above “field”.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We refine the notion of variety over the “field with one element” developed by C. Soul ́e by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soul ́e by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above “field”. |

220. | L'hyperanneau des classes d'adèles Alain Connes J. Théor. Nombres Bordeaux, 23 (1), pp. 71–93, 2011, ISSN: 1246-7405. Journal ArticleAbstract @article{Connes2011b, title = {L'hyperanneau des classes d'adèles}, author = {Alain Connes}, url = {/wp-content/uploads/JTNBordeaux.pdf}, issn = {1246-7405}, year = {2011}, date = {2011-01-01}, journal = {J. Théor. Nombres Bordeaux}, volume = {23}, number = {1}, pages = {71--93}, abstract = {I present here some recent results (obtained in collaboration with C. Consani [3], [4], [5], [6]) about the “characteristic 1” limit case. The main goal is to prove that the adèle class space of a global field, which, up to now, has only been considered as a non- commutative space, has in fact a natural algebraic structure. We will also see that the construction of the Witt ring in characteris- tic p > 1 has a characteristic 1 analogue and that the deformation of the additive structure implies, in a crucial manner, the entropy function.}, keywords = {}, pubstate = {published}, tppubtype = {article} } I present here some recent results (obtained in collaboration with C. Consani [3], [4], [5], [6]) about the “characteristic 1” limit case. The main goal is to prove that the adèle class space of a global field, which, up to now, has only been considered as a non- commutative space, has in fact a natural algebraic structure. We will also see that the construction of the Witt ring in characteris- tic p > 1 has a characteristic 1 analogue and that the deformation of the additive structure implies, in a crucial manner, the entropy function. |

219. | Noncommutative geometric spaces with boundary: spectral action Ali H. Chamseddine, Alain Connes J. Geom. Phys., 61 (1), pp. 317–332, 2011, ISSN: 0393-0440. Journal ArticleAbstract @article{MR2747003, title = {Noncommutative geometric spaces with boundary: spectral action}, author = {Ali H. Chamseddine and Alain Connes}, url = {/wp-content/uploads/boundaryspectral.pdf}, doi = {10.1016/j.geomphys.2010.10.002}, issn = {0393-0440}, year = {2011}, date = {2011-01-01}, journal = {J. Geom. Phys.}, volume = {61}, number = {1}, pages = {317--332}, abstract = {We study spectral action for Riemannian manifolds with boundary, and then generalize this to noncommutative spaces which are products of a Riemannian manifold times a finite space. We determine the boundary conditions consistent with the hermiticity of the Dirac operator. We then define spectral triples of noncommutative spaces with boundary. In particular we eval- uate the spectral action corresponding to the noncommutative space of the standard model and show that the Einstein-Hilbert action gets modified by the addition of the extrinsic curvature terms with the right sign and coefficient necessary for consistency of the Hamiltonian. We also include effects due to the addition of dilaton field.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We study spectral action for Riemannian manifolds with boundary, and then generalize this to noncommutative spaces which are products of a Riemannian manifold times a finite space. We determine the boundary conditions consistent with the hermiticity of the Dirac operator. We then define spectral triples of noncommutative spaces with boundary. In particular we eval- uate the spectral action corresponding to the noncommutative space of the standard model and show that the Einstein-Hilbert action gets modified by the addition of the extrinsic curvature terms with the right sign and coefficient necessary for consistency of the Hamiltonian. We also include effects due to the addition of dilaton field. |

218. | The hyperring of adèle classes Alain Connes, Caterina Consani J. Number Theory, 131 (2), pp. 159–194, 2011, ISSN: 0022-314X. Journal ArticleAbstract @article{MR2736850, title = {The hyperring of adèle classes}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/The-hyperring-of-adeles-classes-2011.pdf}, doi = {10.1016/j.jnt.2010.09.001}, issn = {0022-314X}, year = {2011}, date = {2011-01-01}, journal = {J. Number Theory}, volume = {131}, number = {2}, pages = {159--194}, abstract = {We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK = AK/K× of a global field K. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G ∪ {0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We show that the theory of hyperrings, due to M. Krasner, supplies a perfect framework to understand the algebraic structure of the adèle class space HK = AK/K× of a global field K. After promoting F1 to a hyperfield K, we prove that a hyperring of the form R/G (where R is a ring and G⊂R× is a subgroup of its multiplicative group) is a hyperring extension of K if and only if G ∪ {0} is a subfield of R. This result applies to the adèle class space which thus inherits the structure of a hyperring extension HK of K. We begin to investigate the content of an algebraic geometry over K. The category of commutative hyperring extensions of K is inclusive of: commutative algebras over fields with semi-linear homomorphisms, abelian groups with injective homomorphisms and a rather exotic land comprising homogeneous non-Desarguesian planes. Finally, we show that for a global field K of positive characteristic, the groupoid of the prime elements of the hyperring HK is canonically and equivariantly isomorphic to the groupoid of the loops of the maximal abelian cover of the curve associated to the global field K. |

217. | From monoids to hyperstructures: in search of an absolute arithmetic Alain Connes, Caterina Consani Casimir force, Casimir operators and the Riemann hypothesis, pp. 147–198, Walter de Gruyter, Berlin, 2010. IncollectionAbstract @incollection{MR2777715, title = {From monoids to hyperstructures: in search of an absolute arithmetic}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/From-monoids-to-hyperstructures-2010.pdf}, year = {2010}, date = {2010-01-01}, booktitle = {Casimir force, Casimir operators and the Riemann hypothesis}, pages = {147--198}, publisher = {Walter de Gruyter, Berlin}, abstract = {We show that the trace formula interpretation of the explicit formulas expresses the counting function N.q/ of the hypothetical curve C associated to the Riemann zeta function, as an intersection number involving the scaling action on the adèle class space. Then, we discuss the algebraic structure of the adèle class space both as a monoid and as a hyperring. We construct an extension Rconvex of the hyperfield S of signs, which is the hyperfield analogue of the semifield Rmax of tropical geometry, admitting a one parameter group of automorphisms C fixing S. Finally, we develop function theory over Spec K and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over Spec S.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We show that the trace formula interpretation of the explicit formulas expresses the counting function N.q/ of the hypothetical curve C associated to the Riemann zeta function, as an intersection number involving the scaling action on the adèle class space. Then, we discuss the algebraic structure of the adèle class space both as a monoid and as a hyperring. We construct an extension Rconvex of the hyperfield S of signs, which is the hyperfield analogue of the semifield Rmax of tropical geometry, admitting a one parameter group of automorphisms C fixing S. Finally, we develop function theory over Spec K and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over Spec S. |

216. | Schemes over ?₁ and zeta functions Alain Connes, Caterina Consani Compos. Math., 146 (6), pp. 1383–1415, 2010, ISSN: 0010-437X. Journal ArticleAbstract @article{MR2735370, title = {Schemes over ?₁ and zeta functions}, author = {Alain Connes and Caterina Consani}, url = {/wp-content/uploads/schemesF1zeta.pdf}, doi = {10.1112/S0010437X09004692}, issn = {0010-437X}, year = {2010}, date = {2010-01-01}, journal = {Compos. Math.}, volume = {146}, number = {6}, pages = {1383--1415}, abstract = {We determine the {em real} counting function N(q) (q∈[1,∞)) for the hypothetical "curve" $C=overline {Sp Z}$ over $F_1$, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial $F_1$-schemes which reconciles the previous attempts by C. Soulé and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We determine the {em real} counting function N(q) (q∈[1,∞)) for the hypothetical "curve" $C=overline {Sp Z}$ over $F_1$, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial $F_1$-schemes which reconciles the previous attempts by C. Soulé and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato. |

215. | Alain Connes, Cédric Villani, Daniel Stroock J. Funct. Anal., 259 (11), pp. 2759, 2010, ISSN: 0022-1236. Journal ArticleAbstract @article{MR2719272, title = {The end of an era}, author = {Alain Connes and Cédric Villani and Daniel Stroock}, url = {/wp-content/uploads/endofanera.pdf}, doi = {10.1016/j.jfa.2010.08.001}, issn = {0022-1236}, year = {2010}, date = {2010-01-01}, journal = {J. Funct. Anal.}, volume = {259}, number = {11}, pages = {2759}, abstract = {Paul George Malliavin died on 3 June, 2010. Obviously his absence will be felt most keenlyby his family and friends, but his death also marks the end of an era for this journal. When IrvingM. Segal conceived the idea of creating theJournal of Functional Analysis, he first enlisted thesupport of Ralph S. Phillips and the two of them approached Malliavin at the ICM held in 1962.JFA came into existence shortly thereafter. Now all three of its founders are dead.Both Segal and Phillips died in 1998. Until his death, Segal was the central figure in themanagement of the journal. He oversaw the business arrangements and played a decisive role inthe selection of editors. Fortunately for JFA, Malliavin was ready and able to pick up the looseends which Segal had left behind and to assume the role which he had played. For the last twelveyears, the most important decisions about the journal’s future were made by Malliavin, sittingin the tiny office next to the dining room in his apartment. During those years, Malliavin guidedJFA through some difficult times when, without his guidance, the journal’s very existence mighthave been in jeopardy.Because of his dedication and wisdom, JFA continues to play an important role in the in-ternational mathematics community and promises to be a strong presence in the future. Losing Malliavin is a blow to all of us, but we can take solace and pride in the fact that JFA will live on. Yours,Alain Connes, Cédric Villani, Daniel Stroock. }, keywords = {}, pubstate = {published}, tppubtype = {article} } Paul George Malliavin died on 3 June, 2010. Obviously his absence will be felt most keenlyby his family and friends, but his death also marks the end of an era for this journal. When IrvingM. Segal conceived the idea of creating theJournal of Functional Analysis, he first enlisted thesupport of Ralph S. Phillips and the two of them approached Malliavin at the ICM held in 1962.JFA came into existence shortly thereafter. Now all three of its founders are dead.Both Segal and Phillips died in 1998. Until his death, Segal was the central figure in themanagement of the journal. He oversaw the business arrangements and played a decisive role inthe selection of editors. Fortunately for JFA, Malliavin was ready and able to pick up the looseends which Segal had left behind and to assume the role which he had played. For the last twelveyears, the most important decisions about the journal’s future were made by Malliavin, sittingin the tiny office next to the dining room in his apartment. During those years, Malliavin guidedJFA through some difficult times when, without his guidance, the journal’s very existence mighthave been in jeopardy.Because of his dedication and wisdom, JFA continues to play an important role in the in-ternational mathematics community and promises to be a strong presence in the future. Losing Malliavin is a blow to all of us, but we can take solace and pride in the fact that JFA will live on. Yours,Alain Connes, Cédric Villani, Daniel Stroock. |

214. | Ali H. Chamseddine, Alain Connes Fortschr. Phys., 58 (6), pp. 553–600, 2010, ISSN: 0015-8208. Journal ArticleAbstract @article{MR2674505, title = {Noncommutative geometry as a framework for unification of all fundamental interactions including gravity. Part I}, author = {Ali H. Chamseddine and Alain Connes}, url = {/wp-content/uploads/NCGframeworkCC.pdf}, doi = {10.1002/prop.201000069}, issn = {0015-8208}, year = {2010}, date = {2010-01-01}, journal = {Fortschr. Phys.}, volume = {58}, number = {6}, pages = {553--600}, abstract = {We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geome- try. In particular, this notation is used to determine the spectral data of the standard model. The particle spectrum with all of its symmetries is derived, almost uniquely, under the assumption of irreducibility and of di- mension 6 modulo 8 for the finite space. The reduction from the natural symmetry group SU(2) × SU(2) × SU(4) to U(1) × SU(2) × SU(3) is a consequence of the hypothesis that the two layers of space-time are finite distance apart but is non-dynamical. The square of the Dirac operator, and all geometrical invariants that appear in the calculation of the heat ker- nel expansion are evaluated. We re-derive the leading order terms in the spectral action. The geometrical action yields unification of all fundamental interactions including gravity at very high energies. We make the following predictions: (i) The number of fermions per family is 16. (ii) The symme- try group is U(1) × SU(2) × SU(3). (iii) There are quarks and leptons in the correct representations. (iv) There is a doublet Higgs that breaks the electroweak symmetry to U(1). (v) Top quark mass of 170-175 Gev. (v) There is a right-handed neutrino with a see-saw mechanism. Moreover, the zeroth order spectral action obtained with a cut-off function is consistent with experimental data up to few percent. We discuss a number of open issues. We prepare the ground for computing higher order corrections since the predicted mass of the Higgs field is quite sensitive to the higher order corrections. We speculate on the nature of the noncommutative space at Planckian energies and the possible role of the fundamental group for the problem of generations.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geome- try. In particular, this notation is used to determine the spectral data of the standard model. The particle spectrum with all of its symmetries is derived, almost uniquely, under the assumption of irreducibility and of di- mension 6 modulo 8 for the finite space. The reduction from the natural symmetry group SU(2) × SU(2) × SU(4) to U(1) × SU(2) × SU(3) is a consequence of the hypothesis that the two layers of space-time are finite distance apart but is non-dynamical. The square of the Dirac operator, and all geometrical invariants that appear in the calculation of the heat ker- nel expansion are evaluated. We re-derive the leading order terms in the spectral action. The geometrical action yields unification of all fundamental interactions including gravity at very high energies. We make the following predictions: (i) The number of fermions per family is 16. (ii) The symme- try group is U(1) × SU(2) × SU(3). (iii) There are quarks and leptons in the correct representations. (iv) There is a doublet Higgs that breaks the electroweak symmetry to U(1). (v) Top quark mass of 170-175 Gev. (v) There is a right-handed neutrino with a see-saw mechanism. Moreover, the zeroth order spectral action obtained with a cut-off function is consistent with experimental data up to few percent. We discuss a number of open issues. We prepare the ground for computing higher order corrections since the predicted mass of the Higgs field is quite sensitive to the higher order corrections. We speculate on the nature of the noncommutative space at Planckian energies and the possible role of the fundamental group for the problem of generations. |

213. | The uncanny precision of the spectral action Ali H. Chamseddine, Alain Connes Comm. Math. Phys., 293 (3), pp. 867–897, 2010, ISSN: 0010-3616. Journal ArticleAbstract @article{MR2566165, title = {The uncanny precision of the spectral action}, author = {Ali H. Chamseddine and Alain Connes}, url = {/wp-content/uploads/uncannyprec.pdf}, doi = {10.1007/s00220-009-0949-3}, issn = {0010-3616}, year = {2010}, date = {2010-01-01}, journal = {Comm. Math. Phys.}, volume = {293}, number = {3}, pages = {867--897}, abstract = {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 agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as action functional for the theory. To demon- strate how the spectral action encodes the dynamics of gravity we examine the accuracy of the approximation of the spectral action by its asymptotic 3 expansion in the case of the round sphere S . We find that the two terms corresponding to the cosmological constant and the scalar curvature term already give the full result with remarkable accuracy. This is then applied to the physically relevant case of S3 × S1 where we show that the spectral action in this case is also given, for any test function, by the sum of two terms up to an astronomically small correction, and in particular all higher order terms a2n vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the higher order terms a4 and a6 both vanish due to remarkable cancelations. We also show that the Higgs potential appears as an exact perturbation when the test function used is a smooth cutoff function.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as action functional for the theory. To demon- strate how the spectral action encodes the dynamics of gravity we examine the accuracy of the approximation of the spectral action by its asymptotic 3 expansion in the case of the round sphere S . We find that the two terms corresponding to the cosmological constant and the scalar curvature term already give the full result with remarkable accuracy. This is then applied to the physically relevant case of S3 × S1 where we show that the spectral action in this case is also given, for any test function, by the sum of two terms up to an astronomically small correction, and in particular all higher order terms a2n vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the higher order terms a4 and a6 both vanish due to remarkable cancelations. We also show that the Higgs potential appears as an exact perturbation when the test function used is a smooth cutoff function. |

212. | The Weil proof and the geometry of the adèles class space Alain Connes, Caterina Consani, Matilde Marcolli Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, 269 , pp. 339–405, Birkhäuser Boston, Boston, MA, 2009. IncollectionAbstract @incollection{MR2641176, title = {The Weil proof and the geometry of the adèles class space}, author = {Alain Connes and Caterina Consani and Matilde Marcolli}, url = {/wp-content/uploads/The-Weil-proof-and-the-geometry-2009.pdf}, doi = {10.1007/978-0-8176-4745-2_8}, year = {2009}, date = {2009-01-01}, booktitle = {Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I}, volume = {269}, pages = {339--405}, publisher = {Birkhäuser Boston, Boston, MA}, series = {Progr. Math.}, abstract = {This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adèles class space, which is the noncommutative space underlying Connes’ spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the “restriction map” defined by the inclu- sion of the idèles class group of a global field in the noncommutative adèles class space. Weil’s explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the idèles class group on this cohomology. In this formu- lation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adèles class space, with good working notions of correspondences, degree, and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low tempera- ture KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the idèles class group, that is, to the noncommu- tative spaces on which the geometric side of the trace formula is supported.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adèles class space, which is the noncommutative space underlying Connes’ spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the “restriction map” defined by the inclu- sion of the idèles class group of a global field in the noncommutative adèles class space. Weil’s explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the idèles class group on this cohomology. In this formu- lation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adèles class space, with good working notions of correspondences, degree, and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low tempera- ture KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the idèles class group, that is, to the noncommu- tative spaces on which the geometric side of the trace formula is supported. |

211. | Alain Connes, Caterina Consani, Matilde Marcolli J. Number Theory, 129 (6), pp. 1532–1561, 2009, ISSN: 0022-314X. Journal ArticleAbstract @article{MR2521492, title = {Fun with ?₁}, author = {Alain Connes and Caterina Consani and Matilde Marcolli}, url = {/wp-content/uploads/FunF1.pdf}, doi = {10.1016/j.jnt.2008.08.007}, issn = {0022-314X}, year = {2009}, date = {2009-01-01}, journal = {J. Number Theory}, volume = {129}, number = {6}, pages = {1532--1561}, abstract = {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” to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that “field”, while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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” to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that “field”, while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra. |

210. | Alain Connes, Michael Heller, Shahn Majid, Roger Penrose, John Polkinghorne, Andrew Taylor Cambridge University Press, Cambridge, 2008, ISBN: 978-0-521-88926-1. BookAbstract @book{MR2478456, title = {On space and time}, author = {Alain Connes and Michael Heller and Shahn Majid and Roger Penrose and John Polkinghorne and Andrew Taylor}, doi = {10.1017/CBO9781139644259}, isbn = {978-0-521-88926-1}, year = {2008}, date = {2008-10-20}, pages = {xx+287}, publisher = {Cambridge University Press}, address = {Cambridge}, abstract = {What is the true nature of space and time? These concepts are at the heart of science, but they remain deeply wrapped in mystery. Both house their structure at the smallest pre-subatomic and the largest cosmological levels continues to defy modern physics and may require revolutionary new ideas for which science is still grasping. This unique volume brings together world leaders in cosmology, particle physics, quantum gravity, mathematics, philosophy and theology, to provide fresh insights into the deep structure of space and time. Andrew Taylor, Shahn Majid, Roger Penrose, Alain Connes, Michael Heller, and John Polkinghorne all experts in their respective fields, explain their theories in this outstanding compiled text.}, keywords = {}, pubstate = {published}, tppubtype = {book} } What is the true nature of space and time? These concepts are at the heart of science, but they remain deeply wrapped in mystery. Both house their structure at the smallest pre-subatomic and the largest cosmological levels continues to defy modern physics and may require revolutionary new ideas for which science is still grasping. This unique volume brings together world leaders in cosmology, particle physics, quantum gravity, mathematics, philosophy and theology, to provide fresh insights into the deep structure of space and time. Andrew Taylor, Shahn Majid, Roger Penrose, Alain Connes, Michael Heller, and John Polkinghorne all experts in their respective fields, explain their theories in this outstanding compiled text. |

209. | A unitary invariant in Riemannian geometry Alain Connes Int. J. Geom. Methods Mod. Phys., 5 (8), pp. 1215–1242, 2008, ISSN: 0219-8878. Journal ArticleAbstract @article{MR2484550, title = {A unitary invariant in Riemannian geometry}, author = {Alain Connes}, url = {/wp-content/uploads/unitaryinvariant.pdf}, doi = {10.1142/S0219887808003284}, issn = {0219-8878}, year = {2008}, date = {2008-01-01}, journal = {Int. J. Geom. Methods Mod. Phys.}, volume = {5}, number = {8}, pages = {1215--1242}, abstract = {We introduce an invariant of Riemannian geometry which mea- sures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of the Dirac operator, gives a com- plete invariant of Riemannian geometry. We show that the new invariant plays the same role with respect to the spectral invariant as the Cabibbo–Kobayashi– Maskawa mixing matrix in the Standard Model plays with respect to the list of masses of the quarks.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We introduce an invariant of Riemannian geometry which mea- sures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of the Dirac operator, gives a com- plete invariant of Riemannian geometry. We show that the new invariant plays the same role with respect to the spectral invariant as the Cabibbo–Kobayashi– Maskawa mixing matrix in the Standard Model plays with respect to the list of masses of the quarks. |

208. | On the fine structure of spacetime Alain Connes On space and time, pp. 196–237, Cambridge Univ. Press, Cambridge, 2008. Incollection @incollection{MR2454340, title = {On the fine structure of spacetime}, author = {Alain Connes}, year = {2008}, date = {2008-01-01}, booktitle = {On space and time}, pages = {196--237}, publisher = {Cambridge Univ. Press, Cambridge}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

207. | Alain Connes, Henri Moscovici Traces in number theory, geometry and quantum fields, pp. 57–71, Friedr. Vieweg, Wiesbaden, 2008. IncollectionAbstract @incollection{MR2427588, title = {Type III and spectral triples}, author = {Alain Connes and Henri Moscovici}, url = {/wp-content/uploads/typeIIIspectral.pdf}, year = {2008}, date = {2008-01-01}, booktitle = {Traces in number theory, geometry and quantum fields}, pages = {57--71}, publisher = {Friedr. Vieweg, Wiesbaden}, series = {Aspects Math., E38}, abstract = {We explain how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations. We show that the classical cyclic cohomology valued Chern character of finitely summable spectral triples extends to the twisted case and lands in ordinary (untwisted) cyclic cohomology. The index pairing with ordinary (untwisted) K-theory continues to make sense and the index formula is given by the pairing of the corresponding Chern characters. This opens the road to extending the local index formula to the type III case.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We explain how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations. We show that the classical cyclic cohomology valued Chern character of finitely summable spectral triples extends to the twisted case and lands in ordinary (untwisted) cyclic cohomology. The index pairing with ordinary (untwisted) K-theory continues to make sense and the index formula is given by the pairing of the corresponding Chern characters. This opens the road to extending the local index formula to the type III case. |

206. | A walk in the noncommutative garden Alain Connes, Matilde Marcolli An invitation to noncommutative geometry, pp. 1–128, World Sci. Publ., Hackensack, NJ, 2008. IncollectionAbstract @incollection{MR2408150, title = {A walk in the noncommutative garden}, author = {Alain Connes and Matilde Marcolli}, url = {/wp-content/uploads/walkNCGgarden.pdf}, doi = {10.1142/9789812814333_0001}, year = {2008}, date = {2008-01-01}, booktitle = {An invitation to noncommutative geometry}, pages = {1--128}, publisher = {World Sci. Publ., Hackensack, NJ}, abstract = {If you cleave the hearth of one drop of water a hundred pure oceans emerge from it. (Mahmud Shabistari, Gulshan-i-raz) We have decided to contribute to the volume of the IPM lectures on noncommutative ge- ometry a text that collects a list of examples of noncommutative spaces. As the quote of the Sufi poet here above suggests, it is often better to approach a new subject by analyzing specific examples rather than presenting the general theory. We hope that the diversity of examples the readers will encounter in this text will suffice to convince them of the fact that noncommutative geometry is a very rich field in rapid evolution, full of interesting and yet unexplored landscapes. Many of the examples collected here have not yet been fully explored from the point of view of the general guidelines we propose in Section 2 and the main point of this text is to provide a great number of open questions. The reader should interpret this survey as a suggestion of possible interesting problems to investigate, both in the settings described here, as well as in other examples that are available but did not fit in this list, and in the many more that still await to be discovered. }, keywords = {}, pubstate = {published}, tppubtype = {incollection} } If you cleave the hearth of one drop of water a hundred pure oceans emerge from it. (Mahmud Shabistari, Gulshan-i-raz) We have decided to contribute to the volume of the IPM lectures on noncommutative ge- ometry a text that collects a list of examples of noncommutative spaces. As the quote of the Sufi poet here above suggests, it is often better to approach a new subject by analyzing specific examples rather than presenting the general theory. We hope that the diversity of examples the readers will encounter in this text will suffice to convince them of the fact that noncommutative geometry is a very rich field in rapid evolution, full of interesting and yet unexplored landscapes. Many of the examples collected here have not yet been fully explored from the point of view of the general guidelines we propose in Section 2 and the main point of this text is to provide a great number of open questions. The reader should interpret this survey as a suggestion of possible interesting problems to investigate, both in the settings described here, as well as in other examples that are available but did not fit in this list, and in the many more that still await to be discovered. |

205. | Alain Connes, Michel Dubois-Violette Comm. Math. Phys., 281 (1), pp. 23–127, 2008, ISSN: 0010-3616. Journal ArticleAbstract @article{MR2403605, title = {Noncommutative finite dimensional manifolds II: Moduli space and structure of noncommutative 3-spheres}, author = {Alain Connes and Michel Dubois-Violette}, url = {/wp-content/uploads/NC3spheresMDV.pdf}, doi = {10.1007/s00220-008-0472-y}, issn = {0010-3616}, year = {2008}, date = {2008-01-01}, journal = {Comm. Math. Phys.}, volume = {281}, number = {1}, pages = {23--127}, abstract = {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 algebras, and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic algebras to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show that the above morphism corresponds to a natural ramified covering by a noncommutative 3-dimensional nilmanifold. We then compute the Jacobian of the ramified covering and obtain the answer as the product of a period (of an elliptic integral) by a rational function. We describe the real and complex moduli spaces of noncommutative 3-spheres, relate the real one to root systems and the complex one to the orbits of a birational cubic automorphism of three dimensional projective space. We classify the algebras and establish duality relations between them.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 algebras, and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic algebras to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show that the above morphism corresponds to a natural ramified covering by a noncommutative 3-dimensional nilmanifold. We then compute the Jacobian of the ramified covering and obtain the answer as the product of a period (of an elliptic integral) by a rational function. We describe the real and complex moduli spaces of noncommutative 3-spheres, relate the real one to root systems and the complex one to the orbits of a birational cubic automorphism of three dimensional projective space. We classify the algebras and establish duality relations between them. |

204. | Ali H. Chamseddine, Alain Connes J. Geom. Phys., 58 (1), pp. 38–47, 2008, ISSN: 0393-0440. Journal ArticleAbstract @article{MR2378454, title = {Why the standard model}, author = {Ali H. Chamseddine and Alain Connes}, url = {/wp-content/uploads/whySM.pdf}, doi = {10.1016/j.geomphys.2007.09.011}, issn = {0393-0440}, year = {2008}, date = {2008-01-01}, journal = {J. Geom. Phys.}, volume = {58}, number = {1}, pages = {38--47}, abstract = {The Standard Model is based on the gauge invariance principle with gauge group U(1) × SU(2) × SU(3) and suitable representations for fermions and bosons, which are begging for a conceptual understanding. We propose a purely gravitational explanation: space-time has a fine structure given as a product of a four dimensional continuum by a finite noncommutative geometry F. The raison d’ˆetre for F is to correct the K-theoretic dimension from four to ten (modulo eight). We classify the irreducible finite noncommutative geometries of K-theoretic dimension six and show that the dimension (per generation) is a square of an integer k. Under an additional hypothesis of quaternion linearity, the geometry which reproduces the Standard Model is singled out (and one gets k = 4) with the correct quantum numbers for all fields. The spectral action applied to the product M × F delivers the full Standard Model, with neutrino mixing, coupled to gravity, and makes predictions (the number of generations is still an input).}, keywords = {}, pubstate = {published}, tppubtype = {article} } The Standard Model is based on the gauge invariance principle with gauge group U(1) × SU(2) × SU(3) and suitable representations for fermions and bosons, which are begging for a conceptual understanding. We propose a purely gravitational explanation: space-time has a fine structure given as a product of a four dimensional continuum by a finite noncommutative geometry F. The raison d’ˆetre for F is to correct the K-theoretic dimension from four to ten (modulo eight). We classify the irreducible finite noncommutative geometries of K-theoretic dimension six and show that the dimension (per generation) is a square of an integer k. Under an additional hypothesis of quaternion linearity, the geometry which reproduces the Standard Model is singled out (and one gets k = 4) with the correct quantum numbers for all fields. The spectral action applied to the product M × F delivers the full Standard Model, with neutrino mixing, coupled to gravity, and makes predictions (the number of generations is still an input). |

203. | Noncommutative geometry, quantum fields and motives Alain Connes, Matilde Marcolli American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2007, ISBN: 978-0-8218-4210-2. BookAbstract @book{MR2371808, title = {Noncommutative geometry, quantum fields and motives}, author = {Alain Connes and Matilde Marcolli}, url = {/wp-content/uploads/bookwebfinal-2.pdf}, doi = {10.1090/coll/055}, isbn = {978-0-8218-4210-2}, year = {2007}, date = {2007-12-20}, volume = {55}, pages = {xxii+785}, publisher = {American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi}, series = {Colloquium Publications}, abstract = {The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adele class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, inone and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.}, keywords = {}, pubstate = {published}, tppubtype = {book} } The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adele class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, inone and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function. |

202. | Non-commutative geometry and the spectral model of space-time Alain Connes Quantum spaces, 53 , pp. 203–227, Birkhäuser, Basel, 2007. IncollectionAbstract @incollection{MR2382238, title = {Non-commutative geometry and the spectral model of space-time}, author = {Alain Connes}, url = {/wp-content/uploads/poincareseminar2007.pdf}, doi = {10.1007/978-3-7643-8522-4_5}, year = {2007}, date = {2007-01-01}, booktitle = {Quantum spaces}, volume = {53}, pages = {203--227}, publisher = {Birkhäuser, Basel}, series = {Prog. Math. Phys.}, abstract = {This is a report on our joint work with A. Chamseddine and M. Marcolli. This essay gives a short introduction to a potential application in physics of a new type of geometry based on spectral considerations which is convenient when dealing with non-commutative spaces, i.e., spaces in which the simplifying rule of commutativity is no longer applied to the coordinates. Starting from the phenomenological Lagrangian of gravity coupled with mat- ter one infers, using the spectral action principle, that space-time admits a fine structure which is a subtle mixture of the usual 4-dimensional continuum with a finite discrete structure F . Under the (unrealistic) hypothesis that this structure remains valid (i.e., one does not have any “hyperfine” modifica- tion) until the unification scale, one obtains a number of predictions whose approximate validity is a basic test of the approach.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } This is a report on our joint work with A. Chamseddine and M. Marcolli. This essay gives a short introduction to a potential application in physics of a new type of geometry based on spectral considerations which is convenient when dealing with non-commutative spaces, i.e., spaces in which the simplifying rule of commutativity is no longer applied to the coordinates. Starting from the phenomenological Lagrangian of gravity coupled with mat- ter one infers, using the spectral action principle, that space-time admits a fine structure which is a subtle mixture of the usual 4-dimensional continuum with a finite discrete structure F . Under the (unrealistic) hypothesis that this structure remains valid (i.e., one does not have any “hyperfine” modifica- tion) until the unification scale, one obtains a number of predictions whose approximate validity is a basic test of the approach. |

201. | Gravity and the standard model with neutrino mixing Ali H. Chamseddine, Alain Connes, Matilde Marcolli Adv. Theor. Math. Phys., 11 (6), pp. 991–1089, 2007, ISSN: 1095-0761. Journal ArticleAbstract @article{MR2368941, title = {Gravity and the standard model with neutrino mixing}, author = {Ali H. Chamseddine and Alain Connes and Matilde Marcolli}, url = {/wp-content/uploads/gravityplussm.pdf}, issn = {1095-0761}, year = {2007}, date = {2007-01-01}, journal = {Adv. Theor. Math. Phys.}, volume = {11}, number = {6}, pages = {991--1089}, abstract = {We present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector. |

200. | A property of parallelograms inscribed in ellipses Alain Connes, Don Zagier Amer. Math. Monthly, 114 (10), pp. 909–914, 2007, ISSN: 0002-9890. Journal ArticleAbstract @article{MR2363056, title = {A property of parallelograms inscribed in ellipses}, author = {Alain Connes and Don Zagier}, doi = {10.1080/00029890.2007.11920483}, issn = {0002-9890}, year = {2007}, date = {2007-01-01}, journal = {Amer. Math. Monthly}, volume = {114}, number = {10}, pages = {909--914}, abstract = {The following surprising property of ellipses was observed by the physicist Jean-Marc Richard, in connection with a problem from ballistics [Eur. J. Phys. 25 (2004), no. 6, 835–844 (p. 843)]: Theorem 1. Let ℰ be an ellipse and f(d,d′) the function of two diameters given by the perimeter of the parallelogram with vertices d∩ℰ and d′∩ℰ (Figure 1). Then f(d):=supd′f(d,d′) is constant (independent of d). "Richard proved this result by a direct computational verification. Jean-Pierre Bourguignon told us about the theorem and asked whether one could give a more enlightening proof. In this note we give two simple proofs, one geometric and the other algebraic, as well as a small generalization. We also describe briefly a connection with billiards that was pointed out to us by Sergei Tabachnikov. A different proof of Theorem 1 is given in [M. Berger, Géométrie. Vol. 2, Nathan, Paris, 1990 (p. 350)]." }, keywords = {}, pubstate = {published}, tppubtype = {article} } The following surprising property of ellipses was observed by the physicist Jean-Marc Richard, in connection with a problem from ballistics [Eur. J. Phys. 25 (2004), no. 6, 835–844 (p. 843)]: Theorem 1. Let ℰ be an ellipse and f(d,d′) the function of two diameters given by the perimeter of the parallelogram with vertices d∩ℰ and d′∩ℰ (Figure 1). Then f(d):=supd′f(d,d′) is constant (independent of d). "Richard proved this result by a direct computational verification. Jean-Pierre Bourguignon told us about the theorem and asked whether one could give a more enlightening proof. In this note we give two simple proofs, one geometric and the other algebraic, as well as a small generalization. We also describe briefly a connection with billiards that was pointed out to us by Sergei Tabachnikov. A different proof of Theorem 1 is given in [M. Berger, Géométrie. Vol. 2, Nathan, Paris, 1990 (p. 350)]." |

199. | Conceptual explanation for the algebra in the noncommutative approach to the standard model Ali H. Chamseddine, Alain Connes Phys. Rev. Lett., 99 (19), pp. 191601, 4, 2007, ISSN: 0031-9007. Journal ArticleAbstract @article{MR2362155, title = {Conceptual explanation for the algebra in the noncommutative approach to the standard model}, author = {Ali H. Chamseddine and Alain Connes}, url = {/wp-content/uploads/conceptualSM.pdf}, doi = {10.1103/PhysRevLett.99.191601}, issn = {0031-9007}, year = {2007}, date = {2007-01-01}, journal = {Phys. Rev. Lett.}, volume = {99}, number = {19}, pages = {191601, 4}, abstract = {The purpose of this letter is to remove the arbitrariness of the ad hoc choice of the algebra and its representation in the noncommutative approach to the Standard Model, which was begging for a conceptual explanation. We assume as before that space-time is the product of a four-dimensional manifold by a finite noncommmutative space F. The spectral action is the pure gravitational action for the product space. To remove the above arbitrariness, we classify the irreducibe geometries F consistent with imposing reality and chiral conditions on spinors, to avoid the fermion doubling problem, which amounts to have total dimension 10 (in the K-theoretic sense). It gives, almost uniquely, the Standard Model with all its details, predicting the number of fermions per generation to be 16, their representations and the Higgs breaking mechanism, with very little input. The geometrical model is valid at the unification scale, and has relations connecting the gauge couplings to each other and to the Higgs coupling. This gives a prediction of the Higgs mass of around 170 GeV and a mass relation connecting the sum of the square of the masses of the fermions to the W mass square, which enables us to predict the top quark mass compatible with the measured experimental value. We thus manage to have the advantages of both SO(10) and Kaluza-Klein unification, without paying the price of plethora of Higgs fields or the infinite tower of states.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The purpose of this letter is to remove the arbitrariness of the ad hoc choice of the algebra and its representation in the noncommutative approach to the Standard Model, which was begging for a conceptual explanation. We assume as before that space-time is the product of a four-dimensional manifold by a finite noncommmutative space F. The spectral action is the pure gravitational action for the product space. To remove the above arbitrariness, we classify the irreducibe geometries F consistent with imposing reality and chiral conditions on spinors, to avoid the fermion doubling problem, which amounts to have total dimension 10 (in the K-theoretic sense). It gives, almost uniquely, the Standard Model with all its details, predicting the number of fermions per generation to be 16, their representations and the Higgs breaking mechanism, with very little input. The geometrical model is valid at the unification scale, and has relations connecting the gauge couplings to each other and to the Higgs coupling. This gives a prediction of the Higgs mass of around 170 GeV and a mass relation connecting the sum of the square of the masses of the fermions to the W mass square, which enables us to predict the top quark mass compatible with the measured experimental value. We thus manage to have the advantages of both SO(10) and Kaluza-Klein unification, without paying the price of plethora of Higgs fields or the infinite tower of states. |

198. | Noncommutative geometry and motives: the thermodynamics of endomotives Alain Connes, Caterina Consani, Matilde Marcolli Adv. Math., 214 (2), pp. 761–831, 2007, ISSN: 0001-8708. Journal ArticleAbstract @article{MR2349719, title = {Noncommutative geometry and motives: the thermodynamics of endomotives}, author = {Alain Connes and Caterina Consani and Matilde Marcolli}, url = {/wp-content/uploads/thermodynamicsofendomotives.pdf}, doi = {10.1016/j.aim.2007.03.006}, issn = {0001-8708}, year = {2007}, date = {2007-01-01}, journal = {Adv. Math.}, volume = {214}, number = {2}, pages = {761--831}, abstract = {A few unexpected encounters between noncommutative geometry and the theory of mo- tives have taken place recently. A first instance occurred in [14], where the Weil explicit formulae acquire the geometric meaning of a Lefschetz trace formula over the noncommu- tative space of adele classes. The reason why the construction of [14] should be regarded as motivic is twofold. On the one hand, as we discuss in this paper, the adele class space is obtained from a noncommutative space, the Bost–Connes system, which sits naturally in a category of noncommutative spaces extending the category of Artin motives. Moreover, as we discuss briefly in this paper and in full detail in our forthcoming work [15], it is possible to give a cohomological interpretation of the spectral realization of the zeros of the Riemann zeta function of [14] on the cyclic homology of a noncommutative motive. The reason why this construction takes place in a category of (noncommutative) motives is that the geometric space we need to use is obtained as a cokernel of a morphism of alge- bras, which only exists in a suitable abelian category extending the non-additive category of algebras, exactly as in the context of motives and algebraic varieties.}, keywords = {}, pubstate = {published}, tppubtype = {article} } A few unexpected encounters between noncommutative geometry and the theory of mo- tives have taken place recently. A first instance occurred in [14], where the Weil explicit formulae acquire the geometric meaning of a Lefschetz trace formula over the noncommu- tative space of adele classes. The reason why the construction of [14] should be regarded as motivic is twofold. On the one hand, as we discuss in this paper, the adele class space is obtained from a noncommutative space, the Bost–Connes system, which sits naturally in a category of noncommutative spaces extending the category of Artin motives. Moreover, as we discuss briefly in this paper and in full detail in our forthcoming work [15], it is possible to give a cohomological interpretation of the spectral realization of the zeros of the Riemann zeta function of [14] on the cyclic homology of a noncommutative motive. The reason why this construction takes place in a category of (noncommutative) motives is that the geometric space we need to use is obtained as a cokernel of a morphism of alge- bras, which only exists in a suitable abelian category extending the non-additive category of algebras, exactly as in the context of motives and algebraic varieties. |

197. | Quantum gravity boundary terms from the spectral action of noncommutative space Ali H. Chamseddine, Alain Connes Phys. Rev. Lett., 99 (7), pp. 071302, 4, 2007, ISSN: 0031-9007. Journal ArticleAbstract @article{MR2338507, title = {Quantum gravity boundary terms from the spectral action of noncommutative space}, author = {Ali H. Chamseddine and Alain Connes}, url = {/wp-content/uploads/boundaryterms2007.pdf}, doi = {10.1103/PhysRevLett.99.071302}, issn = {0031-9007}, year = {2007}, date = {2007-01-01}, journal = {Phys. Rev. Lett.}, volume = {99}, number = {7}, pages = {071302, 4}, abstract = {We study the boundary terms of the spectral action of the noncommutative space, defined by the spectral triple dictated by the physical spectrum of the standard model, unifying gravity with all other fundamental interactions. We prove that the spectral action predicts uniquely the gravitational boundary term required for consistency of quantum gravity with the correct sign and coefficient. This is a remarkable result given the lack of freedom in the spectral action to tune this term.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We study the boundary terms of the spectral action of the noncommutative space, defined by the spectral triple dictated by the physical spectrum of the standard model, unifying gravity with all other fundamental interactions. We prove that the spectral action predicts uniquely the gravitational boundary term required for consistency of quantum gravity with the correct sign and coefficient. This is a remarkable result given the lack of freedom in the spectral action to tune this term. |

196. | Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory Alain Connes, Matilde Marcolli Frontiers in number theory, physics, and geometry. II, pp. 617–713, Springer, Berlin, 2007. IncollectionAbstract @incollection{MR2290770, title = {Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory}, author = {Alain Connes and Matilde Marcolli}, url = {/wp-content/uploads/renfinalarxiv.pdf}, doi = {10.1007/978-3-540-30308-4_13}, year = {2007}, date = {2007-01-01}, booktitle = {Frontiers in number theory, physics, and geometry. II}, pages = {617--713}, publisher = {Springer, Berlin}, abstract = {We give here a comprehensive treatment of the mathematical theory of per- turbative renormalization (in the minimal subtraction scheme with dimensional regularization), in the framework of the Riemann–Hilbert correspondence and motivic Galois theory. We give a detailed overview of the work of Connes– Kreimer [31], [32]. We also cover some background material on affine group schemes, Tannakian categories, the Riemann–Hilbert problem in the regular singular and irregular case, and a brief introduction to motives and motivic Ga- lois theory. We then give a complete account of our results on renormalization and motivic Galois theory announced in [35]. }, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We give here a comprehensive treatment of the mathematical theory of per- turbative renormalization (in the minimal subtraction scheme with dimensional regularization), in the framework of the Riemann–Hilbert correspondence and motivic Galois theory. We give a detailed overview of the work of Connes– Kreimer [31], [32]. We also cover some background material on affine group schemes, Tannakian categories, the Riemann–Hilbert problem in the regular singular and irregular case, and a brief introduction to motives and motivic Ga- lois theory. We then give a complete account of our results on renormalization and motivic Galois theory announced in [35]. |

195. | Yang-Mills and some related algebras Alain Connes, Michel Dubois-Violette Rigorous quantum field theory, 251 , pp. 65–78, Birkhäuser, Basel, 2007. IncollectionAbstract @incollection{MR2279211, title = {Yang-Mills and some related algebras}, author = {Alain Connes and Michel Dubois-Violette}, url = {/wp-content/uploads/YangMillsMDV.pdf}, doi = {10.1007/978-3-7643-7434-1_6}, year = {2007}, date = {2007-01-01}, booktitle = {Rigorous quantum field theory}, volume = {251}, pages = {65--78}, publisher = {Birkhäuser, Basel}, series = {Progr. Math.}, abstract = {After a short introduction on the theory of homogeneous algebras we describe the application of this theory to the analysis of the cubic Yang-Mills al- gebra, the quadratic self-duality algebras, their “super” versions as well as to some generalization.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } After a short introduction on the theory of homogeneous algebras we describe the application of this theory to the analysis of the cubic Yang-Mills al- gebra, the quadratic self-duality algebras, their “super” versions as well as to some generalization. |

194. | Transgressions of the Godbillon-Vey class and Rademacher functions Alain Connes, Henri Moscovici Noncommutative geometry and number theory, pp. 79–107, Friedr. Vieweg, Wiesbaden, 2006. IncollectionAbstract @incollection{MR2327300, title = {Transgressions of the Godbillon-Vey class and Rademacher functions}, author = {Alain Connes and Henri Moscovici}, url = {https://fredericc1.sg-host.com/wp-content/uploads/gvtrans.pdf}, doi = {10.1007/978-3-8348-0352-8_4}, year = {2006}, date = {2006-01-01}, booktitle = {Noncommutative geometry and number theory}, pages = {79--107}, publisher = {Friedr. Vieweg, Wiesbaden}, series = {Aspects Math., E37}, abstract = {In earlier work [8, 9] we investigated a surprising interconnection between the transverse geometry of codimension 1 foliations and modular forms. At the core of this interplay lies the Hopf algebra H1, the first in a series of Hopf algebras Hn that were found [6] to determine the affine transverse geometry of codimension n foliations. The periodic Hopf-cyclic cohomology of H1 is generated by two classes, [δ1] for the odd component and [RC1] for the even component. The tautological action of H1 on the ́etale groupoid algebra AG associated to the frame bundle of a codimension 1 foliation preserves (up to a character) the canonical trace on AG , and thus gives rise to a characteristic homomorphism in cyclic cohomology. This homomorphism maps [δ1] to the Godbillon-Vey class and [RC1] to the transverse fundamental class.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } In earlier work [8, 9] we investigated a surprising interconnection between the transverse geometry of codimension 1 foliations and modular forms. At the core of this interplay lies the Hopf algebra H1, the first in a series of Hopf algebras Hn that were found [6] to determine the affine transverse geometry of codimension n foliations. The periodic Hopf-cyclic cohomology of H1 is generated by two classes, [δ1] for the odd component and [RC1] for the even component. The tautological action of H1 on the ́etale groupoid algebra AG associated to the frame bundle of a codimension 1 foliation preserves (up to a character) the canonical trace on AG , and thus gives rise to a characteristic homomorphism in cyclic cohomology. This homomorphism maps [δ1] to the Godbillon-Vey class and [RC1] to the transverse fundamental class. |

193. | Morse inequalities for foliations Alain Connes, Thierry Fack ?*-algebras and elliptic theory, pp. 61–72, Birkhäuser, Basel, 2006. @incollection{MR2276915, title = {Morse inequalities for foliations}, author = {Alain Connes and Thierry Fack}, doi = {10.1007/978-3-7643-7687-1_4}, year = {2006}, date = {2006-01-01}, booktitle = {?*-algebras and elliptic theory}, pages = {61--72}, publisher = {Birkhäuser, Basel}, series = {Trends Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

192. | Noncommutative geometry and the standard model with neutrino mixing Alain Connes J. High Energy Phys., (11), pp. 081, 19, 2006, ISSN: 1126-6708. Journal ArticleAbstract @article{MR2270385, title = {Noncommutative geometry and the standard model with neutrino mixing}, author = {Alain Connes}, url = {/wp-content/uploads/neutrinomixingC.pdf}, doi = {10.1088/1126-6708/2006/11/081}, issn = {1126-6708}, year = {2006}, date = {2006-01-01}, journal = {J. High Energy Phys.}, number = {11}, pages = {081, 19}, abstract = { We show that allowing the metric dimension of a space to be independent of its KO-dimension and turning the finite noncommutative geometry F– whose product with classical 4-dimensional space-time gives the standard model coupled with gravity–into a space of KO-dimension 6 by changing the grading on the antiparticle sector into its opposite, allows to solve three problems of the previous noncommutative geometry interpretation of the standard model of particle physics: The finite geometry F is no longer put in “by hand” but a conceptual understanding of its structure and a classification of its metrics is given. The fermion doubling problem in the fermionic part of the action is resolved. The spectral action of our joint work with Chamseddine now automatically generates the full standard model coupled with gravity with neutrino mixing and see-saw mechanism for neutrino masses. The predictions of the Weinberg angle and the Higgs scattering parameter at unification scale are the same as in our joint work but we also find a mass relation (to be imposed at unification scale).}, keywords = {}, pubstate = {published}, tppubtype = {article} } We show that allowing the metric dimension of a space to be independent of its KO-dimension and turning the finite noncommutative geometry F– whose product with classical 4-dimensional space-time gives the standard model coupled with gravity–into a space of KO-dimension 6 by changing the grading on the antiparticle sector into its opposite, allows to solve three problems of the previous noncommutative geometry interpretation of the standard model of particle physics: The finite geometry F is no longer put in “by hand” but a conceptual understanding of its structure and a classification of its metrics is given. The fermion doubling problem in the fermionic part of the action is resolved. The spectral action of our joint work with Chamseddine now automatically generates the full standard model coupled with gravity with neutrino mixing and see-saw mechanism for neutrino masses. The predictions of the Weinberg angle and the Higgs scattering parameter at unification scale are the same as in our joint work but we also find a mass relation (to be imposed at unification scale). |

191. | Inner fluctuations of the spectral action Alain Connes, Ali H. Chamseddine J. Geom. Phys., 57 (1), pp. 1–21, 2006, ISSN: 0393-0440. Journal ArticleAbstract @article{MR2265456, title = {Inner fluctuations of the spectral action}, author = {Alain Connes and Ali H. Chamseddine}, url = {/wp-content/uploads/innerfluctuations2006.pdf}, doi = {10.1016/j.geomphys.2006.08.003}, issn = {0393-0440}, year = {2006}, date = {2006-01-01}, journal = {J. Geom. Phys.}, volume = {57}, number = {1}, pages = {1--21}, abstract = {We prove in the general framework of noncommutative geometry that the inner fluc- tuations of the spectral action can be computed as residues and give exactly the counterterms for the Feynman graphs with fermionic internal lines. We show that for geometries of dimension less or equal to four the obtained terms add up to a sum of a Yang-Mills action with a Chern-Simons action.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We prove in the general framework of noncommutative geometry that the inner fluc- tuations of the spectral action can be computed as residues and give exactly the counterterms for the Feynman graphs with fermionic internal lines. We show that for geometries of dimension less or equal to four the obtained terms add up to a sum of a Yang-Mills action with a Chern-Simons action. |

190. | KMS states and complex multiplication (Part II) Alain Connes, Matilde Marcolli, Niranjan Ramachandran Operator Algebras: Ŧhe Abel Symposium 2004, 1 , pp. 15–59, Springer, Berlin, 2006. IncollectionAbstract @incollection{MR2265042, title = {KMS states and complex multiplication (Part II)}, author = {Alain Connes and Matilde Marcolli and Niranjan Ramachandran}, url = {/wp-content/uploads/CMR1.pdf}, doi = {10.1007/978-3-540-34197-0_2}, year = {2006}, date = {2006-01-01}, booktitle = {Operator Algebras: Ŧhe Abel Symposium 2004}, volume = {1}, pages = {15--59}, publisher = {Springer, Berlin}, series = {Abel Symp.}, abstract = {Several results point to deep relations between noncommutative geometry and class field theory ([3], [10], [20], [22]). In [3] a quantum statistical mechanical system (BC) is exhibited, with partition func- tion the Riemann zeta function ζ(β), and whose arithmetic properties are related to the Galois theory of the maximal abelian extension of Q. In [10], this system is reinterpreted in terms of the geome- try of commensurable 1-dimensional Q-lattices, and a generalization is constructed for 2-dimensional Q-lattices. The arithmetic properties of this GL2-system and its extremal KMS states at zero tem- perature are related to the Galois theory of the modular field F, that is, the field of elliptic modular functions. These are functions on modular curves, i.e. on moduli spaces of elliptic curves. The low temperature extremal KMS states and the Galois properties of the GL2-system are analyzed in [10] for the generic case of elliptic curves with transcendental j-invariant. As the results of [10] show, one of the main new features of the GL2-system is the presence of symmetries by endomorphism, as in (2.6) below. The full Galois group of the modular field appears then as symmetries, acting on the set of extremal KMSβ states of the system, for large inverse temperature β.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } Several results point to deep relations between noncommutative geometry and class field theory ([3], [10], [20], [22]). In [3] a quantum statistical mechanical system (BC) is exhibited, with partition func- tion the Riemann zeta function ζ(β), and whose arithmetic properties are related to the Galois theory of the maximal abelian extension of Q. In [10], this system is reinterpreted in terms of the geome- try of commensurable 1-dimensional Q-lattices, and a generalization is constructed for 2-dimensional Q-lattices. The arithmetic properties of this GL2-system and its extremal KMS states at zero tem- perature are related to the Galois theory of the modular field F, that is, the field of elliptic modular functions. These are functions on modular curves, i.e. on moduli spaces of elliptic curves. The low temperature extremal KMS states and the Galois properties of the GL2-system are analyzed in [10] for the generic case of elliptic curves with transcendental j-invariant. As the results of [10] show, one of the main new features of the GL2-system is the presence of symmetries by endomorphism, as in (2.6) below. The full Galois group of the modular field appears then as symmetries, acting on the set of extremal KMSβ states of the system, for large inverse temperature β. |

189. | From physics to number theory via noncommutative geometry Alain Connes, Matilde Marcolli Frontiers in number theory, physics, and geometry. I, pp. 269–347, Springer, Berlin, 2006. IncollectionAbstract @incollection{MR2261099, title = {From physics to number theory via noncommutative geometry}, author = {Alain Connes and Matilde Marcolli}, url = {/wp-content/uploads/Qlattices-1.pdf}, year = {2006}, date = {2006-01-01}, booktitle = {Frontiers in number theory, physics, and geometry. I}, pages = {269--347}, publisher = {Springer, Berlin}, abstract = {In this paper we show that the theory of modular Hecke algebras, the spectral realization of zeros of L-functions, and the arithmetic properties of KMS states in quantum statistical mechanics combine into a unique general picture based on the noncommutative geometry of the space of commensurability classes of Q-lattices.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } In this paper we show that the theory of modular Hecke algebras, the spectral realization of zeros of L-functions, and the arithmetic properties of KMS states in quantum statistical mechanics combine into a unique general picture based on the noncommutative geometry of the space of commensurability classes of Q-lattices. |

188. | Scale invariance in the spectral action Ali H. Chamseddine, Alain Connes J. Math. Phys., 47 (6), pp. 063504, 19, 2006, ISSN: 0022-2488. Journal ArticleAbstract @article{MR2239979, title = {Scale invariance in the spectral action}, author = {Ali H. Chamseddine and Alain Connes}, url = {https://fredericc1.sg-host.com/wp-content/uploads/dilatonfinal.pdf}, doi = {10.1063/1.2196748}, issn = {0022-2488}, year = {2006}, date = {2006-01-01}, journal = {J. Math. Phys.}, volume = {47}, number = {6}, pages = {063504, 19}, abstract = {The arbitrary mass scale in the spectral action is made dynamical by in- troducing a scaling dilaton field. We evaluate all the low-energy terms in the spectral action and determine the dilaton couplings. These results are applied to the spectral action of the noncommutative space defined by the standard model. We show that the effective action for all matter couplings is scale invariant, except for the dilaton kinetic term and Einstein-Hilbert term. The resulting action is almost identical to the one proposed for making the standard model scale invariant as well as the model for extended inflation and has the same low-energy limit as the Randall-Sundrum model. Remarkably, all desirable features with correct signs for the relevant terms are obtained uniquely and without any fine tuning}, keywords = {}, pubstate = {published}, tppubtype = {article} } The arbitrary mass scale in the spectral action is made dynamical by in- troducing a scaling dilaton field. We evaluate all the low-energy terms in the spectral action and determine the dilaton couplings. These results are applied to the spectral action of the noncommutative space defined by the standard model. We show that the effective action for all matter couplings is scale invariant, except for the dilaton kinetic term and Einstein-Hilbert term. The resulting action is almost identical to the one proposed for making the standard model scale invariant as well as the model for extended inflation and has the same low-energy limit as the Randall-Sundrum model. Remarkably, all desirable features with correct signs for the relevant terms are obtained uniquely and without any fine tuning |

187. | On the foundations of noncommutative geometry Alain Connes The unity of mathematics, 244 , pp. 173–204, Birkhäuser Boston, Boston, MA, 2006. @incollection{MR2181806, title = {On the foundations of noncommutative geometry}, author = {Alain Connes}, doi = {10.1007/0-8176-4467-9_5}, year = {2006}, date = {2006-01-01}, booktitle = {The unity of mathematics}, volume = {244}, pages = {173--204}, publisher = {Birkhäuser Boston, Boston, MA}, series = {Progr. Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

186. | Alain Connes, Matilde Marcolli J. Geom. Phys., 56 (1), pp. 55–85, 2006, ISSN: 0393-0440. Journal ArticleAbstract @article{MR2170600, title = {Quantum fields and motives}, author = {Alain Connes and Matilde Marcolli}, url = {https://fredericc1.sg-host.com/wp-content/uploads/qftmotives.pdf}, doi = {10.1016/j.geomphys.2005.04.004}, issn = {0393-0440}, year = {2006}, date = {2006-01-01}, journal = {J. Geom. Phys.}, volume = {56}, number = {1}, pages = {55--85}, abstract = {The main idea of renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, labelled by the Feynman graphs that encode the combinatorics of the perturbative expansion of the theory. These counterterms have the effect of cancelling the ultraviolet divergences. Thus, in the procedure of perturbative renormalization, one introduces a counterterm C(Γ) in the initial Lagrangian for every divergent one particle irreducible (1PI) Feynman diagram Γ. In the case of a renormalizable theory, all the necessary counterterms C(Γ) can be obtained by modifying the numerical parameters that appear in the original Lagrangian. It is possible to modify these parameters and replace them by (divergent) series, since they are not observable, unlike actual physical quantities that have to be finite. One of the fundamental difficulties with any renormaliza- tion procedure is a systematic treatment of nested and overlapping divergences in multiloop diagrams.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The main idea of renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, labelled by the Feynman graphs that encode the combinatorics of the perturbative expansion of the theory. These counterterms have the effect of cancelling the ultraviolet divergences. Thus, in the procedure of perturbative renormalization, one introduces a counterterm C(Γ) in the initial Lagrangian for every divergent one particle irreducible (1PI) Feynman diagram Γ. In the case of a renormalizable theory, all the necessary counterterms C(Γ) can be obtained by modifying the numerical parameters that appear in the original Lagrangian. It is possible to modify these parameters and replace them by (divergent) series, since they are not observable, unlike actual physical quantities that have to be finite. One of the fundamental difficulties with any renormaliza- tion procedure is a systematic treatment of nested and overlapping divergences in multiloop diagrams. |

185. | ℚ-lattices: quantum statistical mechanics and Galois theory Alain Connes, Matilde Marcolli J. Geom. Phys., 56 (1), pp. 2–23, 2006, ISSN: 0393-0440. Journal ArticleAbstract @article{MR2170598, title = {ℚ-lattices: quantum statistical mechanics and Galois theory}, author = {Alain Connes and Matilde Marcolli}, doi = {10.1016/j.geomphys.2005.04.010}, issn = {0393-0440}, year = {2006}, date = {2006-01-01}, journal = {J. Geom. Phys.}, volume = {56}, number = {1}, pages = {2--23}, abstract = {In this paper we show that the theory of modular Hecke algebras, the spectral realization of zeros of L-functions, and the arithmetic properties of KMS states in quantum statistical mechanics combine into a unique general picture based on the noncommutative geometry of the space of commensurability classes of Q-lattices.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper we show that the theory of modular Hecke algebras, the spectral realization of zeros of L-functions, and the arithmetic properties of KMS states in quantum statistical mechanics combine into a unique general picture based on the noncommutative geometry of the space of commensurability classes of Q-lattices. |

184. | KMS states and complex multiplication Alain Connes, Matilde Marcolli, Niranjan Ramachandran Selecta Math. (N.S.), 11 (3-4), pp. 325–347, 2005, ISSN: 1022-1824. Journal ArticleAbstract @article{MR2215258, title = {KMS states and complex multiplication}, author = {Alain Connes and Matilde Marcolli and Niranjan Ramachandran}, url = {/wp-content/uploads/cmr0.pdf}, doi = {10.1007/s00029-005-0013-x}, issn = {1022-1824}, year = {2005}, date = {2005-01-01}, journal = {Selecta Math. (N.S.)}, volume = {11}, number = {3-4}, pages = {325--347}, abstract = {The problem originates from the work of Bost–Connes, [3], [4], where a system with all the properties listed above was constructed for K = Q. Important developments in the direction of generalizing the Bost–Connes system to other number fields were obtained by Harari and Leichtnam [10], Cohen [6], Arledge, Laca and Raeburn [1], Laca and van Frankenhuijsen [11]. These results all assume restrictions on the class number of K. It was widely believed that a system satisfying all the properties of Problem 1.1 would exist (supposedly for any number field and certainly at least in the case where K is an imaginary quadratic field). However, a complete construction (without special assumptions on the class number) had not been obtained so far for any case other than Q. The purpose of the present paper is to give a complete solution to Problem 1.1, for K an imaginary quadratic field, without any restriction on the class number of K. In an accompanying paper [8], we explain the geometry underlying and motivating the construction presented in this paper, and we make a detailed comparison between the properties of the system described here, the original Bost–Connes system [4] and the GL2-system of [7].}, keywords = {}, pubstate = {published}, tppubtype = {article} } The problem originates from the work of Bost–Connes, [3], [4], where a system with all the properties listed above was constructed for K = Q. Important developments in the direction of generalizing the Bost–Connes system to other number fields were obtained by Harari and Leichtnam [10], Cohen [6], Arledge, Laca and Raeburn [1], Laca and van Frankenhuijsen [11]. These results all assume restrictions on the class number of K. It was widely believed that a system satisfying all the properties of Problem 1.1 would exist (supposedly for any number field and certainly at least in the case where K is an imaginary quadratic field). However, a complete construction (without special assumptions on the class number) had not been obtained so far for any case other than Q. The purpose of the present paper is to give a complete solution to Problem 1.1, for K an imaginary quadratic field, without any restriction on the class number of K. In an accompanying paper [8], we explain the geometry underlying and motivating the construction presented in this paper, and we make a detailed comparison between the properties of the system described here, the original Bost–Connes system [4] and the GL2-system of [7]. |

183. | ?²-homology for von Neumann algebras Alain Connes, Dimitri Shlyakhtenko J. Reine Angew. Math., 586 , pp. 125–168, 2005, ISSN: 0075-4102. Journal ArticleAbstract @article{MR2180603, title = {?²-homology for von Neumann algebras}, author = {Alain Connes and Dimitri Shlyakhtenko}, url = {/wp-content/uploads/betti.pdf}, doi = {10.1515/crll.2005.2005.586.125}, issn = {0075-4102}, year = {2005}, date = {2005-01-01}, journal = {J. Reine Angew. Math.}, volume = {586}, pages = {125--168}, abstract = {The aim of this paper is to introduce a notion of ?²-homology in the context of von Neumann algebras. Finding a suitable (co)homology theory for von Neumann algebras has been a dream for several generations (see [KR71a, KR71b, JKR72, SS95] and references therein). One’s hope is to have a powerful invariant to distinguish von Neumann algebras. Unfortunately, little positive is known about the Kadison-Ringrose cohomology Hb∗(M,M), except that it vanishes in many cases. Furthermore, there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von Neumann algebra.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The aim of this paper is to introduce a notion of ?²-homology in the context of von Neumann algebras. Finding a suitable (co)homology theory for von Neumann algebras has been a dream for several generations (see [KR71a, KR71b, JKR72, SS95] and references therein). One’s hope is to have a powerful invariant to distinguish von Neumann algebras. Unfortunately, little positive is known about the Kadison-Ringrose cohomology Hb∗(M,M), except that it vanishes in many cases. Furthermore, there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von Neumann algebra. |

182. | Alain Connes Butl. Soc. Catalana Mat., 19 (2), pp. 7–23 (2005), 2004, ISSN: 0214-316X. Journal ArticleAbstract @article{MR2146119, title = {Symmetries}, author = {Alain Connes}, url = {/wp-content/uploads/symetries.pdf}, issn = {0214-316X}, year = {2004}, date = {2004-01-01}, journal = {Butl. Soc. Catalana Mat.}, volume = {19}, number = {2}, pages = {7--23 (2005)}, abstract = {Multiples aspects du concept de symétrie}, keywords = {}, pubstate = {published}, tppubtype = {article} } Multiples aspects du concept de symétrie |

181. | Renormalisation et ambiguïté galoisienne Alain Connes Asterisque, (296), pp. 113–143, 2004, ISSN: 0303-1179, (Analyse complexe, syst`emes dynamiques, sommabilité des séries divergentes et théories galoisiennes. I). IncollectionAbstract @incollection{MR2135686, title = {Renormalisation et ambiguïté galoisienne}, author = {Alain Connes}, url = {/wp-content/uploads/renorm-galois.pdf}, issn = {0303-1179}, year = {2004}, date = {2004-01-01}, booktitle = {Asterisque}, journal = {Astérisque}, number = {296}, pages = {113--143}, abstract = {— This paper contains a detailed ex- position of my joint work with Kreimer on renormalization. The first key result is the identity between the recursive process used by physicists to remove the divergen- cies in quantum field theory and the Birkhoff decomposition of loops with values in a pro-unipotent Lie group. The relevant group for renormalization is the group of diffeographisms which is constructed from Feynmaii graphs. The second key result is the construction of an action of the group of diffeographisms on the dimensionless coupling constants of the theory. The precise link between my work with Kreimer and the Riemann-Hilbert correspondence was obtained in collaboration with M. Marcolli and is explained briefly at the end of the paper. We construct a Riemann-Hilbert correspondence between flat equisingular connections and representations of a spe- cific motivic Galois group U*. This group is the analogue in renormalization of the exponential torus of Ramis in the local theory of irregular singular differential equa- tions. Our work gives a natural candidate for the "cosmic Galois group'envisaged by Cartier as the symmetry underlying renormalization.}, note = {Analyse complexe, syst`emes dynamiques, sommabilité des séries divergentes et théories galoisiennes. I}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } — This paper contains a detailed ex- position of my joint work with Kreimer on renormalization. The first key result is the identity between the recursive process used by physicists to remove the divergen- cies in quantum field theory and the Birkhoff decomposition of loops with values in a pro-unipotent Lie group. The relevant group for renormalization is the group of diffeographisms which is constructed from Feynmaii graphs. The second key result is the construction of an action of the group of diffeographisms on the dimensionless coupling constants of the theory. The precise link between my work with Kreimer and the Riemann-Hilbert correspondence was obtained in collaboration with M. Marcolli and is explained briefly at the end of the paper. We construct a Riemann-Hilbert correspondence between flat equisingular connections and representations of a spe- cific motivic Galois group U*. This group is the analogue in renormalization of the exponential torus of Ramis in the local theory of irregular singular differential equa- tions. Our work gives a natural candidate for the "cosmic Galois group'envisaged by Cartier as the symmetry underlying renormalization. |

180. | Nombres de Betti ?² et facteurs de type II₁ (d'après D. Gaboriau et S. Popa) Alain Connes Astérisque, (294), pp. ix, 321–333, 2004, ISSN: 0303-1179. Journal ArticleAbstract @article{MR2111648, title = {Nombres de Betti ?² et facteurs de type II₁ (d'après D. Gaboriau et S. Popa)}, author = {Alain Connes}, url = {/wp-content/uploads/bourbaki-popa.pdf}, issn = {0303-1179}, year = {2004}, date = {2004-01-01}, journal = {Astérisque}, number = {294}, pages = {ix, 321--333}, abstract = {Seminaire Bourbaki sur les travaux de Sorin Popa.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Seminaire Bourbaki sur les travaux de Sorin Popa. |

179. | Renormalization and motivic Galois theory Alain Connes, Matilde Marcolli Int. Math. Res. Not., (76), pp. 4073–4091, 2004, ISSN: 1073-7928. @article{MR2109986, title = {Renormalization and motivic Galois theory}, author = {Alain Connes and Matilde Marcolli}, doi = {10.1155/S1073792804143122}, issn = {1073-7928}, year = {2004}, date = {2004-01-01}, journal = {Int. Math. Res. Not.}, number = {76}, pages = {4073--4091}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

178. | Rankin-Cohen brackets and the Hopf algebra of transverse geometry Alain Connes, Henri Moscovici Mosc. Math. J., 4 (1), pp. 111–130, 311, 2004, ISSN: 1609-3321. Journal ArticleAbstract @article{MR2074985, title = {Rankin-Cohen brackets and the Hopf algebra of transverse geometry}, author = {Alain Connes and Henri Moscovici}, url = {/wp-content/uploads/RC.pdf}, doi = {10.17323/1609-4514-2004-4-1-111-130}, issn = {1609-3321}, year = {2004}, date = {2004-01-01}, journal = {Mosc. Math. J.}, volume = {4}, number = {1}, pages = {111--130, 311}, abstract = {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 to modular Hecke al- gebras. More generally, our procedure yields such brackets on any associative algebra endowed with an action of the Hopf algebra of transverse geometry in codimension one, such that the derivation cor- responding to the Schwarzian derivative is inner. Moreover, we show in full generality that these Rankin-Cohen brackets give rise to asso- ciative deformations.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 to modular Hecke al- gebras. More generally, our procedure yields such brackets on any associative algebra endowed with an action of the Hopf algebra of transverse geometry in codimension one, such that the derivation cor- responding to the Schwarzian derivative is inner. Moreover, we show in full generality that these Rankin-Cohen brackets give rise to asso- ciative deformations. |

177. | Modular Hecke algebras and their Hopf symmetry Alain Connes, Henri Moscovici Mosc. Math. J., 4 (1), pp. 67–109, 310, 2004, ISSN: 1609-3321. Journal ArticleAbstract @article{MR2074984, title = {Modular Hecke algebras and their Hopf symmetry}, author = {Alain Connes and Henri Moscovici}, url = {/wp-content/uploads/modular.pdf}, doi = {10.17323/1609-4514-2004-4-1-67-109}, issn = {1609-3321}, year = {2004}, date = {2004-01-01}, journal = {Mosc. Math. J.}, volume = {4}, number = {1}, pages = {67--109, 310}, abstract = {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 Hecke operators. At the intuitive level, these are algebras of ‘polynomial coordinates’ for the ‘transverse space’ of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 folia- tions. Its action is shown to span the ‘holomorphic tangent space’ of the noncommutative space, and each of its three basic Hopf cyclic co- cycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin-Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon-Vey cocycle gives rise to a 1-cocycle on PSL(2, Q) with val- ues in Eisenstein series of weight 2, which, when coupled with the ‘period’ cocycle, yields a representative of the Euler class.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 Hecke operators. At the intuitive level, these are algebras of ‘polynomial coordinates’ for the ‘transverse space’ of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 folia- tions. Its action is shown to span the ‘holomorphic tangent space’ of the noncommutative space, and each of its three basic Hopf cyclic co- cycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin-Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon-Vey cocycle gives rise to a 1-cocycle on PSL(2, Q) with val- ues in Eisenstein series of weight 2, which, when coupled with the ‘period’ cocycle, yields a representative of the Euler class. |

176. | Alain Connes, Joachim Cuntz, Eric Guentner, Nigel Higson, Jerry Kaminker, John Roberts Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2004, ISBN: 3-540-20357-5. BookAbstract @book{MR2067646, title = {Noncommutative geometry}, author = {Alain Connes and Joachim Cuntz and Eric Guentner and Nigel Higson and Jerry Kaminker and John Roberts}, editor = {Sergio Doplicher and Roberto Longo}, doi = {10.1007/b94118}, isbn = {3-540-20357-5}, year = {2004}, date = {2004-01-01}, volume = {1831}, pages = {xiv+343}, publisher = {Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence}, series = {Lecture Notes in Mathematics}, abstract = {Lectures given at the C.I.M.E. Summer School held in Martina Franca, September 3--9, 2000}, keywords = {}, pubstate = {published}, tppubtype = {book} } Lectures given at the C.I.M.E. Summer School held in Martina Franca, September 3--9, 2000 |

175. | Cyclic cohomology, noncommutative geometry and quantum group symmetries Alain Connes Noncommutative geometry, 1831 , pp. 1–71, Springer, Berlin, 2004. @incollection{MR2058472, title = {Cyclic cohomology, noncommutative geometry and quantum group symmetries}, author = {Alain Connes}, doi = {10.1007/978-3-540-39702-1_1}, year = {2004}, date = {2004-01-01}, booktitle = {Noncommutative geometry}, volume = {1831}, pages = {1--71}, publisher = {Springer, Berlin}, series = {Lecture Notes in Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

174. | Cyclic cohomology, quantum group symmetries and the local index formula for ???(2) Alain Connes J. Inst. Math. Jussieu, 3 (1), pp. 17–68, 2004, ISSN: 1474-7480. Journal ArticleAbstract @article{MR2036597, title = {Cyclic cohomology, quantum group symmetries and the local index formula for ???(2)}, author = {Alain Connes}, url = {/wp-content/uploads/QuantumgroupSU2-1.pdf}, doi = {10.1017/S1474748004000027}, issn = {1474-7480}, year = {2004}, date = {2004-01-01}, journal = {J. Inst. Math. Jussieu}, volume = {3}, number = {1}, pages = {17--68}, abstract = {We analyse the NC-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 theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudo-differential calculus, the Wodzciki-type residue, and the local cyclic cocycle giving the index formula. The cochain whose coboundary is the difference between the original Chern character and the local one is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. This specific example allows to illustrate the general notion of locality in NCG. The formulas computing the residue are ”local”. Locality by stripping all the expressions from irrelevant details makes them computable. The key fea- ture of this spectral triple is its equivariance, i.e. the SUq(2)-symmetry. We shall explain how this leads naturally to the general concept of invari- ant cyclic cohomology in the framework of quantum group symmetries.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We analyse the NC-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 theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudo-differential calculus, the Wodzciki-type residue, and the local cyclic cocycle giving the index formula. The cochain whose coboundary is the difference between the original Chern character and the local one is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. This specific example allows to illustrate the general notion of locality in NCG. The formulas computing the residue are ”local”. Locality by stripping all the expressions from irrelevant details makes them computable. The key fea- ture of this spectral triple is its equivariance, i.e. the SUq(2)-symmetry. We shall explain how this leads naturally to the general concept of invari- ant cyclic cohomology in the framework of quantum group symmetries. |

173. | Symétries galoisiennes et renormalisation Alain Connes Poincaré Seminar 2002, 30 , pp. 241–264, Birkhäuser, Basel, 2003. IncollectionAbstract @incollection{MR2169915, title = {Symétries galoisiennes et renormalisation}, author = {Alain Connes}, url = {/wp-content/uploads/bourbaphyappeared.pdf}, year = {2003}, date = {2003-01-01}, booktitle = {Poincaré Seminar 2002}, volume = {30}, pages = {241--264}, publisher = {Birkhäuser, Basel}, series = {Prog. Math. Phys.}, abstract = {Nous exposons notre travail en collaboration avec Dirk Kreimer sur les alg`ebres de Hopf et de Lie associ ́ees aux graphes de Feynman, et sur la signification con- ceptuelle de la renormalisation perturbative `a partir du probl`eme de Riemann- Hilbert. Nous interpr ́etons ensuite le groupe de renormalisation comme un groupe d’ambigu ̈ıt ́e et montrons le rˆole que ce groupe devrait jouer pour comprendre la composante connexe du groupe des classes d’id`eles de la th ́eorie du corps de classe comme un groupe de Galois.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } Nous exposons notre travail en collaboration avec Dirk Kreimer sur les alg`ebres de Hopf et de Lie associ ́ees aux graphes de Feynman, et sur la signification con- ceptuelle de la renormalisation perturbative `a partir du probl`eme de Riemann- Hilbert. Nous interpr ́etons ensuite le groupe de renormalisation comme un groupe d’ambigu ̈ıt ́e et montrons le rˆole que ce groupe devrait jouer pour comprendre la composante connexe du groupe des classes d’id`eles de la th ́eorie du corps de classe comme un groupe de Galois. |

172. | Moduli space and structure of noncommutative 3-spheres Alain Connes, Michel Dubois-Violette Lett. Math. Phys., 66 (1-2), pp. 91–121, 2003, ISSN: 0377-9017. Journal ArticleAbstract @article{MR2064594, title = {Moduli space and structure of noncommutative 3-spheres}, author = {Alain Connes and Michel Dubois-Violette}, url = {/wp-content/uploads/modulispace3spheresMDV.pdf}, doi = {10.1023/B:MATH.0000017678.10681.1e}, issn = {0377-9017}, year = {2003}, date = {2003-01-01}, journal = {Lett. Math. Phys.}, volume = {66}, number = {1-2}, pages = {91--121}, abstract = {We analyse the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the C∗-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pairs of points in a symmetric spaceof unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with 8 points in common. We show that generically these curves are the same as the characteristic variety of the associated quadratic algebra. We then apply the general theory of central quadratic forms to show that the noncommutative 3-spheres admit a natural ramified covering π by a noncommutative 3-dimensional nilmanifold. This yields the differential calculus. We then compute the Jacobian of the ramified covering π by pairing the direct image of the fundamental class of the noncommutative 3--dimensional nilmanifold with the Chern character of the defining unitary and obtain the answer as the product of a period (of an elliptic integral) by a rational function}, keywords = {}, pubstate = {published}, tppubtype = {article} } We analyse the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the C∗-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pairs of points in a symmetric spaceof unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with 8 points in common. We show that generically these curves are the same as the characteristic variety of the associated quadratic algebra. We then apply the general theory of central quadratic forms to show that the noncommutative 3-spheres admit a natural ramified covering π by a noncommutative 3-dimensional nilmanifold. This yields the differential calculus. We then compute the Jacobian of the ramified covering π by pairing the direct image of the fundamental class of the noncommutative 3--dimensional nilmanifold with the Chern character of the defining unitary and obtain the answer as the product of a period (of an elliptic integral) by a rational function |

171. | Hommage à Laurent Schwartz Alain Connes Gaz. Math., (94), pp. 7–8, 2002, ISSN: 0224-8999. Journal Article @article{MR2067166, title = {Hommage à Laurent Schwartz}, author = {Alain Connes}, issn = {0224-8999}, year = {2002}, date = {2002-01-01}, journal = {Gaz. Math.}, number = {94}, pages = {7--8}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

170. | Noncommutative geometry year 2000 Alain Connes Highlights of mathematical physics (London, 2000), pp. 49–110, Amer. Math. Soc., Providence, RI, 2002. Incollection @incollection{MR2001573, title = {Noncommutative geometry year 2000}, author = {Alain Connes}, year = {2002}, date = {2002-01-01}, booktitle = {Highlights of mathematical physics (London, 2000)}, pages = {49--110}, publisher = {Amer. Math. Soc., Providence, RI}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

169. | Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples Alain Connes, Michel Dubois-Violette Comm. Math. Phys., 230 (3), pp. 539–579, 2002, ISSN: 0010-3616. Journal ArticleAbstract @article{MR1937657, title = {Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples}, author = {Alain Connes and Michel Dubois-Violette}, url = {/wp-content/uploads/NCmanifolds.pdf}, doi = {10.1007/s00220-002-0715-2}, issn = {0010-3616}, year = {2002}, date = {2002-01-01}, journal = {Comm. Math. Phys.}, volume = {230}, number = {3}, pages = {539--579}, abstract = {We exhibit large classes of examples of noncommutative finite-dimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S3 defined by basic K-theoretic equations. }, keywords = {}, pubstate = {published}, tppubtype = {article} } We exhibit large classes of examples of noncommutative finite-dimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S3 defined by basic K-theoretic equations. |

168. | Alain Connes, Michel Dubois-Violette Lett. Math. Phys., 61 (2), pp. 149–158, 2002, ISSN: 0377-9017. Journal ArticleAbstract @article{MR1936574, title = {Yang-Mills algebra}, author = {Alain Connes and Michel Dubois-Violette}, url = {/wp-content/uploads/yangmills.pdf}, doi = {10.1023/A:1020733628744}, issn = {0377-9017}, year = {2002}, date = {2002-01-01}, journal = {Lett. Math. Phys.}, volume = {61}, number = {2}, pages = {149--158}, abstract = {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 pointed out. This algebra is Gorenstein and Koszul of global dimension 3 but except for s=1 (i.e. in the 2-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin-Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincare series of this algebra A and for the dimension in degree n of the graded Lie algebra of which A is the universal enveloping algebra. In the 4-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie-algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 pointed out. This algebra is Gorenstein and Koszul of global dimension 3 but except for s=1 (i.e. in the 2-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin-Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincare series of this algebra A and for the dimension in degree n of the graded Lie algebra of which A is the universal enveloping algebra. In the 4-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie-algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one. |

167. | Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs Alain Connes, Dirk Kreimer Ann. Henri Poincaré, 3 (3), pp. 411–433, 2002, ISSN: 1424-0637. Journal ArticleAbstract @article{MR1915297, title = {Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs}, author = {Alain Connes and Dirk Kreimer}, url = {/wp-content/uploads/insertion.pdf}, doi = {10.1007/s00023-002-8622-9}, issn = {1424-0637}, year = {2002}, date = {2002-01-01}, journal = {Ann. Henri Poincaré}, volume = {3}, number = {3}, pages = {411--433}, abstract = {The Lie algebra of Feynman graphs gives rise to two natural rep- resentations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The Lie algebra of Feynman graphs gives rise to two natural rep- resentations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine. |

166. | Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry Alain Connes, Henri Moscovici Essays on geometry and related topics, Vol. 1, 2, 38 , pp. 217–255, Enseignement Math., Geneva, 2001. IncollectionAbstract @incollection{MR1929328, title = {Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry}, author = {Alain Connes and Henri Moscovici}, url = {/wp-content/uploads/haefliger.pdf}, year = {2001}, date = {2001-01-01}, booktitle = {Essays on geometry and related topics, Vol. 1, 2}, volume = {38}, pages = {217--255}, publisher = {Enseignement Math., Geneva}, series = {Monogr. Enseign. Math.}, abstract = {We prove a cyclic cohomological analogue of Haefliger’s van Est- type theorem for the groupoid of germs of diffeomorphisms of a man- ifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential operators on that groupoid, which is shown to carry an intrinsic Hopf algebraic structure. We establish a canonical isomorphism between the periodic Hopf cyclic cohomology of this extended Hopf algebra and the Gelfand-Fuchs cohomology of the Lie algebra of formal vector fields. We then show that this iso- morphism can be explicitly implemented at the cochain level, by a cochain map constructed out of a fixed torsion-free linear connection. This allows the direct treatment of the index formula for the hypoelliptic signature operator – representing the diffeomorphism invariant transverse fundamental K-homology class of an oriented manifold – in the general case, when this operator is constructed by means of an arbitrary coupling connection. }, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We prove a cyclic cohomological analogue of Haefliger’s van Est- type theorem for the groupoid of germs of diffeomorphisms of a man- ifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential operators on that groupoid, which is shown to carry an intrinsic Hopf algebraic structure. We establish a canonical isomorphism between the periodic Hopf cyclic cohomology of this extended Hopf algebra and the Gelfand-Fuchs cohomology of the Lie algebra of formal vector fields. We then show that this iso- morphism can be explicitly implemented at the cochain level, by a cochain map constructed out of a fixed torsion-free linear connection. This allows the direct treatment of the index formula for the hypoelliptic signature operator – representing the diffeomorphism invariant transverse fundamental K-homology class of an oriented manifold – in the general case, when this operator is constructed by means of an arbitrary coupling connection. |

165. | From local perturbation theory to Hopf and Lie algebras of Feynman graphs Alain Connes, Dirk Kreimer Mathematical physics in mathematics and physics (Siena, 2000), 30 , pp. 105–114, Amer. Math. Soc., Providence, RI, 2001. @incollection{MR1867549, title = {From local perturbation theory to Hopf and Lie algebras of Feynman graphs}, author = {Alain Connes and Dirk Kreimer}, doi = {10.1007/pl00005547}, year = {2001}, date = {2001-01-01}, booktitle = {Mathematical physics in mathematics and physics (Siena, 2000)}, volume = {30}, pages = {105--114}, publisher = {Amer. Math. Soc., Providence, RI}, series = {Fields Inst. Commun.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

164. | Alain Connes, André Lichnerowicz, Marcel Paul Schützenberger American Mathematical Society, Providence, RI, 2001, ISBN: 0-8218-2614-X. BookAbstract @book{MR1861272, title = {Triangle of thoughts}, author = {Alain Connes and André Lichnerowicz and Marcel Paul Schützenberger}, isbn = {0-8218-2614-X}, year = {2001}, date = {2001-01-01}, pages = {viii+179}, publisher = {American Mathematical Society}, address = {Providence, RI}, abstract = {Our view of the world today is fundamentally influenced by 20th-century results in physics and mathematics. Here, three members of the French Academy of Sciences: Alain Connes, Andre Lichnerowicz, and Marcel Paul Schutzenberger, discuss the relations among mathematics, physics, and philosophy, and other sciences. Written in the form of conversations among three brilliant scientists and deep thinkers, the book touches on, among others, the following questions: is there a primordial truth that exists beyond the realm of what is provable? More generally, is there a distinction between what is true in mathematics and what is provable? How is mathematics different from other sciences? How is it the same? Does mathematics have an object or an object of study, the way physics, chemistry and biology do? Translated from the 2000 French original by Jennifer Gage.}, keywords = {}, pubstate = {published}, tppubtype = {book} } Our view of the world today is fundamentally influenced by 20th-century results in physics and mathematics. Here, three members of the French Academy of Sciences: Alain Connes, Andre Lichnerowicz, and Marcel Paul Schutzenberger, discuss the relations among mathematics, physics, and philosophy, and other sciences. Written in the form of conversations among three brilliant scientists and deep thinkers, the book touches on, among others, the following questions: is there a primordial truth that exists beyond the realm of what is provable? More generally, is there a distinction between what is true in mathematics and what is provable? How is mathematics different from other sciences? How is it the same? Does mathematics have an object or an object of study, the way physics, chemistry and biology do? Translated from the 2000 French original by Jennifer Gage. |

163. | From local perturbation theory to Hopf and Lie algebras of Feynman graphs Alain Connes, Dirk Kreimer 56 (1), pp. 3–15, 2001, ISSN: 0377-9017, (EuroConférence Moshé Flato 2000, Part I (Dijon)). @incollection{MR1848162, title = {From local perturbation theory to Hopf and Lie algebras of Feynman graphs}, author = {Alain Connes and Dirk Kreimer}, doi = {10.1023/A:1010939000212}, issn = {0377-9017}, year = {2001}, date = {2001-01-01}, journal = {Lett. Math. Phys.}, volume = {56}, number = {1}, pages = {3--15}, note = {EuroConférence Moshé Flato 2000, Part I (Dijon)}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

162. | Noncommutative manifolds, the instanton algebra and isospectral deformations Alain Connes, Giovanni Landi Comm. Math. Phys., 221 (1), pp. 141–159, 2001, ISSN: 0010-3616. Journal ArticleAbstract @article{MR1846904, title = {Noncommutative manifolds, the instanton algebra and isospectral deformations}, author = {Alain Connes and Giovanni Landi}, url = {/wp-content/uploads/giannifinal.pdf}, doi = {10.1007/PL00005571}, issn = {0010-3616}, year = {2001}, date = {2001-01-01}, journal = {Comm. Math. Phys.}, volume = {221}, number = {1}, pages = {141--159}, abstract = {We give new examples of noncommutative manifolds with non trivial global properties. }, keywords = {}, pubstate = {published}, tppubtype = {article} } We give new examples of noncommutative manifolds with non trivial global properties. |

161. | Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The ?-function, diffeomorphisms and the renormalization group Alain Connes, Dirk Kreimer Comm. Math. Phys., 216 (1), pp. 215–241, 2001, ISSN: 0010-3616. @article{MR1810779, title = {Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The ?-function, diffeomorphisms and the renormalization group}, author = {Alain Connes and Dirk Kreimer}, doi = {10.1007/PL00005547}, issn = {0010-3616}, year = {2001}, date = {2001-01-01}, journal = {Comm. Math. Phys.}, volume = {216}, number = {1}, pages = {215--241}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

160. | Alain Connes, André Lichnerowicz, Marcel Paul Schützenberger Odile Jacob, Paris, 2000, ISBN: 9780821826140. BookAbstract @book{Connes2000, title = {Triangle de pensées}, author = {Alain Connes and André Lichnerowicz and Marcel Paul Schützenberger}, isbn = {9780821826140}, year = {2000}, date = {2000-01-26}, publisher = {Odile Jacob}, address = {Paris}, abstract = {Les grandes découvertes scientifiques du XXe siècle, telles que la relativité générale, la mécanique quantique ou le théorème de Gödel, modifient en profondeur notre perception de la réalité. Les physiciens sont arrivés en un siècle à un modèle extraordinairement efficace de la réalité matérielle. Aux échelles microscopiques, la mécanique quantique résiste avec succès aux tentatives les plus ingénieuses pour la prendre en défaut. Aux échelles des galaxies, l'observation des pulsars binaires donne des confirmations éclatantes de la relativité générale d'Einstein. Le but de cet ouvrage est de permettre de franchir le décalage croissant entre les subtilités de ces modifications, appréciées des seuls spécialistes, et l'image souvent incroyablement déformée qu'en reçoit le public. Chacun des trois auteurs est l'un des sommets de ce triangle de pensées, au sein duquel le lecteur est invité à trouver sa place. Alain Connes, mathématicien, est professeur au Collège de France à la chaire d'Analyse et Géométrie, membre de l'Académie des sciences et de plusieurs académies étrangères, dont la National Academy of Science des États-Unis. Il a obtenu la médaille Fields, en 1982. André Lichnerowicz, mathématicien, grand géomètre et physicien théoricien, spécialiste de la relativité générale était professeur au Collège de France, membre de l'Académie des sciences et Docteur honoris causa de plusieurs académies et universités étrangères. Marcel Paul Schützenberger, membre de l'Académie des sciences, a contribué de manière éclatante à la combinatoire et à la théorie des graphes. Il était un esprit pluridisciplinaire, à la fois médecin, biologiste, psychiatre, linguiste et algébriste.}, keywords = {}, pubstate = {published}, tppubtype = {book} } Les grandes découvertes scientifiques du XXe siècle, telles que la relativité générale, la mécanique quantique ou le théorème de Gödel, modifient en profondeur notre perception de la réalité. Les physiciens sont arrivés en un siècle à un modèle extraordinairement efficace de la réalité matérielle. Aux échelles microscopiques, la mécanique quantique résiste avec succès aux tentatives les plus ingénieuses pour la prendre en défaut. Aux échelles des galaxies, l'observation des pulsars binaires donne des confirmations éclatantes de la relativité générale d'Einstein. Le but de cet ouvrage est de permettre de franchir le décalage croissant entre les subtilités de ces modifications, appréciées des seuls spécialistes, et l'image souvent incroyablement déformée qu'en reçoit le public. Chacun des trois auteurs est l'un des sommets de ce triangle de pensées, au sein duquel le lecteur est invité à trouver sa place. Alain Connes, mathématicien, est professeur au Collège de France à la chaire d'Analyse et Géométrie, membre de l'Académie des sciences et de plusieurs académies étrangères, dont la National Academy of Science des États-Unis. Il a obtenu la médaille Fields, en 1982. André Lichnerowicz, mathématicien, grand géomètre et physicien théoricien, spécialiste de la relativité générale était professeur au Collège de France, membre de l'Académie des sciences et Docteur honoris causa de plusieurs académies et universités étrangères. Marcel Paul Schützenberger, membre de l'Académie des sciences, a contribué de manière éclatante à la combinatoire et à la théorie des graphes. Il était un esprit pluridisciplinaire, à la fois médecin, biologiste, psychiatre, linguiste et algébriste. |

159. | A lecture on noncommutative geometry Alain Connes (Special Issue), pp. 31–64, 2000, ISSN: 1120-6330, (Mathematics towards the third millennium (Rome, 1999)). Incollection @incollection{MR1839740, title = {A lecture on noncommutative geometry}, author = {Alain Connes}, issn = {1120-6330}, year = {2000}, date = {2000-01-01}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.}, number = {Special Issue}, pages = {31--64}, note = {Mathematics towards the third millennium (Rome, 1999)}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

158. | Noncommutative geometry year 2000 Alain Connes (Special Volume, Part II), pp. 481–559, 2000, ISSN: 1016-443X, (GAFA 2000 (Tel Aviv, 1999)). IncollectionAbstract @incollection{MR1826266, title = {Noncommutative geometry year 2000}, author = {Alain Connes}, url = {/wp-content/uploads/2000.pdf}, doi = {10.1007/978-3-0346-0425-3_3}, issn = {1016-443X}, year = {2000}, date = {2000-01-01}, journal = {Geom. Funct. Anal.}, number = {Special Volume, Part II}, pages = {481--559}, abstract = {Our geometric concepts evolved first through the discovery of NonEuclidean geometry. The discovery of quantum mechanics in the form of the noncommuting coordinates on the phase space of atomic systems entails an equally drastic evolution. We describe a basic construction which extends the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. The 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 noncommuta- tive spaces, such as the transformation θ → 1/θ for the noncommutative torus T2θ which are 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 fundamental role of Poincare duality in K-homology which is the basic reason for the spectral point of view. This leads us, when specializing to 4-geometries to a universal algebra called the ”Instanton algebra”. We describe our joint work with G. Landi which gives noncommutative spheres Sθ4 from representations of the Instanton algebra. We show that any compact Riemannian spin manifold whose isometry group has rank r ≥ 2 admits isospectral deformations to non- commutative geometries. We give a survey of several recent developments. First our joint work with H. Moscovici on the transverse geometry of foliations which yields a diffeomorphism invariant (rather than the usual covariant one) geometry on the bundle of metrics on a manifold and a natural extension of cyclic cohomology to Hopf algebras. Second, our joint 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 for- mula in the Langlands program. We end with a tentalizing connection between the renormalization group and the missing Galois theory at Archimedian places.}, note = {GAFA 2000 (Tel Aviv, 1999)}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } Our geometric concepts evolved first through the discovery of NonEuclidean geometry. The discovery of quantum mechanics in the form of the noncommuting coordinates on the phase space of atomic systems entails an equally drastic evolution. We describe a basic construction which extends the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. The 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 noncommuta- tive spaces, such as the transformation θ → 1/θ for the noncommutative torus T2θ which are 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 fundamental role of Poincare duality in K-homology which is the basic reason for the spectral point of view. This leads us, when specializing to 4-geometries to a universal algebra called the ”Instanton algebra”. We describe our joint work with G. Landi which gives noncommutative spheres Sθ4 from representations of the Instanton algebra. We show that any compact Riemannian spin manifold whose isometry group has rank r ≥ 2 admits isospectral deformations to non- commutative geometries. We give a survey of several recent developments. First our joint work with H. Moscovici on the transverse geometry of foliations which yields a diffeomorphism invariant (rather than the usual covariant one) geometry on the bundle of metrics on a manifold and a natural extension of cyclic cohomology to Hopf algebras. Second, our joint 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 for- mula in the Langlands program. We end with a tentalizing connection between the renormalization group and the missing Galois theory at Archimedian places. |

157. | Cyclic cohomology and Hopf algebra symmetry Alain Connes, Henri Moscovici Conférence Moshé Flato 1999, Vol. I (Dijon), 21 , pp. 121–147, Kluwer Acad. Publ., Dordrecht, 2000. Incollection @incollection{MR1805887, title = {Cyclic cohomology and Hopf algebra symmetry}, author = {Alain Connes and Henri Moscovici}, year = {2000}, date = {2000-01-01}, booktitle = {Conférence Moshé Flato 1999, Vol. I (Dijon)}, volume = {21}, pages = {121--147}, publisher = {Kluwer Acad. Publ., Dordrecht}, series = {Math. Phys. Stud.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

156. | Geometric ?-theory for Lie groups and foliations Paul Baum, Alain Connes Enseign. Math. (2), 46 (1-2), pp. 3–42, 2000, ISSN: 0013-8584. Journal Article @article{MR1769535, title = {Geometric ?-theory for Lie groups and foliations}, author = {Paul Baum and Alain Connes}, issn = {0013-8584}, year = {2000}, date = {2000-01-01}, journal = {Enseign. Math. (2)}, volume = {46}, number = {1-2}, pages = {3--42}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

155. | A short survey of noncommutative geometry Alain Connes J. Math. Phys., 41 (6), pp. 3832–3866, 2000, ISSN: 0022-2488. @article{MR1768641, title = {A short survey of noncommutative geometry}, author = {Alain Connes}, doi = {10.1063/1.533329}, issn = {0022-2488}, year = {2000}, date = {2000-01-01}, journal = {J. Math. Phys.}, volume = {41}, number = {6}, pages = {3832--3866}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

154. | Noncommutative geometry and the Riemann zeta function Alain Connes Mathematics: frontiers and perspectives, pp. 35–54, Amer. Math. Soc., Providence, RI, 2000. Incollection @incollection{MR1754766, title = {Noncommutative geometry and the Riemann zeta function}, author = {Alain Connes}, year = {2000}, date = {2000-01-01}, booktitle = {Mathematics: frontiers and perspectives}, pages = {35--54}, publisher = {Amer. Math. Soc., Providence, RI}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

153. | Alain Connes, Dirk Kreimer Comm. Math. Phys., 210 (1), pp. 249–273, 2000, ISSN: 0010-3616. Journal ArticleAbstract @article{MR1748177, title = {Renormalization in quantum field theory and the Riemann-Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem}, author = {Alain Connes and Dirk Kreimer}, url = {/wp-content/uploads/RH1.pdf}, doi = {10.1007/s002200050779}, issn = {0010-3616}, year = {2000}, date = {2000-01-01}, journal = {Comm. Math. Phys.}, volume = {210}, number = {1}, pages = {249--273}, abstract = {This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann- Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra H which is commutative as an algebra. It is the dual Hopf algebra of the envelopping algebra of a Lie algebra G whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of H. We shall then show that, using dimensional regularization, the bare (unrenormal- ized) theory gives rise to a loop γ(z) ∈ G , z ∈ C where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z = D of the holomorphic part γ+ of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann- Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra H which is commutative as an algebra. It is the dual Hopf algebra of the envelopping algebra of a Lie algebra G whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of H. We shall then show that, using dimensional regularization, the bare (unrenormal- ized) theory gives rise to a loop γ(z) ∈ G , z ∈ C where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z = D of the holomorphic part γ+ of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title. |

152. | Hopf algebras, renormalization and noncommutative geometry Alain Connes, Dirk Kreimer Quantum field theory: perspective and prospective (Les Houches, 1998), 530 , pp. 59–108, Kluwer Acad. Publ., Dordrecht, 1999. IncollectionAbstract @incollection{MR1725011, title = {Hopf algebras, renormalization and noncommutative geometry}, author = {Alain Connes and Dirk Kreimer}, url = {/wp-content/uploads/ncgk.pdf}, year = {1999}, date = {1999-01-01}, booktitle = {Quantum field theory: perspective and prospective (Les Houches, 1998)}, volume = {530}, pages = {59--108}, publisher = {Kluwer Acad. Publ., Dordrecht}, series = {NATO Sci. Ser. C Math. Phys. Sci.}, abstract = {We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations. |

151. | Renormalization in quantum field theory and the Riemann-Hilbert problem Alain Connes, Dirk Kreimer J. High Energy Phys., (9), pp. Paper 24, 8, 1999, ISSN: 1126-6708. @article{MR1720691, title = {Renormalization in quantum field theory and the Riemann-Hilbert problem}, author = {Alain Connes and Dirk Kreimer}, doi = {10.1088/1126-6708/1999/09/024}, issn = {1126-6708}, year = {1999}, date = {1999-01-01}, journal = {J. High Energy Phys.}, number = {9}, pages = {Paper 24, 8}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

150. | Cyclic cohomology and Hopf algebras Alain Connes, Henri Moscovici 48 (1), pp. 97–108, 1999, ISSN: 0377-9017, (Moshé Flato (1937--1998)). @incollection{MR1718047, title = {Cyclic cohomology and Hopf algebras}, author = {Alain Connes and Henri Moscovici}, doi = {10.1023/A:1007527510226}, issn = {0377-9017}, year = {1999}, date = {1999-01-01}, journal = {Lett. Math. Phys.}, volume = {48}, number = {1}, pages = {97--108}, note = {Moshé Flato (1937--1998)}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

149. | Lessons from quantum field theory: Hopf algebras and spacetime geometries Alain Connes, Dirk Kreimer 48 (1), pp. 85–96, 1999, ISSN: 0377-9017, (Moshé Flato (1937--1998)). IncollectionAbstract @incollection{MR1718046, title = {Lessons from quantum field theory: Hopf algebras and spacetime geometries}, author = {Alain Connes and Dirk Kreimer}, url = {/wp-content/uploads/lessonsQFT.pdf}, doi = {10.1023/A:1007523409317}, issn = {0377-9017}, year = {1999}, date = {1999-01-01}, journal = {Lett. Math. Phys.}, volume = {48}, number = {1}, pages = {85--96}, abstract = {We discuss the prominence of Hopf algebras in recent progress in Quan- tum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a concep- tual understanding of the subtraction procedure. We shall then describe several occurences of this, or closely related Hopf algebras, in other math- ematical domains, such as foliations, Runge-Kutta methods, iterated in- tegrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.}, note = {Moshé Flato (1937--1998)}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We discuss the prominence of Hopf algebras in recent progress in Quan- tum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a concep- tual understanding of the subtraction procedure. We shall then describe several occurences of this, or closely related Hopf algebras, in other math- ematical domains, such as foliations, Runge-Kutta methods, iterated in- tegrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT. |

148. | Hypoelliptic operators, Hopf algebras and cyclic cohomology Alain Connes Algebraic ?-theory and its applications (Trieste, 1997), pp. 164–205, World Sci. Publ., River Edge, NJ, 1999. Incollection @incollection{MR1715875, title = {Hypoelliptic operators, Hopf algebras and cyclic cohomology}, author = {Alain Connes}, year = {1999}, date = {1999-01-01}, booktitle = {Algebraic ?-theory and its applications (Trieste, 1997)}, pages = {164--205}, publisher = {World Sci. Publ., River Edge, NJ}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

147. | Trace formula on the adèle class space and Weil positivity Alain Connes Current developments in mathematics, 1997 (Cambridge, MA), pp. 5–64, Int. Press, Boston, MA, 1999. Incollection @incollection{MR1698852, title = {Trace formula on the adèle class space and Weil positivity}, author = {Alain Connes}, year = {1999}, date = {1999-01-01}, booktitle = {Current developments in mathematics, 1997 (Cambridge, MA)}, pages = {5--64}, publisher = {Int. Press, Boston, MA}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

146. | Trace formula in noncommutative geometry and the zeros of the Riemann zeta function Alain Connes Selecta Math. (N.S.), 5 (1), pp. 29–106, 1999, ISSN: 1022-1824. Journal ArticleAbstract @article{MR1694895, title = {Trace formula in noncommutative geometry and the zeros of the Riemann zeta function}, author = {Alain Connes}, url = {/wp-content/uploads/selecta.ps-2.pdf}, doi = {10.1007/s000290050042}, issn = {1022-1824}, year = {1999}, date = {1999-01-01}, journal = {Selecta Math. (N.S.)}, volume = {5}, number = {1}, pages = {29--106}, abstract = {We give a spectral interpretation of the critical zeros of the Rie- mann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometric interpretation of the explicit formulas of number theory as a trace formula on the noncommutative space of Adele classes. This reduces the Riemann hypothesis to the validity of the trace formula and eliminates the parameter δ of our previous approach.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We give a spectral interpretation of the critical zeros of the Rie- mann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometric interpretation of the explicit formulas of number theory as a trace formula on the noncommutative space of Adele classes. This reduces the Riemann hypothesis to the validity of the trace formula and eliminates the parameter δ of our previous approach. |

145. | A new proof of Morley's theorem Alain Connes Les relations entre les mathématiques et la physique théorique, pp. 43–46, Inst. Hautes Études Sci., Bures-sur-Yvette, 1998. IncollectionAbstract @incollection{MR1667897, title = {A new proof of Morley's theorem}, author = {Alain Connes}, url = {/wp-content/uploads/morley.pdf}, year = {1998}, date = {1998-01-01}, booktitle = {Les relations entre les mathématiques et la physique théorique}, pages = {43--46}, publisher = {Inst. Hautes Études Sci., Bures-sur-Yvette}, abstract = {Around 1899, F. Morley proved a remarkable the- orem on the elementary geometry of Euclidean trian- gles: “Given a triangle A, B, C the pairwise intersec- tions α, β, γ of the trisectors form the vertices of an equilateral triangle” }, keywords = {}, pubstate = {published}, tppubtype = {incollection} } Around 1899, F. Morley proved a remarkable the- orem on the elementary geometry of Euclidean trian- gles: “Given a triangle A, B, C the pairwise intersec- tions α, β, γ of the trisectors form the vertices of an equilateral triangle” |

144. | Hopf algebras, renormalization and noncommutative geometry Alain Connes, Dirk Kreimer Comm. Math. Phys., 199 (1), pp. 203–242, 1998, ISSN: 0010-3616. @article{MR1660199, title = {Hopf algebras, renormalization and noncommutative geometry}, author = {Alain Connes and Dirk Kreimer}, doi = {10.1007/s002200050499}, issn = {0010-3616}, year = {1998}, date = {1998-01-01}, journal = {Comm. Math. Phys.}, volume = {199}, number = {1}, pages = {203--242}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

143. | Hopf algebras, cyclic cohomology and the transverse index theorem Alain Connes, Henri Moscovici Comm. Math. Phys., 198 (1), pp. 199–246, 1998, ISSN: 0010-3616. @article{MR1657389, title = {Hopf algebras, cyclic cohomology and the transverse index theorem}, author = {Alain Connes and Henri Moscovici}, doi = {10.1007/s002200050477}, issn = {0010-3616}, year = {1998}, date = {1998-01-01}, journal = {Comm. Math. Phys.}, volume = {198}, number = {1}, pages = {199--246}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

142. | Noncommutative differential geometry and the structure of space-time Alain Connes The geometric universe (Oxford, 1996), pp. 49–80, Oxford Univ. Press, Oxford, 1998. Incollection @incollection{MR1634504, title = {Noncommutative differential geometry and the structure of space-time}, author = {Alain Connes}, year = {1998}, date = {1998-01-01}, booktitle = {The geometric universe (Oxford, 1996)}, pages = {49--80}, publisher = {Oxford Univ. Press, Oxford}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

141. | Noncommutative geometry: the spectral aspect Alain Connes Symétries quantiques (Les Ħouches, 1995), pp. 643–686, North-Holland, Amsterdam, 1998. Incollection @incollection{MR1616407, title = {Noncommutative geometry: the spectral aspect}, author = {Alain Connes}, year = {1998}, date = {1998-01-01}, booktitle = {Symétries quantiques (Les Ħouches, 1995)}, pages = {643--686}, publisher = {North-Holland, Amsterdam}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

140. | Noncommutative geometry and matrix theory: compactification on tori Alain Connes, Michael R. Douglas, Albert Schwarz J. High Energy Phys., (2), pp. Paper 3, 35, 1998, ISSN: 1126-6708. Journal ArticleAbstract @article{MR1613978, title = {Noncommutative geometry and matrix theory: compactification on tori}, author = {Alain Connes and Michael R. Douglas and Albert Schwarz}, url = {/wp-content/uploads/CDS.pdf}, doi = {10.1088/1126-6708/1998/02/003}, issn = {1126-6708}, year = {1998}, date = {1998-01-01}, journal = {J. High Energy Phys.}, number = {2}, pages = {Paper 3, 35}, abstract = {We study toroidal compactification of Matrix theory, using ideas and results of non-commutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that they correspond in supergravity to tori with constant background three-form tensor field. The paper includes an introduction for mathematicians to the IKKT formulation of Matrix theory and its relation to the BFSS Matrix theory.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We study toroidal compactification of Matrix theory, using ideas and results of non-commutative geometry. We generalize this to compactification on the noncommutative torus, explain the classification of these backgrounds, and argue that they correspond in supergravity to tori with constant background three-form tensor field. The paper includes an introduction for mathematicians to the IKKT formulation of Matrix theory and its relation to the BFSS Matrix theory. |

139. | Brisure de symétrie spontanée et géométrie du point de vue spectral Alain Connes Fields Medallists' lectures, 5 , pp. 340–371, World Sci. Publ., River Edge, NJ, 1997. @incollection{MR1622944, title = {Brisure de symétrie spontanée et géométrie du point de vue spectral}, author = {Alain Connes}, doi = {10.1142/9789812385215_0038}, year = {1997}, date = {1997-01-01}, booktitle = {Fields Medallists' lectures}, volume = {5}, pages = {340--371}, publisher = {World Sci. Publ., River Edge, NJ}, series = {World Sci. Ser. 20th Century Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

138. | Noncommutative differential geometry and the structure of space time Alain Connes Quantum fields and quantum space time (Cargèse, 1996), 364 , pp. 45–72, Plenum, New York, 1997. Incollection @incollection{MR1614991, title = {Noncommutative differential geometry and the structure of space time}, author = {Alain Connes}, year = {1997}, date = {1997-01-01}, booktitle = {Quantum fields and quantum space time (Cargèse, 1996)}, volume = {364}, pages = {45--72}, publisher = {Plenum, New York}, series = {NATO Adv. Sci. Inst. Ser. B Phys.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

137. | Noncommutative differential geometry and the structure of space time Alain Connes Operator algebras and quantum field theory (Rome, 1996), pp. 330–358, Int. Press, Cambridge, MA, 1997. @incollection{MR1491126, title = {Noncommutative differential geometry and the structure of space time}, author = {Alain Connes}, doi = {10.1007/s002200050126}, year = {1997}, date = {1997-01-01}, booktitle = {Operator algebras and quantum field theory (Rome, 1996)}, pages = {330--358}, publisher = {Int. Press, Cambridge, MA}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

136. | Brisure de symétrie spontanée et géométrie du point de vue spectral Alain Connes J. Geom. Phys., 23 (3-4), pp. 206–234, 1997, ISSN: 0393-0440. @article{MR1484588, title = {Brisure de symétrie spontanée et géométrie du point de vue spectral}, author = {Alain Connes}, doi = {10.1016/S0393-0440(97)80001-0}, issn = {0393-0440}, year = {1997}, date = {1997-01-01}, journal = {J. Geom. Phys.}, volume = {23}, number = {3-4}, pages = {206--234}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

135. | Trace formula in noncommutative geometry and the zeros of the Riemann zeta function Alain Connes Journées "Équations aux Dérivées Partielles" (Saint-Jean-de-Monts, 1997), pp. Exp. No. IV, 28, École Polytech., Palaiseau, 1997. IncollectionAbstract @incollection{MR1482271, title = {Trace formula in noncommutative geometry and the zeros of the Riemann zeta function}, author = {Alain Connes}, url = {/wp-content/uploads/traceformulaEDP.pdf}, year = {1997}, date = {1997-01-01}, booktitle = {Journées "Équations aux Dérivées Partielles" (Saint-Jean-de-Monts, 1997)}, pages = {Exp. No. IV, 28}, publisher = {École Polytech., Palaiseau}, abstract = {We give a spectral interpretation of the critical zeros of the Rie- mann zeta function, and a geometric interpretation of the explicit formulas of number theory as a trace formula on a noncommutative space. This reduces the Riemann hypothesis to the validity of the trace formula.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } We give a spectral interpretation of the critical zeros of the Rie- mann zeta function, and a geometric interpretation of the explicit formulas of number theory as a trace formula on a noncommutative space. This reduces the Riemann hypothesis to the validity of the trace formula. |

134. | Noncommutative differential geometry and the structure of space time Alain Connes Deformation theory and symplectic geometry (Ascona, 1996), 20 , pp. 1–33, Kluwer Acad. Publ., Dordrecht, 1997. Incollection @incollection{MR1480713, title = {Noncommutative differential geometry and the structure of space time}, author = {Alain Connes}, year = {1997}, date = {1997-01-01}, booktitle = {Deformation theory and symplectic geometry (Ascona, 1996)}, volume = {20}, pages = {1--33}, publisher = {Kluwer Acad. Publ., Dordrecht}, series = {Math. Phys. Stud.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

133. | Noncommutative differential geometry and the structure of space time Alain Connes Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), 17 , pp. 17–42, Amer. Math. Soc., Providence, RI, 1997. @incollection{MR1478699, title = {Noncommutative differential geometry and the structure of space time}, author = {Alain Connes}, doi = {10.1007/s002200050126}, year = {1997}, date = {1997-01-01}, booktitle = {Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995)}, volume = {17}, pages = {17--42}, publisher = {Amer. Math. Soc., Providence, RI}, series = {Fields Inst. Commun.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

132. | Brisure de symétrie spontanée et géométrie du point de vue spectral Alain Connes (241), pp. Exp. No. 816, 5, 313–349, 1997, ISSN: 0303-1179, (Séminaire Bourbaki, Vol. 1995/96). IncollectionAbstract @incollection{MR1472544, title = {Brisure de symétrie spontanée et géométrie du point de vue spectral}, author = {Alain Connes}, url = {/wp-content/uploads/brisuresymspectral.pdf}, issn = {0303-1179}, year = {1997}, date = {1997-01-01}, journal = {Astérisque}, number = {241}, pages = {Exp. No. 816, 5, 313--349}, abstract = {Séminaire Bourbaki}, note = {Séminaire Bourbaki, Vol. 1995/96}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } Séminaire Bourbaki |

131. | Ali H. Chamseddine, Alain Connes Comm. Math. Phys., 186 (3), pp. 731–750, 1997, ISSN: 0010-3616. Journal ArticleAbstract @article{MR1463819, title = {The spectral action principle}, author = {Ali H. Chamseddine and Alain Connes}, url = {/wp-content/uploads/spectralaction.pdf}, doi = {10.1007/s002200050126}, issn = {0010-3616}, year = {1997}, date = {1997-01-01}, journal = {Comm. Math. Phys.}, volume = {186}, number = {3}, pages = {731--750}, abstract = {We propose a new action principle to be associated with a noncommutative space (A, H, D). The universal formula for the spectral action is (ψ, Dψ) + Trace(χ(D/ Λ)) where ψ is a spinor on the Hilbert space, Λ is a scale and χ a positive func- tion. When this principle is applied to the noncommutative space defined by the spectrum of the standard model one obtains the standard model action coupled to Einstein plus Weyl gravity. There are relations between the gauge coupling con- stants identical to those of SU(5) as well as the Higgs self-coupling, to be taken at a fixed high energy scale.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We propose a new action principle to be associated with a noncommutative space (A, H, D). The universal formula for the spectral action is (ψ, Dψ) + Trace(χ(D/ Λ)) where ψ is a spinor on the Hilbert space, Λ is a scale and χ a positive func- tion. When this principle is applied to the noncommutative space defined by the spectrum of the standard model one obtains the standard model action coupled to Einstein plus Weyl gravity. There are relations between the gauge coupling con- stants identical to those of SU(5) as well as the Higgs self-coupling, to be taken at a fixed high energy scale. |

130. | Aspherical gravitational monopoles Alain Connes, Thibault Damour, Pierre Fayet Nuclear Phys. B, 490 (1-2), pp. 391–431, 1997, ISSN: 0550-3213. Journal ArticleAbstract @article{MR1452330, title = {Aspherical gravitational monopoles}, author = {Alain Connes and Thibault Damour and Pierre Fayet}, url = {/wp-content/uploads/aspherical.pdf}, doi = {10.1016/S0550-3213(97)00041-2}, issn = {0550-3213}, year = {1997}, date = {1997-01-01}, journal = {Nuclear Phys. B}, volume = {490}, number = {1-2}, pages = {391--431}, abstract = {We show how to construct non-spherically-symmetric extended bodies of uniform density behaving exactly as pointlike masses. These “gravitational monopoles” have the following equivalent properties: (i) they generate, outside them, a spherically-symmetric gravitational potential M/| x − xO| ; (ii) their interaction energy with an external gravitational potential U(x) is −M U(xO); and (iii) all their mul- tipole moments (of order l ≥ 1) with respect to their center of mass O vanish identically. The method applies for any number of space dimensions. The free parameters entering the construction are: (1) an arbitrary surface Σ bounding a connected open subset Ω of R3; (2) the arbitrary choice of the center of mass O within Ω ; and (3) the total volume of the body. An extension of the method allows one to construct homogeneous bodies which are gravitationally equivalent (in the sense of having exactly the same multipole moments) to any given body.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We show how to construct non-spherically-symmetric extended bodies of uniform density behaving exactly as pointlike masses. These “gravitational monopoles” have the following equivalent properties: (i) they generate, outside them, a spherically-symmetric gravitational potential M/| x − xO| ; (ii) their interaction energy with an external gravitational potential U(x) is −M U(xO); and (iii) all their mul- tipole moments (of order l ≥ 1) with respect to their center of mass O vanish identically. The method applies for any number of space dimensions. The free parameters entering the construction are: (1) an arbitrary surface Σ bounding a connected open subset Ω of R3; (2) the arbitrary choice of the center of mass O within Ω ; and (3) the total volume of the body. An extension of the method allows one to construct homogeneous bodies which are gravitationally equivalent (in the sense of having exactly the same multipole moments) to any given body. |

129. | Matrix Vieta theorem revisited Alain Connes, Albert Schwarz Lett. Math. Phys., 39 (4), pp. 349–353, 1997, ISSN: 0377-9017. Journal ArticleAbstract @article{MR1449580, title = {Matrix Vieta theorem revisited}, author = {Alain Connes and Albert Schwarz}, url = {/wp-content/uploads/CONNES-SCHWARZ1997_Article_MatrixVietaTheoremRevisited.pdf}, doi = {10.1023/A:1007373114601}, issn = {0377-9017}, year = {1997}, date = {1997-01-01}, journal = {Lett. Math. Phys.}, volume = {39}, number = {4}, pages = {349--353}, abstract = {We give another proof of the noncommutative analog of the Vieta theorem. This proof gives a little bit stronger statement and leads to some generalizations.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We give another proof of the noncommutative analog of the Vieta theorem. This proof gives a little bit stronger statement and leads to some generalizations. |

128. | Gravity coupled with matter and the foundation of non-commutative geometry Alain Connes Comm. Math. Phys., 182 (1), pp. 155–176, 1996, ISSN: 0010-3616. Journal ArticleAbstract @article{MR1441908, title = {Gravity coupled with matter and the foundation of non-commutative geometry}, author = {Alain Connes}, url = {/wp-content/uploads/foundation96.pdf}, issn = {0010-3616}, year = {1996}, date = {1996-01-01}, journal = {Comm. Math. Phys.}, volume = {182}, number = {1}, pages = {155--176}, abstract = {We first exhibit in the commutative case the simple al- gebraic relations between the algebra of functions on a manifold and its infinitesimal length element ds. Its unitary representations cor- respond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = ×—× = D−1 where D is the Dirac operator. We extend these simple relations to the non commutative case using Tomita’s involution J. We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standard Model will be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in the slightly non commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group. }, keywords = {}, pubstate = {published}, tppubtype = {article} } We first exhibit in the commutative case the simple al- gebraic relations between the algebra of functions on a manifold and its infinitesimal length element ds. Its unitary representations cor- respond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = ×—× = D−1 where D is the Dirac operator. We extend these simple relations to the non commutative case using Tomita’s involution J. We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standard Model will be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in the slightly non commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group. |

127. | Formule de trace en géométrie non-commutative et hypothèse de Riemann Alain Connes C. R. Acad. Sci. Paris Sér. I Math., 323 (12), pp. 1231–1236, 1996, ISSN: 0764-4442. Journal Article @article{MR1428542, title = {Formule de trace en géométrie non-commutative et hypothèse de Riemann}, author = {Alain Connes}, issn = {0764-4442}, year = {1996}, date = {1996-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {323}, number = {12}, pages = {1231--1236}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

126. | Universal formula for noncommutative geometry actions: unification of gravity and the standard model Ali H. Chamseddine, Alain Connes Phys. Rev. Lett., 77 (24), pp. 4868–4871, 1996, ISSN: 0031-9007. @article{MR1419931, title = {Universal formula for noncommutative geometry actions: unification of gravity and the standard model}, author = {Ali H. Chamseddine and Alain Connes}, doi = {10.1103/PhysRevLett.77.4868}, issn = {0031-9007}, year = {1996}, date = {1996-01-01}, journal = {Phys. Rev. Lett.}, volume = {77}, number = {24}, pages = {4868--4871}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

125. | Polarized modules and Fredholm modules Jacek Brodzki, Alain Connes, David Ellwood Mat. Fiz. Anal. Geom., 2 (1), pp. 15–24, 1995, ISSN: 1027-1767. Journal Article @article{MR1484114, title = {Polarized modules and Fredholm modules}, author = {Jacek Brodzki and Alain Connes and David Ellwood}, issn = {1027-1767}, year = {1995}, date = {1995-01-01}, journal = {Mat. Fiz. Anal. Geom.}, volume = {2}, number = {1}, pages = {15--24}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

124. | Non-commutative geometry and physics Alain Connes Gravitation et quantifications (Les Ħouches, 1992), pp. 805–950, North-Holland, Amsterdam, 1995. Incollection @incollection{MR1461287, title = {Non-commutative geometry and physics}, author = {Alain Connes}, year = {1995}, date = {1995-01-01}, booktitle = {Gravitation et quantifications (Les Ħouches, 1992)}, pages = {805--950}, publisher = {North-Holland, Amsterdam}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

123. | Round table: physics and mathematics Joel Lebowitz, Michael Atiyah, Edouard Brézin, Alain Connes, Juerg Fröhlich, David Gross, Arthur Jaffe, Leo Kadanoff, David Ruelle XIth International Congress of Mathematical Physics (Paris, 1994), pp. 691–705, Int. Press, Cambridge, MA, 1995. Incollection @incollection{MR1370725, title = {Round table: physics and mathematics}, author = {Joel Lebowitz and Michael Atiyah and Edouard Brézin and Alain Connes and Juerg Fröhlich and David Gross and Arthur Jaffe and Leo Kadanoff and David Ruelle}, year = {1995}, date = {1995-01-01}, booktitle = {XIth International Congress of Mathematical Physics (Paris, 1994)}, pages = {691--705}, publisher = {Int. Press, Cambridge, MA}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

122. | Quantized calculus and applications Alain Connes XIth International Congress of Mathematical Physics (Paris, 1994), pp. 15–36, Int. Press, Cambridge, MA, 1995. Incollection @incollection{MR1370665, title = {Quantized calculus and applications}, author = {Alain Connes}, year = {1995}, date = {1995-01-01}, booktitle = {XIth International Congress of Mathematical Physics (Paris, 1994)}, pages = {15--36}, publisher = {Int. Press, Cambridge, MA}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

121. | Jean-Benoît Bost, Alain Connes Selecta Math. (N.S.), 1 (3), pp. 411–457, 1995, ISSN: 1022-1824. Journal ArticleAbstract @article{MR1366621, title = {Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory}, author = {Jean-Benoît Bost and Alain Connes}, url = {/wp-content/uploads/bostconnesscan.pdf}, doi = {10.1007/BF01589495}, issn = {1022-1824}, year = {1995}, date = {1995-01-01}, journal = {Selecta Math. (N.S.)}, volume = {1}, number = {3}, pages = {411--457}, abstract = {We construct a C* dynamical system whose partition function is the Riemann zeta function.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We construct a C* dynamical system whose partition function is the Riemann zeta function. |

120. | Noncommutative geometry and reality Alain Connes J. Math. Phys., 36 (11), pp. 6194–6231, 1995, ISSN: 0022-2488. Journal ArticleAbstract @article{MR1355905, title = {Noncommutative geometry and reality}, author = {Alain Connes}, url = {/wp-content/uploads/reality.pdf}, doi = {10.1063/1.531241}, issn = {0022-2488}, year = {1995}, date = {1995-01-01}, journal = {J. Math. Phys.}, volume = {36}, number = {11}, pages = {6194--6231}, abstract = {We introduce the notion of real structure in our spectral geometry. This notion is motivated by Atiyah’s KR-theory and by Tomita’s involution J. It allows us to remove two unpleasant features of the “Connes-Lott” description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincare duality and in the unimodularity condition. }, keywords = {}, pubstate = {published}, tppubtype = {article} } We introduce the notion of real structure in our spectral geometry. This notion is motivated by Atiyah’s KR-theory and by Tomita’s involution J. It allows us to remove two unpleasant features of the “Connes-Lott” description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincare duality and in the unimodularity condition. |

119. | Geometry from the spectral point of view Alain Connes Lett. Math. Phys., 34 (3), pp. 203–238, 1995, ISSN: 0377-9017. @article{MR1345552, title = {Geometry from the spectral point of view}, author = {Alain Connes}, doi = {10.1007/BF01872777}, issn = {0377-9017}, year = {1995}, date = {1995-01-01}, journal = {Lett. Math. Phys.}, volume = {34}, number = {3}, pages = {203--238}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

118. | The local index formula in noncommutative geometry Alain Connes, Henri Moscovici Geom. Funct. Anal., 5 (2), pp. 174–243, 1995, ISSN: 1016-443X. Journal ArticleAbstract @article{MR1334867, title = {The local index formula in noncommutative geometry}, author = {Alain Connes and Henri Moscovici}, url = {/wp-content/uploads/localtrace95.ps_.pdf}, doi = {10.1007/BF01895667}, issn = {1016-443X}, year = {1995}, date = {1995-01-01}, journal = {Geom. Funct. Anal.}, volume = {5}, number = {2}, pages = {174--243}, abstract = {In noncommutative geometry a geometric space is described from a spectral point of view, as a triple (A,H,D) consisting of a ∗-algebra A represented in a Hilbert space H together with an unbounded selfadjoint operator D, with compact resolvent, which interacts with the algebra in a bounded fashion. This paper contributes to the advancement of this point of view in two significant ways: (1) by showing that any pseudogroup of transformations of a manifold gives rise to such a spectral triple of finite summability degree, and (2) by proving a general, in some sense universal, local index formula for an arbitrary spectral triple of finite summability degree, in terms of the Dixmier trace and its residue-type extension. }, keywords = {}, pubstate = {published}, tppubtype = {article} } In noncommutative geometry a geometric space is described from a spectral point of view, as a triple (A,H,D) consisting of a ∗-algebra A represented in a Hilbert space H together with an unbounded selfadjoint operator D, with compact resolvent, which interacts with the algebra in a bounded fashion. This paper contributes to the advancement of this point of view in two significant ways: (1) by showing that any pseudogroup of transformations of a manifold gives rise to such a spectral triple of finite summability degree, and (2) by proving a general, in some sense universal, local index formula for an arbitrary spectral triple of finite summability degree, in terms of the Dixmier trace and its residue-type extension. |

117. | Conversations on mind, matter, and mathematics Jean-Pierre Changeux, Alain Connes Princeton University Press, Princeton, NJ, 1995, ISBN: 0-691-08759-8. BookAbstract @book{MR1327523, title = {Conversations on mind, matter, and mathematics}, author = {Jean-Pierre Changeux and Alain Connes}, isbn = {0-691-08759-8}, year = {1995}, date = {1995-01-01}, pages = {xii+261}, publisher = {Princeton University Press}, address = {Princeton, NJ}, abstract = {Do numbers and the other objects of mathematics enjoy a timeless existence independent of human minds, or are they the products of cerebral invention? Do we discover them, as Plato supposed and many others have believed since, or do we construct them? Does mathematics constitute a universal language that in principle would permit human beings to communicate with extraterrestrial civilizations elsewhere in the universe, or is it merely an earthly language that owes its accidental existence to the peculiar evolution of neuronal networks in our brains? Does the physical world actually obey mathematical laws, or does it seem to conform to them simply because physicists have increasingly been able to make mathematical sense of it? Jean-Pierre Changeux, an internationally renowned neurobiologist, and Alain Connes, one of the most eminent living mathematicians, find themselves deeply divided by these questions. The problematic status of mathematical objects leads Changeux and Connes to the organization and function of the brain, the ways in which its embryonic and post-natal development influences the unfolding of mathematical reasoning and other kinds of thinking, and whether human intelligence can be simulated, modeled, --or actually reproduced-- by mechanical means. The two men go on to pose ethical questions, inquiring into the natural foundations of morality and the possibility that it may have a neural basis underlying its social manifestations. This vivid record of profound disagreement and, at the same time, sincere search for mutual understanding, follows in the tradition of Poincare, Hadamard, and von Neumann in probing the limits of human experience and intellectual possibility. Why order should exist in the world at all, and why it should be comprehensible to human beings, is the question that lies at the heart of these remarkable dialogues. Edited and translated from the 1989 French original by M. B. DeBevoise.}, keywords = {}, pubstate = {published}, tppubtype = {book} } Do numbers and the other objects of mathematics enjoy a timeless existence independent of human minds, or are they the products of cerebral invention? Do we discover them, as Plato supposed and many others have believed since, or do we construct them? Does mathematics constitute a universal language that in principle would permit human beings to communicate with extraterrestrial civilizations elsewhere in the universe, or is it merely an earthly language that owes its accidental existence to the peculiar evolution of neuronal networks in our brains? Does the physical world actually obey mathematical laws, or does it seem to conform to them simply because physicists have increasingly been able to make mathematical sense of it? Jean-Pierre Changeux, an internationally renowned neurobiologist, and Alain Connes, one of the most eminent living mathematicians, find themselves deeply divided by these questions. The problematic status of mathematical objects leads Changeux and Connes to the organization and function of the brain, the ways in which its embryonic and post-natal development influences the unfolding of mathematical reasoning and other kinds of thinking, and whether human intelligence can be simulated, modeled, --or actually reproduced-- by mechanical means. The two men go on to pose ethical questions, inquiring into the natural foundations of morality and the possibility that it may have a neural basis underlying its social manifestations. This vivid record of profound disagreement and, at the same time, sincere search for mutual understanding, follows in the tradition of Poincare, Hadamard, and von Neumann in probing the limits of human experience and intellectual possibility. Why order should exist in the world at all, and why it should be comprehensible to human beings, is the question that lies at the heart of these remarkable dialogues. Edited and translated from the 1989 French original by M. B. DeBevoise. |

116. | Alain Connes Academic Press, Inc., San Diego, CA, 1994, ISBN: 0-12-185860-X. BookAbstract @book{MR1303779, title = {Noncommutative geometry}, author = {Alain Connes}, url = {/wp-content/uploads/book94bigpdf.pdf}, isbn = {0-12-185860-X}, year = {1994}, date = {1994-11-22}, pages = {xiv+661}, publisher = {Academic Press, Inc.}, address = {San Diego, CA}, abstract = {This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by Alain Connes. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields.}, keywords = {}, pubstate = {published}, tppubtype = {book} } This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by Alain Connes. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields. |

115. | Alain Connes, Carlo Rovelli Classical Quantum Gravity, 11 (12), pp. 2899–2917, 1994, ISSN: 0264-9381. Journal ArticleAbstract @article{MR1307019, title = {Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories}, author = {Alain Connes and Carlo Rovelli}, url = {/wp-content/uploads/carlotime.pdf}, issn = {0264-9381}, year = {1994}, date = {1994-01-01}, journal = {Classical Quantum Gravity}, volume = {11}, number = {12}, pages = {2899--2917}, abstract = {We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in partic- ular, we address the problem of relating the “timelessness” of the hypothetical fundamental general covariant quantum field theory with the “evidence” of the flow of time. By using the algebraic formulation of quantum theory, we pro- pose a unifying perspective on these problems, based on the hypothesis that in a generally covariant quantum theory the physical time-flow is not a universal property of the mechanical theory, but rather it is determined by the thermo- dynamical state of the system (“thermal time hypothesis”). We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita-Takesaki theorem, which allows to derive a time-flow, namely a one- parameter group of automorphisms of the observable algebra, from a generic thermal physical state. We study this time-flow, its classical limit, and we re- late it to various characteristic theoretical facts, as the Unruh temperature and the Hawking radiation. We also point out the existence of a state-independent notion of “time”, given by the canonical one-parameter subgroup of outer au- tomorphisms provided by Connes Cocycle Radon-Nikodym theorem. }, keywords = {}, pubstate = {published}, tppubtype = {article} } We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in partic- ular, we address the problem of relating the “timelessness” of the hypothetical fundamental general covariant quantum field theory with the “evidence” of the flow of time. By using the algebraic formulation of quantum theory, we pro- pose a unifying perspective on these problems, based on the hypothesis that in a generally covariant quantum theory the physical time-flow is not a universal property of the mechanical theory, but rather it is determined by the thermo- dynamical state of the system (“thermal time hypothesis”). We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita-Takesaki theorem, which allows to derive a time-flow, namely a one- parameter group of automorphisms of the observable algebra, from a generic thermal physical state. We study this time-flow, its classical limit, and we re- late it to various characteristic theoretical facts, as the Unruh temperature and the Hawking radiation. We also point out the existence of a state-independent notion of “time”, given by the canonical one-parameter subgroup of outer au- tomorphisms provided by Connes Cocycle Radon-Nikodym theorem. |

114. | Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes Alain Connes, Dennis Sullivan, Nicolas Teleman Topology, 33 (4), pp. 663–681, 1994, ISSN: 0040-9383. Journal ArticleAbstract @article{MR1293305, title = {Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes}, author = {Alain Connes and Dennis Sullivan and Nicolas Teleman}, url = {/wp-content/uploads/quasiconformal.pdf}, doi = {10.1016/0040-9383(94)90003-5}, issn = {0040-9383}, year = {1994}, date = {1994-01-01}, journal = {Topology}, volume = {33}, number = {4}, pages = {663--681}, abstract = {THEOREM1.1. Given quasiconformal M2’ with a bounded measurable conformal structure or, more generally, a bounded measurable *-structure on the /-forms, the local construction yields an operator S which is determined by * up to the ideal I(G). Moreover, any such S satisjies: (i) S agrees mod Z(e) with the identity on exact /-forms; (ii) S anticommutes modulo I(e) with the involution y associated to * (y = * if8 is even, y = i* if& is odd).}, keywords = {}, pubstate = {published}, tppubtype = {article} } THEOREM1.1. Given quasiconformal M2’ with a bounded measurable conformal structure or, more generally, a bounded measurable *-structure on the /-forms, the local construction yields an operator S which is determined by * up to the ideal I(G). Moreover, any such S satisjies: (i) S agrees mod Z(e) with the identity on exact /-forms; (ii) S anticommutes modulo I(e) with the involution y associated to * (y = * if8 is even, y = i* if& is odd). |

113. | Classifying space for proper actions and ?-theory of group ?*-algebras Paul Baum, Alain Connes, Nigel Higson ?*-algebras: 1943--1993 (San Antonio, TX, 1993), 167 , pp. 240–291, Amer. Math. Soc., Providence, RI, 1994. @incollection{MR1292018, title = { Classifying space for proper actions and ?-theory of group ?*-algebras}, author = {Paul Baum and Alain Connes and Nigel Higson}, doi = {10.1090/conm/167/1292018}, year = {1994}, date = {1994-01-01}, booktitle = {?*-algebras: 1943--1993 (San Antonio, TX, 1993)}, volume = {167}, pages = {240--291}, publisher = {Amer. Math. Soc., Providence, RI}, series = {Contemp. Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

112. | Interprétation géométrique du modèle standard de la physique des particules et structure fine de l'espace-temps Alain Connes C. R. Acad. Sci. Paris Sér. Gén. Vie Sci., 10 (3), pp. 223–234, 1993, ISSN: 0762-0969. Journal Article @article{MR1251586, title = {Interprétation géométrique du modèle standard de la physique des particules et structure fine de l'espace-temps}, author = {Alain Connes}, issn = {0762-0969}, year = {1993}, date = {1993-01-01}, journal = {C. R. Acad. Sci. Paris Sér. Gén. Vie Sci.}, volume = {10}, number = {3}, pages = {223--234}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

111. | Formules locales pour les classes de Pontrjagin topologiques Alain Connes, Dennis Sullivan, Nicolas Teleman C. R. Acad. Sci. Paris Sér. I Math., 317 (5), pp. 521–526, 1993, ISSN: 0764-4442. Journal Article @article{MR1239041, title = {Formules locales pour les classes de Pontrjagin topologiques}, author = {Alain Connes and Dennis Sullivan and Nicolas Teleman}, issn = {0764-4442}, year = {1993}, date = {1993-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {317}, number = {5}, pages = {521--526}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

110. | Transgression and the Chern character of finite-dimensional ?-cycles Alain Connes, Henri Moscovici Comm. Math. Phys., 155 (1), pp. 103–122, 1993, ISSN: 0010-3616. @article{MR1228528, title = {Transgression and the Chern character of finite-dimensional ?-cycles}, author = {Alain Connes and Henri Moscovici}, url = {http://projecteuclid.org/euclid.cmp/1104253202}, issn = {0010-3616}, year = {1993}, date = {1993-01-01}, journal = {Comm. Math. Phys.}, volume = {155}, number = {1}, pages = {103--122}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

109. | Editorial (dedication of this issue to Huzihiro Araki) Alain Connes, Moshé Flato, Heisuke Hironaka, Arthur Jaffe, Vaughan Jones Comm. Math. Phys., 155 (1), pp. 1–2, 1993, ISSN: 0010-3616. @article{MR1228522, title = {Editorial (dedication of this issue to Huzihiro Araki)}, author = {Alain Connes and Moshé Flato and Heisuke Hironaka and Arthur Jaffe and Vaughan Jones}, url = {http://projecteuclid.org/euclid.cmp/1104253196}, issn = {0010-3616}, year = {1993}, date = {1993-01-01}, journal = {Comm. Math. Phys.}, volume = {155}, number = {1}, pages = {1--2}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

108. | Group cohomology with Lipschitz control and higher signatures Alain Connes, Mikhaïl Gromov, Henri Moscovici Geom. Funct. Anal., 3 (1), pp. 1–78, 1993, ISSN: 1016-443X. @article{MR1204787, title = {Group cohomology with Lipschitz control and higher signatures}, author = {Alain Connes and Mikhaïl Gromov and Henri Moscovici}, doi = {10.1007/BF01895513}, issn = {1016-443X}, year = {1993}, date = {1993-01-01}, journal = {Geom. Funct. Anal.}, volume = {3}, number = {1}, pages = {1--78}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

107. | Géométrie non commutative et physique quantique Alain Connes Mathématiques quantiques, 1992 , pp. 20, Soc. Math. France, Paris, 1992. Incollection @incollection{MR1484740, title = {Géométrie non commutative et physique quantique}, author = {Alain Connes}, year = {1992}, date = {1992-01-01}, booktitle = {Mathématiques quantiques}, volume = {1992}, pages = {20}, publisher = {Soc. Math. France, Paris}, series = {SMF Journ. Annu.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

106. | Mathématiques quantiques. Géométrie non commutative et physique quantique. Sur l'équation de Schrödinger. Alain Connes, Bernard Helffer Journée annuelle SMF, 23 mai 1992, 1992 , pp. 52, 1992. Journal Article @article{MR1484739, title = {Mathématiques quantiques. Géométrie non commutative et physique quantique. Sur l'équation de Schrödinger.}, author = {Alain Connes and Bernard Helffer}, year = {1992}, date = {1992-01-01}, journal = {Journée annuelle SMF, 23 mai 1992}, volume = {1992}, pages = {52}, publisher = {Société Mathématique de France, Paris}, series = {SMF Journée Annuelle [SMF Annual Conference]}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

105. | Noncommutative geometry Alain Connes In the forest of symbols (Finnish), pp. 244–263, Art House, Helsinki, 1992. Incollection @incollection{MR1444865, title = {Noncommutative geometry}, author = {Alain Connes}, year = {1992}, date = {1992-01-01}, booktitle = {In the forest of symbols (Finnish)}, pages = {244--263}, publisher = {Art House, Helsinki}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

104. | Round-table discussion Alain Connes, Gerd Faltings, Vaughan Jones, Stephen Smale, René Thom, Jorge Wagensberg Mathematical research today and tomorrow (Barcelona, 1991), 1525 , pp. 87–108, Springer, Berlin, 1992. @incollection{MR1247058, title = {Round-table discussion}, author = {Alain Connes and Gerd Faltings and Vaughan Jones and Stephen Smale and René Thom and Jorge Wagensberg}, doi = {10.1007/BFb0089208}, year = {1992}, date = {1992-01-01}, booktitle = {Mathematical research today and tomorrow (Barcelona, 1991)}, volume = {1525}, pages = {87--108}, publisher = {Springer, Berlin}, series = {Lecture Notes in Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

103. | Noncommutative geometry Alain Connes Mathematical research today and tomorrow (Barcelona, 1991), 1525 , pp. 40–58, Springer, Berlin, 1992. @incollection{MR1247054, title = {Noncommutative geometry}, author = {Alain Connes}, doi = {10.1007/BFb0089204}, year = {1992}, date = {1992-01-01}, booktitle = {Mathematical research today and tomorrow (Barcelona, 1991)}, volume = {1525}, pages = {40--58}, publisher = {Springer, Berlin}, series = {Lecture Notes in Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

102. | The metric aspect of noncommutative geometry Alain Connes, John Lott New symmetry principles in quantum field theory (Carg`ese, 1991), 295 , pp. 53–93, Plenum, New York, 1992. Incollection @incollection{MR1204452, title = {The metric aspect of noncommutative geometry}, author = {Alain Connes and John Lott}, year = {1992}, date = {1992-01-01}, booktitle = {New symmetry principles in quantum field theory (Carg`ese, 1991)}, volume = {295}, pages = {53--93}, publisher = {Plenum, New York}, series = {NATO Adv. Sci. Inst. Ser. B Phys.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

101. | Produits eulériens et facteurs de type III Jean-Benoît Bost, Alain Connes C. R. Acad. Sci. Paris Sér. I Math., 315 (3), pp. 279–284, 1992, ISSN: 0764-4442. Journal Article @article{MR1179720, title = {Produits eulériens et facteurs de type III}, author = {Jean-Benoît Bost and Alain Connes}, issn = {0764-4442}, year = {1992}, date = {1992-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {315}, number = {3}, pages = {279--284}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

100. | Closed star products and cyclic cohomology Alain Connes, Moshé Flato, Daniel Sternheimer Lett. Math. Phys., 24 (1), pp. 1–12, 1992, ISSN: 0377-9017. @article{MR1162894, title = {Closed star products and cyclic cohomology}, author = {Alain Connes and Moshé Flato and Daniel Sternheimer}, doi = {10.1007/BF00429997}, issn = {0377-9017}, year = {1992}, date = {1992-01-01}, journal = {Lett. Math. Phys.}, volume = {24}, number = {1}, pages = {1--12}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

99. | Sur la nature de la réalité mathématique Alain Connes Elem. Math., 47 (1), pp. 19–26, 1992, ISSN: 0013-6018. Journal Article @article{MR1158144, title = {Sur la nature de la réalité mathématique}, author = {Alain Connes}, issn = {0013-6018}, year = {1992}, date = {1992-01-01}, journal = {Elem. Math.}, volume = {47}, number = {1}, pages = {19--26}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

98. | On the Chern character of ? summable Fredholm modules Alain Connes Comm. Math. Phys., 139 (1), pp. 171–181, 1991, ISSN: 0010-3616. @article{MR1116414, title = {On the Chern character of ? summable Fredholm modules}, author = {Alain Connes}, url = {http://projecteuclid.org/euclid.cmp/1104203140}, issn = {0010-3616}, year = {1991}, date = {1991-01-01}, journal = {Comm. Math. Phys.}, volume = {139}, number = {1}, pages = {171--181}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

97. | Caractères des représentations ?-sommables des groupes discrets Alain Connes C. R. Acad. Sci. Paris Sér. I Math., 312 (9), pp. 661–666, 1991, ISSN: 0764-4442. Journal Article @article{MR1105621, title = {Caractères des représentations ?-sommables des groupes discrets}, author = {Alain Connes}, issn = {0764-4442}, year = {1991}, date = {1991-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {312}, number = {9}, pages = {661--666}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

96. | Particle models and noncommutative geometry Alain Connes, John Lott Recent advances in field theory (Annecy-le-Vieux, 1990), 18B , pp. 29–47 (1991), 1990, ISSN: 0920-5632. @incollection{MR1128127, title = {Particle models and noncommutative geometry}, author = {Alain Connes and John Lott}, doi = {10.1016/0920-5632(91)90120-4}, issn = {0920-5632}, year = {1990}, date = {1990-01-01}, booktitle = {Recent advances in field theory (Annecy-le-Vieux, 1990)}, journal = {Nuclear Phys. B Proc. Suppl.}, volume = {18B}, pages = {29--47 (1991)}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

95. | Essay on physics and noncommutative geometry Alain Connes The interface of mathematics and particle physics (Oxford, 1988), 24 , pp. 9–48, Oxford Univ. Press, New York, 1990. Incollection @incollection{MR1103128, title = {Essay on physics and noncommutative geometry}, author = {Alain Connes}, year = {1990}, date = {1990-01-01}, booktitle = {The interface of mathematics and particle physics (Oxford, 1988)}, volume = {24}, pages = {9--48}, publisher = {Oxford Univ. Press, New York}, series = {Inst. Math. Appl. Conf. Ser. New Ser.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

94. | Alain Connes InterEditions, Paris, 1990, ISBN: 2-7296-0284-4. BookAbstract @book{MR1079062, title = {Géométrie non commutative}, author = {Alain Connes}, isbn = {2-7296-0284-4}, year = {1990}, date = {1990-01-01}, pages = {240}, publisher = {InterEditions}, address = {Paris}, abstract = {Alain Connes, Médaille Fields 1982 et Médaille d'or du CNRS 2004, explore ici les relations existant entre physique théorique et mathématiques. Au terme de sa réflexion, il propose un espace "double" permettant de rendre compte du modèle de Weinberg - Salam, et notamment de l'existence des bosons de Higgs. Un ouvrage véritablement novateur.}, keywords = {}, pubstate = {published}, tppubtype = {book} } Alain Connes, Médaille Fields 1982 et Médaille d'or du CNRS 2004, explore ici les relations existant entre physique théorique et mathématiques. Au terme de sa réflexion, il propose un espace "double" permettant de rendre compte du modèle de Weinberg - Salam, et notamment de l'existence des bosons de Higgs. Un ouvrage véritablement novateur. |

93. | Introduction à la géométrie non-commutative Alain Connes The legacy of John von Neumann (Hempstead, NY, 1988), 50 , pp. 91–118, Amer. Math. Soc., Providence, RI, 1990. @incollection{MR1067754, title = {Introduction à la géométrie non-commutative}, author = {Alain Connes}, doi = {10.1090/pspum/050/1067754}, year = {1990}, date = {1990-01-01}, booktitle = {The legacy of John von Neumann (Hempstead, NY, 1988)}, volume = {50}, pages = {91--118}, publisher = {Amer. Math. Soc., Providence, RI}, series = {Proc. Sympos. Pure Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

92. | Cyclic cohomology, the Novikov conjecture and hyperbolic groups Alain Connes, Henri Moscovici Topology, 29 (3), pp. 345–388, 1990, ISSN: 0040-9383. Journal ArticleAbstract @article{MR1066176, title = {Cyclic cohomology, the Novikov conjecture and hyperbolic groups}, author = {Alain Connes and Henri Moscovici}, url = {/wp-content/uploads/novikov-1.pdf}, doi = {10.1016/0040-9383(90)90003-3}, issn = {0040-9383}, year = {1990}, date = {1990-01-01}, journal = {Topology}, volume = {29}, number = {3}, pages = {345--388}, abstract = {In this paper we present a new and more direct method for attacking the Novikov conjecture, which yields a proof of the conjecture for Gromov’s (word) hyperbolic groups [ 183.These groups form an extremely rich and interesting classof finitely presented groups, which differs significantly, both in size and in nature, from the groups for which Novikov’s conjecture was previously known. First of all, as pointed out by Gromov [18], they are “generic” among all finitely presented groups in the following sense:the ratio between the number of hyperbolic groups and all groups with a fixed number of generators and a fixed number of relations, each of length at most 1,tends to 1 when I+co [18,0.2(A)]. Secondly, when adding at random relations to a (non-elementary) hyperbolic group, one obtains again a hyperbolic group [18,5.5]. Thirdly, the cohomology of any finite polyhedron can be embedded into the cohomology of a hyperbolic group [18, 0.2(c)]. Also, many of the hyperbolic groups exhibit “exotic” properties, like Kazhdan’s property T [ 18,5.61or being non-linear (in a non-trivial way). }, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper we present a new and more direct method for attacking the Novikov conjecture, which yields a proof of the conjecture for Gromov’s (word) hyperbolic groups [ 183.These groups form an extremely rich and interesting classof finitely presented groups, which differs significantly, both in size and in nature, from the groups for which Novikov’s conjecture was previously known. First of all, as pointed out by Gromov [18], they are “generic” among all finitely presented groups in the following sense:the ratio between the number of hyperbolic groups and all groups with a fixed number of generators and a fixed number of relations, each of length at most 1,tends to 1 when I+co [18,0.2(A)]. Secondly, when adding at random relations to a (non-elementary) hyperbolic group, one obtains again a hyperbolic group [18,5.5]. Thirdly, the cohomology of any finite polyhedron can be embedded into the cohomology of a hyperbolic group [18, 0.2(c)]. Also, many of the hyperbolic groups exhibit “exotic” properties, like Kazhdan’s property T [ 18,5.61or being non-linear (in a non-trivial way). |

91. | Déformations, morphismes asymptotiques et ?-théorie bivariante Alain Connes, Nigel Higson C. R. Acad. Sci. Paris Sér. I Math., 311 (2), pp. 101–106, 1990, ISSN: 0764-4442. Journal Article @article{MR1065438, title = {Déformations, morphismes asymptotiques et ?-théorie bivariante}, author = {Alain Connes and Nigel Higson}, issn = {0764-4442}, year = {1990}, date = {1990-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {311}, number = {2}, pages = {101--106}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

90. | Conjecture de Novikov et fibrés presque plats Alain Connes, Mikhaïl Gromov, Henri Moscovici C. R. Acad. Sci. Paris Sér. I Math., 310 (5), pp. 273–277, 1990, ISSN: 0764-4442. Journal Article @article{MR1042862, title = {Conjecture de Novikov et fibrés presque plats}, author = {Alain Connes and Mikhaïl Gromov and Henri Moscovici}, issn = {0764-4442}, year = {1990}, date = {1990-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {310}, number = {5}, pages = {273--277}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

89. | Jean-Pierre Changeux, Alain Connes Odile Jacob, Paris, 1989, ISBN: 9782738100733. BookAbstract @book{MR1246560, title = {Matière à pensée}, author = {Jean-Pierre Changeux and Alain Connes}, isbn = {9782738100733}, year = {1989}, date = {1989-10-01}, pages = {268}, publisher = {Odile Jacob}, address = {Paris}, abstract = {Dialogue entre un mathématicien et un neurobiologiste sur les liens qui peuvent exister entre le cerveau et les objets mathématiques, débouchant sur les rapports entre science et éthique.}, keywords = {}, pubstate = {published}, tppubtype = {book} } Dialogue entre un mathématicien et un neurobiologiste sur les liens qui peuvent exister entre le cerveau et les objets mathématiques, débouchant sur les rapports entre science et éthique. |

88. | Compact metric spaces, Fredholm modules, and hyperfiniteness Alain Connes Ergodic Theory Dynam. Systems, 9 (2), pp. 207–220, 1989, ISSN: 0143-3857. @article{MR1007407, title = {Compact metric spaces, Fredholm modules, and hyperfiniteness}, author = {Alain Connes}, doi = {10.1017/S0143385700004934}, issn = {0143-3857}, year = {1989}, date = {1989-01-01}, journal = {Ergodic Theory Dynam. Systems}, volume = {9}, number = {2}, pages = {207--220}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

87. | Embedding of U(1)-current algebras in noncommutative algebras of classical statistical mechanics Alain Connes, David E. Evans Comm. Math. Phys., 121 (3), pp. 507–525, 1989, ISSN: 0010-3616. @article{MR990778, title = {Embedding of U(1)-current algebras in noncommutative algebras of classical statistical mechanics}, author = {Alain Connes and David E. Evans}, url = {http://projecteuclid.org/euclid.cmp/1104178144}, issn = {0010-3616}, year = {1989}, date = {1989-01-01}, journal = {Comm. Math. Phys.}, volume = {121}, number = {3}, pages = {507--525}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

86. | Hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks Alain Connes, Edward James Woods Pacific J. Math., 137 (2), pp. 225–243, 1989, ISSN: 0030-8730. @article{MR990212, title = {Hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks}, author = {Alain Connes and Edward James Woods}, url = {http://projecteuclid.org/euclid.pjm/1102650384}, issn = {0030-8730}, year = {1989}, date = {1989-01-01}, journal = {Pacific J. Math.}, volume = {137}, number = {2}, pages = {225--243}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

85. | Noncommutative geometry Alain Connes Nonperturbative quantum field theory (Carg`ese, 1987), 185 , pp. 33–69, Plenum, New York, 1988. Incollection @incollection{MR1008275, title = {Noncommutative geometry}, author = {Alain Connes}, year = {1988}, date = {1988-01-01}, booktitle = {Nonperturbative quantum field theory (Carg`ese, 1987)}, volume = {185}, pages = {33--69}, publisher = {Plenum, New York}, series = {NATO Adv. Sci. Inst. Ser. B Phys.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

84. | Trace de Dixmier, modules de Fredholm et géométrie riemannienne Alain Connes 5B , pp. 65–70, 1988, ISSN: 0920-5632, (Conformal field theories and related topics (Annecy-le-Vieux, 1988)). @incollection{MR1002957, title = {Trace de Dixmier, modules de Fredholm et géométrie riemannienne}, author = {Alain Connes}, doi = {10.1016/0920-5632(88)90369-6}, issn = {0920-5632}, year = {1988}, date = {1988-01-01}, journal = {Nuclear Phys. B Proc. Suppl.}, volume = {5B}, pages = {65--70}, note = {Conformal field theories and related topics (Annecy-le-Vieux, 1988)}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

83. | ?-theory for discrete groups Paul Baum, Alain Connes Operator algebras and applications, Vol. 1, 135 , pp. 1–20, Cambridge Univ. Press, Cambridge, 1988. @incollection{MR996437, title = {?-theory for discrete groups}, author = {Paul Baum and Alain Connes}, doi = {10.1007/978-1-4612-3762-4_1}, year = {1988}, date = {1988-01-01}, booktitle = {Operator algebras and applications, Vol. 1}, volume = {135}, pages = {1--20}, publisher = {Cambridge Univ. Press, Cambridge}, series = {London Math. Soc. Lecture Note Ser.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

82. | Caractere multiplicatif d'un module de Fredholm Alain Connes, Max Karoubi ?-Theory, 2 (3), pp. 431–463, 1988, ISSN: 0920-3036. Journal ArticleAbstract @article{MR972606, title = {Caractere multiplicatif d'un module de Fredholm}, author = {Alain Connes and Max Karoubi}, url = {/wp-content/uploads/CK.pdf}, doi = {10.1007/BF00533391}, issn = {0920-3036}, year = {1988}, date = {1988-01-01}, journal = {?-Theory}, volume = {2}, number = {3}, pages = {431--463}, abstract = {The main object of this paper is to define a pairing EllP(A) x Kp+I(A) ~ C*, where K,(A) is the algebraic K-theory of A and EIIP(A) is the group generated by Fredholm modules of dimension p. We relate this pairing to Fredholm determinants and central extensions of loop groups and the group of diffeomorphisms of the circle}, keywords = {}, pubstate = {published}, tppubtype = {article} } The main object of this paper is to define a pairing EllP(A) x Kp+I(A) ~ C*, where K,(A) is the algebraic K-theory of A and EIIP(A) is the group generated by Fredholm modules of dimension p. We relate this pairing to Fredholm determinants and central extensions of loop groups and the group of diffeomorphisms of the circle |

81. | Conjecture de Novikov et groupes hyperboliques Alain Connes, Henri Moscovici C. R. Acad. Sci. Paris Sér. I Math., 307 (9), pp. 475–480, 1988, ISSN: 0249-6291. Journal Article @article{MR964110, title = {Conjecture de Novikov et groupes hyperboliques}, author = {Alain Connes and Henri Moscovici}, issn = {0249-6291}, year = {1988}, date = {1988-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {307}, number = {9}, pages = {475--480}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

80. | Cyclic cohomology and noncommutative differential geometry Alain Connes Géométrie différentielle (Paris, 1986), 33 , pp. 33–50, Hermann, Paris, 1988. Incollection @incollection{MR955850, title = {Cyclic cohomology and noncommutative differential geometry}, author = {Alain Connes}, year = {1988}, date = {1988-01-01}, booktitle = {Géométrie différentielle (Paris, 1986)}, volume = {33}, pages = {33--50}, publisher = {Hermann, Paris}, series = {Travaux en Cours}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

79. | Entire cyclic cohomology of Banach algebras and characters of ?-summable Fredholm modules Alain Connes ?-Theory, 1 (6), pp. 519–548, 1988, ISSN: 0920-3036. @article{MR953915, title = { Entire cyclic cohomology of Banach algebras and characters of ?-summable Fredholm modules}, author = {Alain Connes}, doi = {10.1007/BF00533785}, issn = {0920-3036}, year = {1988}, date = {1988-01-01}, journal = {?-Theory}, volume = {1}, number = {6}, pages = {519--548}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

78. | The action functional in noncommutative geometry Alain Connes Comm. Math. Phys., 117 (4), pp. 673–683, 1988, ISSN: 0010-3616. @article{MR953826, title = {The action functional in noncommutative geometry}, author = {Alain Connes}, url = {http://projecteuclid.org/euclid.cmp/1104161823}, issn = {0010-3616}, year = {1988}, date = {1988-01-01}, journal = {Comm. Math. Phys.}, volume = {117}, number = {4}, pages = {673--683}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

77. | Quasi homomorphismes, cohomologie cyclique et positivité Alain Connes, Joachim Cuntz Comm. Math. Phys., 114 (3), pp. 515–526, 1988, ISSN: 0010-3616. @article{MR929142, title = {Quasi homomorphismes, cohomologie cyclique et positivité}, author = {Alain Connes and Joachim Cuntz}, url = {http://projecteuclid.org/euclid.cmp/1104160692}, issn = {0010-3616}, year = {1988}, date = {1988-01-01}, journal = {Comm. Math. Phys.}, volume = {114}, number = {3}, pages = {515--526}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

76. | Chern character for discrete groups Paul Baum, Alain Connes A fête of topology, pp. 163–232, Academic Press, Boston, MA, 1988. @incollection{MR928402, title = {Chern character for discrete groups}, author = {Paul Baum and Alain Connes}, doi = {10.1016/B978-0-12-480440-1.50015-0}, year = {1988}, date = {1988-01-01}, booktitle = {A fête of topology}, pages = {163--232}, publisher = {Academic Press, Boston, MA}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

75. | Cyclic cohomology and noncommutative differential geometry Alain Connes Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pp. 879–889, Amer. Math. Soc., Providence, RI, 1987. @inproceedings{MR934290, title = {Cyclic cohomology and noncommutative differential geometry}, author = {Alain Connes}, doi = {10.1007/bf01225381}, year = {1987}, date = {1987-01-01}, booktitle = {Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986)}, pages = {879--889}, publisher = {Amer. Math. Soc., Providence, RI}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } |

74. | Dynamical Entropy of ?* Algebras and von Neumann Algebras Alain Connes, Heide Narnhofer, Walter Thirring Comm. Math. Phys., 112 (4), pp. 691–719, 1987, ISSN: 0010-3616. Journal ArticleAbstract @article{MR910587, title = {Dynamical Entropy of ?* Algebras and von Neumann Algebras}, author = {Alain Connes and Heide Narnhofer and Walter Thirring}, url = {/wp-content/uploads/entropy.pdf}, issn = {0010-3616}, year = {1987}, date = {1987-01-01}, journal = {Comm. Math. Phys.}, volume = {112}, number = {4}, pages = {691--719}, abstract = {The definition of the dynamical entropy is extended for automorph- ism groups of C* algebras A. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The definition of the dynamical entropy is extended for automorph- ism groups of C* algebras A. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density. |

73. | Yang-Mills for noncommutative two-tori Alain Connes, Marc A. Rieffel Operator algebras and mathematical physics (Iowa City, Iowa, 1985), 62 , pp. 237–266, Amer. Math. Soc., Providence, RI, 1987. @incollection{MR878383, title = {Yang-Mills for noncommutative two-tori}, author = {Alain Connes and Marc A. Rieffel}, doi = {10.1090/conm/062/878383}, year = {1987}, date = {1987-01-01}, booktitle = {Operator algebras and mathematical physics (Iowa City, Iowa, 1985)}, volume = {62}, pages = {237--266}, publisher = {Amer. Math. Soc., Providence, RI}, series = {Contemp. Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

72. | Transgression du caractère de Chern et cohomologie cyclique Alain Connes, Henri Moscovici C. R. Acad. Sci. Paris Sér. I Math., 303 (18), pp. 913–918, 1986, ISSN: 0249-6291. Journal Article @article{MR873393, title = {Transgression du caractère de Chern et cohomologie cyclique}, author = {Alain Connes and Henri Moscovici}, issn = {0249-6291}, year = {1986}, date = {1986-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {303}, number = {18}, pages = {913--918}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

71. | Cyclic cohomology and the transverse fundamental class of a foliation Alain Connes Geometric methods in operator algebras (Kyoto, 1983), 123 , pp. 52–144, Longman Sci. Tech., Harlow, 1986. IncollectionAbstract @incollection{MR866491, title = {Cyclic cohomology and the transverse fundamental class of a foliation}, author = {Alain Connes}, url = {/wp-content/uploads/transfund.pdf}, year = {1986}, date = {1986-01-01}, booktitle = {Geometric methods in operator algebras (Kyoto, 1983)}, volume = {123}, pages = {52--144}, publisher = {Longman Sci. Tech., Harlow}, series = {Pitman Res. Notes Math. Ser.}, abstract = {In this paper we shall prove that for any transversally oriented foliated man- ifold (V, F ), integration on the transverse fundamental class does yield a well defined map of K∗(V/F) to C. Here K∗(V/F) is by definition the K theory of the C∗ algebra C∗(V, F ) canonically associated to (V, F ).}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } In this paper we shall prove that for any transversally oriented foliated man- ifold (V, F ), integration on the transverse fundamental class does yield a well defined map of K∗(V/F) to C. Here K∗(V/F) is by definition the K theory of the C∗ algebra C∗(V, F ) canonically associated to (V, F ). |

70. | Indice des sous facteurs, algèbres de Hecke et théorie des nœuds (d'après Vaughan Jones) Alain Connes (133-134), pp. 289–308, 1986, ISSN: 0303-1179, (Seminar Bourbaki, Vol. 1984/85). Incollection @incollection{MR837226, title = {Indice des sous facteurs, algèbres de Hecke et théorie des nœuds (d'après Vaughan Jones)}, author = {Alain Connes}, issn = {0303-1179}, year = {1986}, date = {1986-01-01}, journal = {Astérisque}, number = {133-134}, pages = {289--308}, note = {Seminar Bourbaki, Vol. 1984/85}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

69. | Leafwise homotopy equivalence and rational Pontrjagin classes Paul Baum, Alain Connes Foliations (Tokyo, 1983), 5 , pp. 1–14, North-Holland, Amsterdam, 1985. @incollection{MR877325, title = {Leafwise homotopy equivalence and rational Pontrjagin classes}, author = {Paul Baum and Alain Connes}, doi = {10.2969/aspm/00510001}, year = {1985}, date = {1985-01-01}, booktitle = {Foliations (Tokyo, 1983)}, volume = {5}, pages = {1--14}, publisher = {North-Holland, Amsterdam}, series = {Adv. Stud. Pure Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

68. | Noncommutative differential geometry Alain Connes Inst. Hautes Études Sci. Publ. Math., (62), pp. 257–360, 1985, ISSN: 0073-8301. Journal ArticleAbstract @article{MR823176, title = {Noncommutative differential geometry}, author = {Alain Connes}, url = {/wp-content/uploads/noncommutative_differential_geometry.pdf}, issn = {0073-8301}, year = {1985}, date = {1985-01-01}, journal = {Inst. Hautes Études Sci. Publ. Math.}, number = {62}, pages = {257--360}, abstract = {This is the introduction to a series of papers in which we shall extend the calculus of differential forms and the de Rham homology of currents beyond their customary framework of manifolds, in order to deal with spaces of a more elaborate nature, such as, a) the space of leaves of a foliation, b) the dual space of a finitely generated non-abelian discrete group (or Lie group), c) the orbit space of the action of a discrete group (or Lie group) on a manifold. What such spaces have in common is to be, in general, badly behaved as point sets, so that the usual tools of measure theory, topology and differential geometry lose their pertinence. These spaces are much better understood by means of a canonically associated algebra which is the group convolution algebra in case b). When the space V is an ordinary manifold, the associated algebra is commutative. It is an algebra of complex-valued functions on V, endowed with the pointwise operations of sum and product.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This is the introduction to a series of papers in which we shall extend the calculus of differential forms and the de Rham homology of currents beyond their customary framework of manifolds, in order to deal with spaces of a more elaborate nature, such as, a) the space of leaves of a foliation, b) the dual space of a finitely generated non-abelian discrete group (or Lie group), c) the orbit space of the action of a discrete group (or Lie group) on a manifold. What such spaces have in common is to be, in general, badly behaved as point sets, so that the usual tools of measure theory, topology and differential geometry lose their pertinence. These spaces are much better understood by means of a canonically associated algebra which is the group convolution algebra in case b). When the space V is an ordinary manifold, the associated algebra is commutative. It is an algebra of complex-valued functions on V, endowed with the pointwise operations of sum and product. |

67. | Diameters of state spaces of type III factors Alain Connes, Uffe Haagerup, Erling Størmer Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), 1132 , pp. 91–116, Springer, Berlin, 1985. @incollection{MR799565, title = {Diameters of state spaces of type III factors}, author = {Alain Connes and Uffe Haagerup and Erling Størmer}, doi = {10.1007/BFb0074881}, year = {1985}, date = {1985-01-01}, booktitle = {Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983)}, volume = {1132}, pages = {91--116}, publisher = {Springer, Berlin}, series = {Lecture Notes in Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

66. | Entropie de Kolmogoroff-Sinai et mécanique statistique quantique Alain Connes C. R. Acad. Sci. Paris Sér. I Math., 301 (1), pp. 1–6, 1985, ISSN: 0249-6291. Journal Article @article{MR797804, title = {Entropie de Kolmogoroff-Sinai et mécanique statistique quantique}, author = {Alain Connes}, issn = {0249-6291}, year = {1985}, date = {1985-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {301}, number = {1}, pages = {1--6}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

65. | Introduction to noncommutative differential geometry Alain Connes Workshop Bonn 1984 (Bonn, 1984), 1111 , pp. 3–16, Springer, Berlin, 1985. @incollection{MR797413, title = {Introduction to noncommutative differential geometry}, author = {Alain Connes}, doi = {10.1007/BFb0084582}, year = {1985}, date = {1985-01-01}, booktitle = {Workshop Bonn 1984 (Bonn, 1984)}, volume = {1111}, pages = {3--16}, publisher = {Springer, Berlin}, series = {Lecture Notes in Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

64. | Approximately transitive flows and ITPFI factors Alain Connes, Edward James Woods Ergodic Theory Dynam. Systems, 5 (2), pp. 203–236, 1985, ISSN: 0143-3857. @article{MR796751, title = {Approximately transitive flows and ITPFI factors}, author = {Alain Connes and Edward James Woods}, doi = {10.1017/S0143385700002868}, issn = {0143-3857}, year = {1985}, date = {1985-01-01}, journal = {Ergodic Theory Dynam. Systems}, volume = {5}, number = {2}, pages = {203--236}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

63. | Factors of type III₁, property ?' Alain Connes J. Operator Theory, 14 (1), pp. 189–211, 1985, ISSN: 0379-4024. Journal Article @article{MR789385, title = {Factors of type III₁, property ?'_{λ} and closure of inner automorphisms}, author = {Alain Connes}, issn = {0379-4024}, year = {1985}, date = {1985-01-01}, journal = {J. Operator Theory}, volume = {14}, number = {1}, pages = {189--211}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

62. | Property T for von Neumann algebras Alain Connes, Vaughan Jones Bull. London Math. Soc., 17 (1), pp. 57–62, 1985, ISSN: 0024-6093. @article{MR766450, title = {Property T for von Neumann algebras}, author = {Alain Connes and Vaughan Jones}, doi = {10.1112/blms/17.1.57}, issn = {0024-6093}, year = {1985}, date = {1985-01-01}, journal = {Bull. London Math. Soc.}, volume = {17}, number = {1}, pages = {57--62}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

61. | The longitudinal index theorem for foliations Alain Connes, Georges Skandalis Publ. Res. Inst. Math. Sci., 20 (6), pp. 1139–1183, 1984, ISSN: 0034-5318. Journal ArticleAbstract @article{MR775126, title = {The longitudinal index theorem for foliations}, author = {Alain Connes and Georges Skandalis}, url = {/wp-content/uploads/longitudinal.pdf}, doi = {10.2977/prims/1195180375}, issn = {0034-5318}, year = {1984}, date = {1984-01-01}, journal = {Publ. Res. Inst. Math. Sci.}, volume = {20}, number = {6}, pages = {1139--1183}, abstract = {In this paper , we use the bivariant K theory of Kasparov ([19]) as a basic tool to prove the K-theoretical version of the index theorem for longitudinal elliptic differential operators for foliations which is stated as a problem in [10], Section 10. When the foliation is by the fibers of a fibration, this theorem reduces to the Atiyah-Singer index theorem for families ([2], Theorem 3.1). It implies the index theorem for measured foliations ([9], Theorem, p. 136) and unlike the latter makes sense for arbitrary foliations, not necessarily gifted with a holonomy invariant transverse measure.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper , we use the bivariant K theory of Kasparov ([19]) as a basic tool to prove the K-theoretical version of the index theorem for longitudinal elliptic differential operators for foliations which is stated as a problem in [10], Section 10. When the foliation is by the fibers of a fibration, this theorem reduces to the Atiyah-Singer index theorem for families ([2], Theorem 3.1). It implies the index theorem for measured foliations ([9], Theorem, p. 136) and unlike the latter makes sense for arbitrary foliations, not necessarily gifted with a holonomy invariant transverse measure. |

60. | Caractère multiplicatif d'un module de Fredholm Alain Connes, Max Karoubi C. R. Acad. Sci. Paris Sér. I Math., 299 (19), pp. 963–968, 1984, ISSN: 0249-6291. Journal Article @article{MR774679, title = {Caractère multiplicatif d'un module de Fredholm}, author = {Alain Connes and Max Karoubi}, issn = {0249-6291}, year = {1984}, date = {1984-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {299}, number = {19}, pages = {963--968}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

59. | A connection between the classical and the quantum mechanical entropies Alain Connes, Erling Størmer Operator algebras and group representations, Vol. I (Neptun, 1980), 17 , pp. 113–123, Pitman, Boston, MA, 1984. Incollection @incollection{MR731767, title = {A connection between the classical and the quantum mechanical entropies}, author = {Alain Connes and Erling Størmer}, year = {1984}, date = {1984-01-01}, booktitle = {Operator algebras and group representations, Vol. I (Neptun, 1980)}, volume = {17}, pages = {113--123}, publisher = {Pitman, Boston, MA}, series = {Monogr. Stud. Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

58. | Cohomologie cyclique et foncteurs Ext Alain Connes C. R. Acad. Sci. Paris Sér. I Math., 296 (23), pp. 953–958, 1983, ISSN: 0249-6291. Journal ArticleAbstract @article{MR777584, title = {Cohomologie cyclique et foncteurs Ext^{?}}, author = {Alain Connes}, url = {/wp-content/uploads/n83.pdf}, issn = {0249-6291}, year = {1983}, date = {1983-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {296}, number = {23}, pages = {953--958}, abstract = {We construct a functor A → A♮ from the category of (non commutative) algebras over a field k, to an abelian category and show that the cyclic cohomology Hλn(A), which we introduced and studied in [5] and [6] coincides with Extn(A♮,k♮).}, keywords = {}, pubstate = {published}, tppubtype = {article} } We construct a functor A → A♮ from the category of (non commutative) algebras over a field k, to an abelian category and show that the cyclic cohomology Hλn(A), which we introduced and studied in [5] and [6] coincides with Extn(A♮,k♮). |

57. | A survey of foliations and operator algebras Alain Connes Operator algebras and applications, Part I (Kingston, Ont., 1980), 38 , pp. 521–628, Amer. Math. Soc., Providence, R.I., 1982. IncollectionAbstract @incollection{MR679730, title = {A survey of foliations and operator algebras}, author = {Alain Connes}, url = {/wp-content/uploads/foliationsfine.pdf}, year = {1982}, date = {1982-01-01}, booktitle = {Operator algebras and applications, Part I (Kingston, Ont., 1980)}, volume = {38}, pages = {521--628}, publisher = {Amer. Math. Soc., Providence, R.I.}, series = {Proc. Sympos. Pure Math.}, abstract = {Survey of the role of the geometry of foliations through the associated operator algebras.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } Survey of the role of the geometry of foliations through the associated operator algebras. |

56. | ?²-index theory on homogeneous spaces and discrete series representations Alain Connes, Henri Moscovici Operator algebras and applications, Part I (Kingston, Ont., 1980), 38 , pp. 419–433, Amer. Math. Soc., Providence, R.I., 1982. @incollection{MR679725, title = {?²-index theory on homogeneous spaces and discrete series representations}, author = {Alain Connes and Henri Moscovici}, doi = {10.1016/0016-0032(81)90055-7}, year = {1982}, date = {1982-01-01}, booktitle = {Operator algebras and applications, Part I (Kingston, Ont., 1980)}, volume = {38}, pages = {419--433}, publisher = {Amer. Math. Soc., Providence, R.I.}, series = {Proc. Sympos. Pure Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

55. | Classification des facteurs Alain Connes Operator algebras and applications, Part 2 (Kingston, Ont., 1980), 38 , pp. 43–109, Amer. Math. Soc., Providence, R.I., 1982. Incollection @incollection{MR679497, title = {Classification des facteurs}, author = {Alain Connes}, year = {1982}, date = {1982-01-01}, booktitle = {Operator algebras and applications, Part 2 (Kingston, Ont., 1980)}, volume = {38}, pages = {43--109}, publisher = {Amer. Math. Soc., Providence, R.I.}, series = {Proc. Sympos. Pure Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

54. | The ?²-index theorem for homogeneous spaces of Lie groups Alain Connes, Henri Moscovici Ann. of Math. (2), 115 (2), pp. 291–330, 1982, ISSN: 0003-486X. @article{MR647808, title = {The ?²-index theorem for homogeneous spaces of Lie groups}, author = {Alain Connes and Henri Moscovici}, doi = {10.2307/1971393}, issn = {0003-486X}, year = {1982}, date = {1982-01-01}, journal = {Ann. of Math. (2)}, volume = {115}, number = {2}, pages = {291--330}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

53. | A II₁ factor with two nonconjugate Cartan subalgebras Alain Connes, Vaughan Jones Bull. Amer. Math. Soc. (N.S.), 6 (2), pp. 211–212, 1982, ISSN: 0273-0979. @article{MR640947, title = {A II₁ factor with two nonconjugate Cartan subalgebras}, author = {Alain Connes and Vaughan Jones}, doi = {10.1090/S0273-0979-1982-14981-3}, issn = {0273-0979}, year = {1982}, date = {1982-01-01}, journal = {Bull. Amer. Math. Soc. (N.S.)}, volume = {6}, number = {2}, pages = {211--212}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

52. | An amenable equivalence relation is generated by a single transformation Alain Connes, Jacob Feldman, Benjamin Weiss Ergodic Theory Dynam. Systems, 1 (4), pp. 431–450 (1982), 1981, ISSN: 0143-3857. @article{MR662736, title = {An amenable equivalence relation is generated by a single transformation}, author = {Alain Connes and Jacob Feldman and Benjamin Weiss}, doi = {10.1017/s014338570000136x}, issn = {0143-3857}, year = {1981}, date = {1981-01-01}, journal = {Ergodic Theory Dynam. Systems}, volume = {1}, number = {4}, pages = {431--450 (1982)}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

51. | Feuilletages et algèbres d'opérateurs Alain Connes Bourbaki Seminar, Vol. 1979/80, 842 , pp. 139–155, Springer, Berlin-New York, 1981. Incollection @incollection{MR636521, title = {Feuilletages et algèbres d'opérateurs}, author = {Alain Connes}, year = {1981}, date = {1981-01-01}, booktitle = {Bourbaki Seminar, Vol. 1979/80}, volume = {842}, pages = {139--155}, publisher = {Springer, Berlin-New York}, series = {Lecture Notes in Math.}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

50. | Théorème de l'indice pour les feuilletages Alain Connes, Georges Skandalis C. R. Acad. Sci. Paris Sér. I Math., 292 (18), pp. 871–876, 1981, ISSN: 0249-6291. Journal Article @article{MR623519, title = {Théorème de l'indice pour les feuilletages}, author = {Alain Connes and Georges Skandalis}, issn = {0249-6291}, year = {1981}, date = {1981-01-01}, journal = {C. R. Acad. Sci. Paris Sér. I Math.}, volume = {292}, number = {18}, pages = {871--876}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

49. | An analogue of the Thom isomorphism for crossed products of a ?*-algebra by an action of ℝ Alain Connes Adv. in Math., 39 (1), pp. 31–55, 1981, ISSN: 0001-8708. @article{MR605351, title = {An analogue of the Thom isomorphism for crossed products of a ?*-algebra by an action of ℝ}, author = {Alain Connes}, doi = {10.1016/0001-8708(81)90056-6}, issn = {0001-8708}, year = {1981}, date = {1981-01-01}, journal = {Adv. in Math.}, volume = {39}, number = {1}, pages = {31--55}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

48. | Property T and asymptotically invariant sequences Alain Connes, Benjamin Weiss Israel J. Math., 37 (3), pp. 209–210, 1980, ISSN: 0021-2172. @article{MR599455, title = {Property T and asymptotically invariant sequences}, author = {Alain Connes and Benjamin Weiss}, doi = {10.1007/BF02760962}, issn = {0021-2172}, year = {1980}, date = {1980-01-01}, journal = {Israel J. Math.}, volume = {37}, number = {3}, pages = {209--210}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

47. | A factor of type II₁ with countable fundamental group Alain Connes J. Operator Theory, 4 (1), pp. 151–153, 1980, ISSN: 0379-4024. Journal Article @article{MR587372, title = {A factor of type II₁ with countable fundamental group}, author = {Alain Connes}, issn = {0379-4024}, year = {1980}, date = {1980-01-01}, journal = {J. Operator Theory}, volume = {4}, number = {1}, pages = {151--153}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

46. | A construction of approximately finite-dimensional non-ITPFI factors Alain Connes, Edward James Woods Canad. Math. Bull., 23 (2), pp. 227–230, 1980, ISSN: 0008-4395. @article{MR576103, title = {A construction of approximately finite-dimensional non-ITPFI factors}, author = {Alain Connes and Edward James Woods}, doi = {10.4153/CMB-1980-030-5}, issn = {0008-4395}, year = {1980}, date = {1980-01-01}, journal = {Canad. Math. Bull.}, volume = {23}, number = {2}, pages = {227--230}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

45. | ?*-algèbres et géométrie différentielle Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 290 (13), pp. A599–A604, 1980, ISSN: 0151-0509. Journal ArticleAbstract @article{MR572645, title = {?*-algèbres et géométrie différentielle}, author = {Alain Connes}, url = {/wp-content/uploads/note80.pdf /wp-content/uploads/notetranslated.pdf}, issn = {0151-0509}, year = {1980}, date = {1980-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {290}, number = {13}, pages = {A599--A604}, abstract = {This note is at the foundation of noncommutative geometry, it gives the analogues of connections and curvature for C* dynamical systems associated to a Lie group action on a C*-algebra. It also contains integrality results of the Chern character and the first construction of finite projective modules over the noncommutative torus such as the Schwartz space S(R). An English translation can also be downloaded.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This note is at the foundation of noncommutative geometry, it gives the analogues of connections and curvature for C* dynamical systems associated to a Lie group action on a C*-algebra. It also contains integrality results of the Chern character and the first construction of finite projective modules over the noncommutative torus such as the Schwartz space S(R). An English translation can also be downloaded. |

44. | Von Neumann algebras Alain Connes Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 97–109, Acad. Sci. Fennica, Helsinki, 1980. Inproceedings @inproceedings{MR562600, title = {Von Neumann algebras}, author = {Alain Connes}, year = {1980}, date = {1980-01-01}, booktitle = {Proceedings of the International Congress of Mathematicians (Helsinki, 1978)}, pages = {97--109}, publisher = {Acad. Sci. Fennica, Helsinki}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } |

43. | On the spatial theory of von Neumann algebras Alain Connes J. Functional Analysis, 35 (2), pp. 153–164, 1980, ISSN: 0022-1236. @article{MR561983, title = {On the spatial theory of von Neumann algebras}, author = {Alain Connes}, doi = {10.1016/0022-1236(80)90002-6}, issn = {0022-1236}, year = {1980}, date = {1980-01-01}, journal = {J. Functional Analysis}, volume = {35}, number = {2}, pages = {153--164}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

42. | On the equivalence between injectivity and semidiscreteness for operator algebras Alain Connes Algèbres d'opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), pp. 107–112, CNRS, Paris, 1979. Inproceedings @inproceedings{MR560628, title = {On the equivalence between injectivity and semidiscreteness for operator algebras}, author = {Alain Connes}, year = {1979}, date = {1979-01-01}, booktitle = {Algèbres d'opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977)}, volume = {274}, pages = {107--112}, publisher = {CNRS, Paris}, series = {Colloq. Internat. CNRS}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } |

41. | Sur la théorie non commutative de l'intégration Alain Connes Algèbres d'opérateurs (Sém., Les Plans-sur-Bex, 1978), 725 , pp. 19–143, Springer, Berlin, 1979. IncollectionAbstract @incollection{MR548112, title = {Sur la théorie non commutative de l'intégration}, author = {Alain Connes}, url = {/wp-content/uploads/ThNonComm.pdf}, year = {1979}, date = {1979-01-01}, booktitle = {Algèbres d'opérateurs (Sém., Les Plans-sur-Bex, 1978)}, volume = {725}, pages = {19--143}, publisher = {Springer, Berlin}, series = {Lecture Notes in Math.}, abstract = {Dans cet article, nous développons d’abord une généralisation de la théorie usuelle de l’intégration, qui, mieux adaptée à létude d’espaces singuliers comme l’espace des feuilles d’un feuilletage, l’espace des géodésiques d’une variété Riemannienne compacte ou celui des représentations irréductibles d’une C∗-algèbre, permet de donner une valeur finie à une intégrale comme celle qui définit βi. Nous l’appliquons ensuite pour d ́emontrer la version "feuilletage mesuré" du théorème de l’indice d’Atiyah Singer (cf. No VIII). Ce résultat s’inscrit dans la direction proposée par Singer dans [43] [44] et déjà illustrée par Atiyah dans [2].}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } Dans cet article, nous développons d’abord une généralisation de la théorie usuelle de l’intégration, qui, mieux adaptée à létude d’espaces singuliers comme l’espace des feuilles d’un feuilletage, l’espace des géodésiques d’une variété Riemannienne compacte ou celui des représentations irréductibles d’une C∗-algèbre, permet de donner une valeur finie à une intégrale comme celle qui définit βi. Nous l’appliquons ensuite pour d ́emontrer la version "feuilletage mesuré" du théorème de l’indice d’Atiyah Singer (cf. No VIII). Ce résultat s’inscrit dans la direction proposée par Singer dans [43] [44] et déjà illustrée par Atiyah dans [2]. |

40. | The ?²-index theorem for homogeneous spaces Alain Connes, Henri Moscovici Bull. Amer. Math. Soc. (N.S.), 1 (4), pp. 688–690, 1979, ISSN: 0273-0979. @article{MR532554, title = {The ?²-index theorem for homogeneous spaces}, author = {Alain Connes and Henri Moscovici}, doi = {10.1090/S0273-0979-1979-14670-6}, issn = {0273-0979}, year = {1979}, date = {1979-01-01}, journal = {Bull. Amer. Math. Soc. (N.S.)}, volume = {1}, number = {4}, pages = {688--690}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

39. | The von Neumann algebra of a foliation Alain Connes Mathematical problems in theoretical physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977), pp. 145–151, Springer, Berlin-New York, 1978. Inproceedings @inproceedings{MR518430, title = {The von Neumann algebra of a foliation}, author = {Alain Connes}, year = {1978}, date = {1978-01-01}, booktitle = {Mathematical problems in theoretical physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977)}, volume = {80}, pages = {145--151}, publisher = {Springer, Berlin-New York}, series = {Lecture Notes in Phys.}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } |

38. | Errata: "The flow of weights on factors of type III" Alain Connes, Masamichi Takesaki Tohoku Math. J. (2), 30 (4), pp. 653–655, 1978, ISSN: 0040-8735. @article{MR516896, title = {Errata: "The flow of weights on factors of type III"}, author = {Alain Connes and Masamichi Takesaki}, doi = {10.2748/tmj/1178229923}, issn = {0040-8735}, year = {1978}, date = {1978-01-01}, journal = {Tohoku Math. J. (2)}, volume = {30}, number = {4}, pages = {653--655}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

37. | On the cohomology of operator algebras Alain Connes J. Functional Analysis, 28 (2), pp. 248–253, 1978. @article{MR0493383, title = {On the cohomology of operator algebras}, author = {Alain Connes}, doi = {10.1016/0022-1236(78)90088-5}, year = {1978}, date = {1978-01-01}, journal = {J. Functional Analysis}, volume = {28}, number = {2}, pages = {248--253}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

36. | Homogeneity of the state space of factors of type III₁ Alain Connes, Erling Størmer J. Functional Analysis, 28 (2), pp. 187–196, 1978. @article{MR0470689, title = {Homogeneity of the state space of factors of type III₁}, author = {Alain Connes and Erling Størmer}, doi = {10.1016/0022-1236(78)90085-x}, year = {1978}, date = {1978-01-01}, journal = {J. Functional Analysis}, volume = {28}, number = {2}, pages = {187--196}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

35. | The flow of weights on factors of type III Alain Connes, Masamichi Takesaki Tohoku Math. J. (2), 29 (4), pp. 473–575, 1977, ISSN: 0040-8735. @article{MR480760, title = {The flow of weights on factors of type III}, author = {Alain Connes and Masamichi Takesaki}, doi = {10.2748/tmj/1178240493}, issn = {0040-8735}, year = {1977}, date = {1977-01-01}, journal = {Tohoku Math. J. (2)}, volume = {29}, number = {4}, pages = {473--575}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

34. | Periodic automorphisms of the hyperfinite factor of type II₁ Alain Connes Acta Sci. Math. (Szeged), 39 (1-2), pp. 39–66, 1977, ISSN: 0001-6969. Journal ArticleAbstract @article{MR448101, title = {Periodic automorphisms of the hyperfinite factor of type II₁}, author = {Alain Connes}, url = {/wp-content/uploads/szego.pdf}, issn = {0001-6969}, year = {1977}, date = {1977-01-01}, journal = {Acta Sci. Math. (Szeged)}, volume = {39}, number = {1-2}, pages = {39--66}, abstract = {Classification of periodic automorphisms of the hyperfinite factor of type II1}, keywords = {}, pubstate = {published}, tppubtype = {article} } Classification of periodic automorphisms of the hyperfinite factor of type II1 |

33. | Measure space automorphisms, the normalizers of their full groups, and approximate finiteness Alain Connes, Wolfgang Krieger J. Functional Analysis, 24 (4), pp. 336–352, 1977. @article{MR0444900, title = {Measure space automorphisms, the normalizers of their full groups, and approximate finiteness}, author = {Alain Connes and Wolfgang Krieger}, doi = {10.1016/0022-1236(77)90062-3}, year = {1977}, date = {1977-01-01}, journal = {J. Functional Analysis}, volume = {24}, number = {4}, pages = {336--352}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

32. | The Tomita-Takesaki theory and classification of type-III factors Alain Connes ?*-algebras and their applications to statistical mechanics and quantum field theory (Proc. Internat. School of Physics "Enrico Fermi", Course LX, Varenna, 1973), pp. 29–46, 1976. Inproceedings @inproceedings{MR0633775, title = {The Tomita-Takesaki theory and classification of type-III factors}, author = {Alain Connes}, year = {1976}, date = {1976-01-01}, booktitle = {?*-algebras and their applications to statistical mechanics and quantum field theory (Proc. Internat. School of Physics "Enrico Fermi", Course LX, Varenna, 1973)}, pages = {29--46}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } |

31. | Classification of injective factors. Cases II₁, II Alain Connes Ann. of Math. (2), 104 (1), pp. 73–115, 1976, ISSN: 0003-486X. @article{MR454659, title = { Classification of injective factors. Cases II₁, II_{∞}, III_{λ}, λ ≠ 1}, author = {Alain Connes}, doi = {10.2307/1971057}, issn = {0003-486X}, year = {1976}, date = {1976-01-01}, journal = {Ann. of Math. (2)}, volume = {104}, number = {1}, pages = {73--115}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

30. | On the classification of von Neumann algebras and their automorphisms Alain Connes Symposia Mathematica, Vol. XX (Convegno sulle Algebre ?* e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria ?, INDAM, Rome, 1975), pp. 435–478, 1976. Incollection @incollection{MR0450988, title = {On the classification of von Neumann algebras and their automorphisms}, author = {Alain Connes}, year = {1976}, date = {1976-01-01}, booktitle = {Symposia Mathematica, Vol. XX (Convegno sulle Algebre ?* e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria ?, INDAM, Rome, 1975)}, pages = {435--478}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

29. | Outer conjugacy of automorphisms of factors Alain Connes Symposia Mathematica, Vol. XX (Convegno sulle Algebre ?* e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria ?, INDAM, Rome, 1975), pp. 149–159, 1976. Incollection @incollection{MR0450987, title = {Outer conjugacy of automorphisms of factors}, author = {Alain Connes}, year = {1976}, date = {1976-01-01}, booktitle = {Symposia Mathematica, Vol. XX (Convegno sulle Algebre ?* e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria ?, INDAM, Rome, 1975)}, pages = {149--159}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

28. | Entropy for automorphisms of II₁ von Neumann algebras Alain Connes, Erling Størmer Acta Math., 134 (3-4), pp. 289–306, 1975, ISSN: 0001-5962. @article{MR454657, title = {Entropy for automorphisms of II₁ von Neumann algebras}, author = {Alain Connes and Erling Størmer}, doi = {10.1007/BF02392105}, issn = {0001-5962}, year = {1975}, date = {1975-01-01}, journal = {Acta Math.}, volume = {134}, number = {3-4}, pages = {289--306}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

27. | Structure theory for Type III factors Alain Connes Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, pp. 87–91, 1975. Inproceedings @inproceedings{MR0440378, title = {Structure theory for Type III factors}, author = {Alain Connes}, year = {1975}, date = {1975-01-01}, booktitle = {Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2}, pages = {87--91}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } |

26. | A factor not anti-isomorphic to itself Alain Connes Bull. London Math. Soc., 7 , pp. 171–174, 1975, ISSN: 0024-6093. @article{MR435864, title = {A factor not anti-isomorphic to itself}, author = {Alain Connes}, doi = {10.1112/blms/7.2.171}, issn = {0024-6093}, year = {1975}, date = {1975-01-01}, journal = {Bull. London Math. Soc.}, volume = {7}, pages = {171--174}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

25. | On the hierarchy of W. Krieger Alain Connes Illinois J. Math., 19 , pp. 428–432, 1975, ISSN: 0019-2082. @article{MR412825, title = {On the hierarchy of W. Krieger}, author = {Alain Connes}, url = {http://projecteuclid.org/euclid.ijm/1256050743}, issn = {0019-2082}, year = {1975}, date = {1975-01-01}, journal = {Illinois J. Math.}, volume = {19}, pages = {428--432}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

24. | Outer conjugacy classes of automorphisms of factors Alain Connes Ann. Sci. École Norm. Sup. (4), 8 (3), pp. 383–419, 1975, ISSN: 0012-9593. Journal ArticleAbstract @article{MR394228, title = {Outer conjugacy classes of automorphisms of factors}, author = {Alain Connes}, url = {/wp-content/uploads/automorphismes.pdf}, issn = {0012-9593}, year = {1975}, date = {1975-01-01}, journal = {Ann. Sci. École Norm. Sup. (4)}, volume = {8}, number = {3}, pages = {383--419}, abstract = {Two automorphisms a and P of a von Neumann algebra M are called outer conjugate when their classes e (a), e (?) modulo inner automorphisms of M, are conjugate in the group Out M = Aut M/Int M. The outer period po (a) of an automorphism a of M is by definition the period of £ (a) inOutM,andisequalto0ifnopowere(a)",n ^ 0isequalto 1. The obstruction y (a) of an automorphism a of M is the root of 1, y in C such that o^o(a) = Ad U => a(U) = y U for U unitary in M. This definition makes sense when M is a factor, moreover y(a)^0(a) = 1and y(a) = 1if PQ(a) = 0. In [8], theorem 1.5, we showed that/?o and y are complete invariants of outer conjugacy for automorphisms of the hyperfinite factor of type IIi :R, which are periodic. In this paper we shall show that the restriction of periodicky is unnecessary, that is: Any two automorphisms a and P of R such that po (a) = PQ(?) = 0 are outer conjugate. It shows that Out R is a simple group with only countably many conjugacy classes.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Two automorphisms a and P of a von Neumann algebra M are called outer conjugate when their classes e (a), e (?) modulo inner automorphisms of M, are conjugate in the group Out M = Aut M/Int M. The outer period po (a) of an automorphism a of M is by definition the period of £ (a) inOutM,andisequalto0ifnopowere(a)",n ^ 0isequalto 1. The obstruction y (a) of an automorphism a of M is the root of 1, y in C such that o^o(a) = Ad U => a(U) = y U for U unitary in M. This definition makes sense when M is a factor, moreover y(a)^0(a) = 1and y(a) = 1if PQ(a) = 0. In [8], theorem 1.5, we showed that/?o and y are complete invariants of outer conjugacy for automorphisms of the hyperfinite factor of type IIi :R, which are periodic. In this paper we shall show that the restriction of periodicky is unnecessary, that is: Any two automorphisms a and P of R such that po (a) = PQ(?) = 0 are outer conjugate. It shows that Out R is a simple group with only countably many conjugacy classes. |

23. | Classification of automorphisms of hyperfinite factors of type II₁ and II Alain Connes Bull. Amer. Math. Soc., 81 (6), pp. 1090–1092, 1975, ISSN: 0002-9904. @article{MR388117, title = { Classification of automorphisms of hyperfinite factors of type II₁ and II_{∞} and application to type III factors}, author = {Alain Connes}, doi = {10.1090/S0002-9904-1975-13929-2}, issn = {0002-9904}, year = {1975}, date = {1975-01-01}, journal = {Bull. Amer. Math. Soc.}, volume = {81}, number = {6}, pages = {1090--1092}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

22. | Sur la classification des facteurs de type II Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 281 (1), pp. Aii, A13–A15, 1975, ISSN: 0151-0509. Journal Article @article{MR377534, title = {Sur la classification des facteurs de type II}, author = {Alain Connes}, issn = {0151-0509}, year = {1975}, date = {1975-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {281}, number = {1}, pages = {Aii, A13--A15}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

21. | On hyperfinite factors of type III₀ and Krieger's factors Alain Connes J. Functional Analysis, 18 , pp. 318–327, 1975. @article{MR0372635, title = {On hyperfinite factors of type III₀ and Krieger's factors}, author = {Alain Connes}, doi = {10.1016/0022-1236(75)90019-1}, year = {1975}, date = {1975-01-01}, journal = {J. Functional Analysis}, volume = {18}, pages = {318--327}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

20. | A factor not anti-isomorphic to itself Alain Connes Ann. of Math. (2), 101 , pp. 536–554, 1975, ISSN: 0003-486X. @article{MR370209, title = {A factor not anti-isomorphic to itself}, author = {Alain Connes}, doi = {10.2307/1970940}, issn = {0003-486X}, year = {1975}, date = {1975-01-01}, journal = {Ann. of Math. (2)}, volume = {101}, pages = {536--554}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

19. | Flots des poids sur les facteurs de type III Alain Connes, Masamichi Takesaki C. R. Acad. Sci. Paris Sér. A, 278 , pp. 945–948, 1974, ISSN: 0302-8429. Journal Article @article{MR355625, title = {Flots des poids sur les facteurs de type III}, author = {Alain Connes and Masamichi Takesaki}, issn = {0302-8429}, year = {1974}, date = {1974-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A}, volume = {278}, pages = {945--948}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

18. | Caractérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von Neumann Alain Connes Ann. Inst. Fourier (Grenoble), 24 (4), pp. x, 121–155 (1975), 1974, ISSN: 0373-0956. Journal ArticleAbstract @article{MR377533, title = {Caractérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von Neumann}, author = {Alain Connes}, url = {/wp-content/uploads/caracevord.pdf}, issn = {0373-0956}, year = {1974}, date = {1974-01-01}, journal = {Ann. Inst. Fourier (Grenoble)}, volume = {24}, number = {4}, pages = {x, 121--155 (1975)}, abstract = {We characterize the ordered vector spaces underlying von Neumann algebras.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We characterize the ordered vector spaces underlying von Neumann algebras. |

17. | Existence de facteurs infinis asymtotiquement abéliens Alain Connes, Edward James Woods C. R. Acad. Sci. Paris Sér. A, 279 , pp. 189–191, 1974, ISSN: 0302-8429. Journal Article @article{MR361811, title = {Existence de facteurs infinis asymtotiquement abéliens}, author = {Alain Connes and Edward James Woods}, issn = {0302-8429}, year = {1974}, date = {1974-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A}, volume = {279}, pages = {189--191}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

16. | Almost periodic states and factors of type III₁ Alain Connes J. Functional Analysis, 16 , pp. 415–445, 1974. @article{MR0358374, title = {Almost periodic states and factors of type III₁}, author = {Alain Connes}, doi = {10.1016/0022-1236(74)90059-7}, year = {1974}, date = {1974-01-01}, journal = {J. Functional Analysis}, volume = {16}, pages = {415--445}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

15. | Sur le théorème de Radon-Nikodym pour les poids normaux fidèles semi-finis Alain Connes Bull. Sci. Math. (2), 97 , pp. 253–258 (1974), 1973, ISSN: 0007-4497. Journal Article @article{MR358375, title = {Sur le théorème de Radon-Nikodym pour les poids normaux fidèles semi-finis}, author = {Alain Connes}, issn = {0007-4497}, year = {1973}, date = {1973-01-01}, journal = {Bull. Sci. Math. (2)}, volume = {97}, pages = {253--258 (1974)}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

14. | Une classification des facteurs de type III Alain Connes Ann. Sci. École Norm. Sup. (4), 6 , pp. 133–252, 1973, ISSN: 0012-9593. @article{MR341115, title = {Une classification des facteurs de type III}, author = {Alain Connes}, url = {/wp-content/uploads/classificationfacteurs.pdf}, issn = {0012-9593}, year = {1973}, date = {1973-01-01}, journal = {Ann. Sci. École Norm. Sup. (4)}, volume = {6}, pages = {133--252}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

13. | The group property of the invariant ? of von Neumann algebras Alain Connes, Alfons van Daele Math. Scand., 32 , pp. 187–192 (1974), 1973, ISSN: 0025-5521. @article{MR336362, title = {The group property of the invariant ? of von Neumann algebras}, author = {Alain Connes and Alfons van Daele}, doi = {10.7146/math.scand.a-11454}, issn = {0025-5521}, year = {1973}, date = {1973-01-01}, journal = {Math. Scand.}, volume = {32}, pages = {187--192 (1974)}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

12. | Ordres faibles et localisation de zéros de polynômes Alain Connes Séminaire Delange-Pisot-Poitou (12e année: 1970/71), Théorie des nombres, Exp. No. 18, pp. 11, 1972. Incollection @incollection{MR0432609, title = {Ordres faibles et localisation de zéros de polynômes}, author = {Alain Connes}, year = {1972}, date = {1972-01-01}, booktitle = {Séminaire Delange-Pisot-Poitou (12e année: 1970/71), Théorie des nombres, Exp. No. 18}, pages = {11}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

11. | Une classification des facteurs de type III Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 275 , pp. A523–A525, 1972, ISSN: 0151-0509. Journal Article @article{MR303306, title = {Une classification des facteurs de type III}, author = {Alain Connes}, issn = {0151-0509}, year = {1972}, date = {1972-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {275}, pages = {A523--A525}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

10. | Groupe modulaire d'une algèbre de von Neumann Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 274 , pp. A1923–A1926, 1972, ISSN: 0151-0509. Journal Article @article{MR303305, title = {Groupe modulaire d'une algèbre de von Neumann}, author = {Alain Connes}, issn = {0151-0509}, year = {1972}, date = {1972-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {274}, pages = {A1923--A1926}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

9. | États presque périodiques sur une algèbre de von Neumann Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 274 , pp. A1402–A1405, 1972, ISSN: 0151-0509. Journal Article @article{MR295092, title = {États presque périodiques sur une algèbre de von Neumann}, author = {Alain Connes}, issn = {0151-0509}, year = {1972}, date = {1972-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {274}, pages = {A1402--A1405}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

8. | Calcul des deux invariants d'Araki et Woods par la théorie de Tomita et Takesaki Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 274 , pp. A175–A177, 1972, ISSN: 0151-0509. @article{MR293406, title = {Calcul des deux invariants d'Araki et Woods par la théorie de Tomita et Takesaki}, author = {Alain Connes}, url = {/wp-content/uploads/calculinvariantsAW.pdf}, issn = {0151-0509}, year = {1972}, date = {1972-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {274}, pages = {A175--A177}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

7. | Un théorème de décomposition d'applications mesurables Alain Connes Séminaire Choquet, 10e année (1970/71), Initiation à l'analyse, Fasc. 1, Exp. No. 12, pp. 7, 1971. Incollection @incollection{MR0482187, title = {Un théorème de décomposition d'applications mesurables}, author = {Alain Connes}, year = {1971}, date = {1971-01-01}, booktitle = {Séminaire Choquet, 10e année (1970/71), Initiation à l'analyse, Fasc. 1, Exp. No. 12}, pages = {7}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

6. | Un nouvel invariant pour les algèbres de von Neumann Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 273 , pp. A900–A903, 1971, ISSN: 0151-0509. Journal Article @article{MR290119, title = {Un nouvel invariant pour les algèbres de von Neumann}, author = {Alain Connes}, issn = {0151-0509}, year = {1971}, date = {1971-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {273}, pages = {A900--A903}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

5. | Ultrapuissances et applications dans le cadre de l'analyse non standard Alain Connes Séminaire Choquet: 1969/70, Initiation `a lÁnalyse, pp. Fasc. 1, Exp. 8, 25, Secrétariat mathématique, Paris, 1970. Incollection @incollection{MR0286646, title = {Ultrapuissances et applications dans le cadre de l'analyse non standard}, author = {Alain Connes}, year = {1970}, date = {1970-01-01}, booktitle = {Séminaire Choquet: 1969/70, Initiation `a lÁnalyse}, pages = {Fasc. 1, Exp. 8, 25}, publisher = {Secrétariat mathématique, Paris}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

4. | Détermination de modèles minimaux en analyse non standard et application Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 271 , pp. A969–A971, 1970, ISSN: 0151-0509. Journal Article @article{MR286645, title = {Détermination de modèles minimaux en analyse non standard et application}, author = {Alain Connes}, issn = {0151-0509}, year = {1970}, date = {1970-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {271}, pages = {A969--A971}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

3. | Ordres faibles et localisation de zéros de polynômes Alain Connes Séminaire Choquet: 1968/69, Initiation à l'Analyse, Exp. 5, pp. 27, Secrétariat mathématique, Paris, 1969. Incollection @incollection{MR0280463, title = {Ordres faibles et localisation de zéros de polynômes}, author = {Alain Connes}, year = {1969}, date = {1969-01-01}, booktitle = {Séminaire Choquet: 1968/69, Initiation à l'Analyse, Exp. 5}, pages = {27}, publisher = {Secrétariat mathématique, Paris}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |

2. | Ordres faibles et localisation des zéros de polynomes Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 269 , pp. A373–A376, 1969, ISSN: 0151-0509. @article{MR249413, title = {Ordres faibles et localisation des zéros de polynomes}, author = {Alain Connes}, url = {/wp-content/uploads/ordresfaibles.pdf}, issn = {0151-0509}, year = {1969}, date = {1969-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {269}, pages = {A373--A376}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

1. | Sur une généralisation de la notion de corps ordonné Alain Connes C. R. Acad. Sci. Paris Sér. A-B, 269 , pp. A337–A340, 1969, ISSN: 0151-0509. @article{MR249412, title = {Sur une généralisation de la notion de corps ordonné}, author = {Alain Connes}, url = {/wp-content/uploads/crascorpsordonnes.pdf}, issn = {0151-0509}, year = {1969}, date = {1969-01-01}, journal = {C. R. Acad. Sci. Paris Sér. A-B}, volume = {269}, pages = {A337--A340}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

# Publications

269. | Spectral Triples and Zeta-CyclesForthcoming Enseignement Mathématique, Forthcoming. Journal ArticleAbstract |

268. | Quasi-inner functions and local factors Journal of Number Theory, 226 , pp. 139-167, 2021. Journal ArticleAbstract |

267. | On absolute algebraic geometry, the affine case Forthcoming Advances in Mathematics, pp. 1–45, Forthcoming. Journal ArticleAbstract |

266. | Weil positivity and trace formula, the archimedean placeForthcoming Selecta Mathematica, Forthcoming. Journal ArticleAbstract |

265. | Noncommutative Geometry, the Spectral Standpoint Cambridge University Press (Ed.): Cambridge University Press, 2021. Book ChapterAbstract |

264. | Spectral truncation in noncommutative geometry and operator systems Communication in Mathematical Physics, 383 , pp. 2021–2067, 2021. Journal ArticleAbstract |

263. | Segal's Gamma Rings and Universal Arithmetic Quarterly Journal of Mathematics, pp. 1-29, 2020. Journal ArticleAbstract |

262. | Journal of Mathematics and Music, 15 (1), pp. 1–16, 2020, ISSN: 1745-9737. Journal ArticleAbstract |

261. | J. Operator Theory, 85 (1), pp. 257–276, 2019, ISSN: 0379-4024. Journal ArticleAbstract |

260. | Sir Michael Atiyah, a Knight Mathematician Notices Amer. Math. Soc., 66 (10), pp. 1660–1671, 2019, ISSN: 0002-9920. Journal ArticleAbstract |

259. | CNRS Éditions, Paris, 2019, ISBN: 978-2-271-12712-9. BookAbstract |

258. | Theory Appl. Categ., 35 (6), pp. 155–178, 2019. Journal ArticleAbstract |

257. | The term ?₄ in the heat kernel expansion of noncommutative tori Münster J. Math., 12 (2), pp. 239–410, 2019, ISSN: 1867-5778. Journal ArticleAbstract |

256. | Conformal trace theorem for Julia sets of quadratic polynomials Ergodic Theory Dynam. Systems, 39 (9), pp. 2481–2506, 2019, ISSN: 0143-3857. Journal ArticleAbstract |

255. | Noncommutative geometry for symmetric non-self-adjoint operators J. Funct. Anal., 277 (3), pp. 889–936, 2019, ISSN: 0022-1236. Journal ArticleAbstract |

254. | Homological algebra in characteristic one High. Struct., 3 (1), pp. 155–247, 2019. Journal ArticleAbstract |

253. | Iteration of the exterior power on representation rings J. Geom. Phys., 141 , pp. 1–10, 2019, ISSN: 0393-0440. Journal ArticleAbstract |

252. | J. Number Theory, 194 , pp. 1–7, 2018, ISSN: 0022-314X. Journal ArticleAbstract |

251. | Entropy and the spectral action Comm. Math. Phys., 373 (2), pp. 457–471, 2018, ISSN: 0010-3616. Journal ArticleAbstract |

250. | 2018, ISBN: 9782738141361. BookAbstract |

249. | Foundations of mathematics and physics one century after Ħilbert, pp. 159–196, Springer, Cham, 2018. IncollectionAbstract |

248. | Trace theorem for quasi-Fuchsian groups Mat. Sb., 208 (10), pp. 59–90, 2017, ISSN: 0368-8666. Journal ArticleAbstract |

247. | Selecta Math. (N.S.), 23 (3), pp. 1803–1850, 2017, ISSN: 1022-1824. Journal ArticleAbstract |

246. | An essay on the Riemann hypothesis Open problems in mathematics, pp. 225–257, Springer, [Cham], 2016. IncollectionAbstract |

245. | Geometry of the arithmetic site Adv. Math., 291 , pp. 274–329, 2016, ISSN: 0001-8708. Journal ArticleAbstract |

244. | Absolute algebra and Segal's Γ-rings J. Number Theory, 162 , pp. 518–551, 2016, ISSN: 0022-314X. Journal ArticleAbstract |

243. | C. R. Math. Acad. Sci. Paris, 354 (1), pp. 1–6, 2016, ISSN: 1631-073X. Journal ArticleAbstract |

242. | Notices Amer. Math. Soc., 63 (1), pp. 48–49, 2016, ISSN: 0002-9920. Journal ArticleAbstract |

241. | Grand unification in the spectral Pati–Salam model J. High Energy Phys., (11), pp. 011, front matter+12, 2015, ISSN: 1126-6708. Journal ArticleAbstract |

240. | Quanta of geometry: noncommutative aspects Phys. Rev. Lett., 114 (9), pp. 091302, 5, 2015, ISSN: 0031-9007. Journal ArticleAbstract |

239. | The cyclic and epicyclic sites Rend. Semin. Mat. Univ. Padova, 134 , pp. 197–237, 2015, ISSN: 0041-8994. Journal ArticleAbstract |

238. | Universal thickening of the field of real numbers Advances in the theory of numbers, 77 , pp. 11–74, Fields Inst. Res. Math. Sci., Toronto, ON, 2015. IncollectionAbstract |

237. | Projective geometry in characteristic one and the epicyclic category Nagoya Math. J., 217 , pp. 95–132, 2015, ISSN: 0027-7630. Journal ArticleAbstract |

236. | Cyclic structures and the topos of simplicial sets J. Pure Appl. Algebra, 219 (4), pp. 1211–1235, 2015, ISSN: 0022-4049. Journal ArticleAbstract |

235. | Geometry and the quantum: basics J. High Energy Phys., (12), pp. 098, front matter+24, 2014, ISSN: 1126-6708. Journal ArticleAbstract |

234. | C. R. Math. Acad. Sci. Paris, 352 (12), pp. 971–975, 2014, ISSN: 1631-073X. Journal ArticleAbstract |

233. | On the arithmetic of the BC-system J. Noncommut. Geom., 8 (3), pp. 873–945, 2014, ISSN: 1661-6952. Journal ArticleAbstract |

232. | Cyclic homology, Serre's local factors and λ-operations J. K-Theory, 14 (1), pp. 1–45, 2014, ISSN: 1865-2433. Journal ArticleAbstract |

231. | Modular curvature for noncommutative two-tori J. Amer. Math. Soc., 27 (3), pp. 639–684, 2014, ISSN: 0894-0347. Journal ArticleAbstract |

230. | 2013, ISBN: 9782738129833. BookAbstract |

229. | Inner fluctuations in noncommutative geometry without the first order condition J. Geom. Phys., 73 , pp. 222–234, 2013, ISSN: 0393-0440. Journal ArticleAbstract |

228. | On the spectral characterization of manifolds J. Noncommut. Geom., 7 (1), pp. 1–82, 2013, ISSN: 1661-6952. Journal ArticleAbstract |

227. | Resilience of the spectral standard model J. High Energy Phys., (9), pp. 104, front matter+10, 2012, ISSN: 1126-6708. Journal ArticleAbstract |

226. | Spectral action for Robertson-Walker metrics J. High Energy Phys., (10), pp. 101, front matter + 29, 2012, ISSN: 1126-6708. Journal ArticleAbstract |

225. | The Gauss-Bonnet theorem for the noncommutative two torus Noncommutative geometry, arithmetic, and related topics, pp. 141–158, Johns Hopkins Univ. Press, Baltimore, MD, 2011. IncollectionAbstract |

224. | Characteristic 1, entropy and the absolute point Noncommutative geometry, arithmetic, and related topics, pp. 75–139, Johns Hopkins Univ. Press, Baltimore, MD, 2011. IncollectionAbstract |

223. | Jpn. J. Math., 6 (1), pp. 1–44, 2011, ISSN: 0289-2316. Journal ArticleAbstract |

222. | The Witt construction in characteristic one and quantization Noncommutative geometry and global analysis, 546 , pp. 83–113, Amer. Math. Soc., Providence, RI, 2011. IncollectionAbstract |

221. | On the notion of geometry over ?₁ J. Algebraic Geom., 20 (3), pp. 525–557, 2011, ISSN: 1056-3911. Journal ArticleAbstract |

220. | L'hyperanneau des classes d'adèles J. Théor. Nombres Bordeaux, 23 (1), pp. 71–93, 2011, ISSN: 1246-7405. Journal ArticleAbstract |

219. | Noncommutative geometric spaces with boundary: spectral action J. Geom. Phys., 61 (1), pp. 317–332, 2011, ISSN: 0393-0440. Journal ArticleAbstract |

218. | The hyperring of adèle classes J. Number Theory, 131 (2), pp. 159–194, 2011, ISSN: 0022-314X. Journal ArticleAbstract |

217. | From monoids to hyperstructures: in search of an absolute arithmetic Casimir force, Casimir operators and the Riemann hypothesis, pp. 147–198, Walter de Gruyter, Berlin, 2010. IncollectionAbstract |

216. | Schemes over ?₁ and zeta functions Compos. Math., 146 (6), pp. 1383–1415, 2010, ISSN: 0010-437X. Journal ArticleAbstract |

215. | J. Funct. Anal., 259 (11), pp. 2759, 2010, ISSN: 0022-1236. Journal ArticleAbstract |

214. | Fortschr. Phys., 58 (6), pp. 553–600, 2010, ISSN: 0015-8208. Journal ArticleAbstract |

213. | The uncanny precision of the spectral action Comm. Math. Phys., 293 (3), pp. 867–897, 2010, ISSN: 0010-3616. Journal ArticleAbstract |

212. | The Weil proof and the geometry of the adèles class space Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, 269 , pp. 339–405, Birkhäuser Boston, Boston, MA, 2009. IncollectionAbstract |

211. | J. Number Theory, 129 (6), pp. 1532–1561, 2009, ISSN: 0022-314X. Journal ArticleAbstract |

210. | Cambridge University Press, Cambridge, 2008, ISBN: 978-0-521-88926-1. BookAbstract |

209. | A unitary invariant in Riemannian geometry Int. J. Geom. Methods Mod. Phys., 5 (8), pp. 1215–1242, 2008, ISSN: 0219-8878. Journal ArticleAbstract |

208. | On the fine structure of spacetime On space and time, pp. 196–237, Cambridge Univ. Press, Cambridge, 2008. Incollection |

207. | Traces in number theory, geometry and quantum fields, pp. 57–71, Friedr. Vieweg, Wiesbaden, 2008. IncollectionAbstract |

206. | A walk in the noncommutative garden An invitation to noncommutative geometry, pp. 1–128, World Sci. Publ., Hackensack, NJ, 2008. IncollectionAbstract |

205. | Comm. Math. Phys., 281 (1), pp. 23–127, 2008, ISSN: 0010-3616. Journal ArticleAbstract |

204. | J. Geom. Phys., 58 (1), pp. 38–47, 2008, ISSN: 0393-0440. Journal ArticleAbstract |

203. | Noncommutative geometry, quantum fields and motives American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2007, ISBN: 978-0-8218-4210-2. BookAbstract |

202. | Non-commutative geometry and the spectral model of space-time Quantum spaces, 53 , pp. 203–227, Birkhäuser, Basel, 2007. IncollectionAbstract |

201. | Gravity and the standard model with neutrino mixing Adv. Theor. Math. Phys., 11 (6), pp. 991–1089, 2007, ISSN: 1095-0761. Journal ArticleAbstract |

200. | A property of parallelograms inscribed in ellipses Amer. Math. Monthly, 114 (10), pp. 909–914, 2007, ISSN: 0002-9890. Journal ArticleAbstract |

199. | Conceptual explanation for the algebra in the noncommutative approach to the standard model Phys. Rev. Lett., 99 (19), pp. 191601, 4, 2007, ISSN: 0031-9007. Journal ArticleAbstract |

198. | Noncommutative geometry and motives: the thermodynamics of endomotives Adv. Math., 214 (2), pp. 761–831, 2007, ISSN: 0001-8708. Journal ArticleAbstract |

197. | Quantum gravity boundary terms from the spectral action of noncommutative space Phys. Rev. Lett., 99 (7), pp. 071302, 4, 2007, ISSN: 0031-9007. Journal ArticleAbstract |

196. | Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory Frontiers in number theory, physics, and geometry. II, pp. 617–713, Springer, Berlin, 2007. IncollectionAbstract |

195. | Yang-Mills and some related algebras Rigorous quantum field theory, 251 , pp. 65–78, Birkhäuser, Basel, 2007. IncollectionAbstract |

194. | Transgressions of the Godbillon-Vey class and Rademacher functions Noncommutative geometry and number theory, pp. 79–107, Friedr. Vieweg, Wiesbaden, 2006. IncollectionAbstract |

193. | Morse inequalities for foliations ?*-algebras and elliptic theory, pp. 61–72, Birkhäuser, Basel, 2006. |

192. | Noncommutative geometry and the standard model with neutrino mixing J. High Energy Phys., (11), pp. 081, 19, 2006, ISSN: 1126-6708. Journal ArticleAbstract |

191. | Inner fluctuations of the spectral action J. Geom. Phys., 57 (1), pp. 1–21, 2006, ISSN: 0393-0440. Journal ArticleAbstract |

190. | KMS states and complex multiplication (Part II) Operator Algebras: Ŧhe Abel Symposium 2004, 1 , pp. 15–59, Springer, Berlin, 2006. IncollectionAbstract |

189. | From physics to number theory via noncommutative geometry Frontiers in number theory, physics, and geometry. I, pp. 269–347, Springer, Berlin, 2006. IncollectionAbstract |

188. | Scale invariance in the spectral action J. Math. Phys., 47 (6), pp. 063504, 19, 2006, ISSN: 0022-2488. Journal ArticleAbstract |

187. | On the foundations of noncommutative geometry The unity of mathematics, 244 , pp. 173–204, Birkhäuser Boston, Boston, MA, 2006. |

186. | J. Geom. Phys., 56 (1), pp. 55–85, 2006, ISSN: 0393-0440. Journal ArticleAbstract |

185. | ℚ-lattices: quantum statistical mechanics and Galois theory J. Geom. Phys., 56 (1), pp. 2–23, 2006, ISSN: 0393-0440. Journal ArticleAbstract |

184. | KMS states and complex multiplication Selecta Math. (N.S.), 11 (3-4), pp. 325–347, 2005, ISSN: 1022-1824. Journal ArticleAbstract |

183. | ?²-homology for von Neumann algebras J. Reine Angew. Math., 586 , pp. 125–168, 2005, ISSN: 0075-4102. Journal ArticleAbstract |

182. | Butl. Soc. Catalana Mat., 19 (2), pp. 7–23 (2005), 2004, ISSN: 0214-316X. Journal ArticleAbstract |

181. | Renormalisation et ambiguïté galoisienne Asterisque, (296), pp. 113–143, 2004, ISSN: 0303-1179, (Analyse complexe, syst`emes dynamiques, sommabilité des séries divergentes et théories galoisiennes. I). IncollectionAbstract |

180. | Nombres de Betti ?² et facteurs de type II₁ (d'après D. Gaboriau et S. Popa) Astérisque, (294), pp. ix, 321–333, 2004, ISSN: 0303-1179. Journal ArticleAbstract |

179. | Renormalization and motivic Galois theory Int. Math. Res. Not., (76), pp. 4073–4091, 2004, ISSN: 1073-7928. |

178. | Rankin-Cohen brackets and the Hopf algebra of transverse geometry Mosc. Math. J., 4 (1), pp. 111–130, 311, 2004, ISSN: 1609-3321. Journal ArticleAbstract |

177. | Modular Hecke algebras and their Hopf symmetry Mosc. Math. J., 4 (1), pp. 67–109, 310, 2004, ISSN: 1609-3321. Journal ArticleAbstract |

176. | Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2004, ISBN: 3-540-20357-5. BookAbstract |

175. | Cyclic cohomology, noncommutative geometry and quantum group symmetries Noncommutative geometry, 1831 , pp. 1–71, Springer, Berlin, 2004. |

174. | Cyclic cohomology, quantum group symmetries and the local index formula for ???(2) J. Inst. Math. Jussieu, 3 (1), pp. 17–68, 2004, ISSN: 1474-7480. Journal ArticleAbstract |

173. | Symétries galoisiennes et renormalisation Poincaré Seminar 2002, 30 , pp. 241–264, Birkhäuser, Basel, 2003. IncollectionAbstract |

172. | Moduli space and structure of noncommutative 3-spheres Lett. Math. Phys., 66 (1-2), pp. 91–121, 2003, ISSN: 0377-9017. Journal ArticleAbstract |

171. | Hommage à Laurent Schwartz Gaz. Math., (94), pp. 7–8, 2002, ISSN: 0224-8999. Journal Article |

170. | Noncommutative geometry year 2000 Highlights of mathematical physics (London, 2000), pp. 49–110, Amer. Math. Soc., Providence, RI, 2002. Incollection |

169. | Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples Comm. Math. Phys., 230 (3), pp. 539–579, 2002, ISSN: 0010-3616. Journal ArticleAbstract |

168. | Lett. Math. Phys., 61 (2), pp. 149–158, 2002, ISSN: 0377-9017. Journal ArticleAbstract |

167. | Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs Ann. Henri Poincaré, 3 (3), pp. 411–433, 2002, ISSN: 1424-0637. Journal ArticleAbstract |

166. | Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry Essays on geometry and related topics, Vol. 1, 2, 38 , pp. 217–255, Enseignement Math., Geneva, 2001. IncollectionAbstract |

165. | From local perturbation theory to Hopf and Lie algebras of Feynman graphs Mathematical physics in mathematics and physics (Siena, 2000), 30 , pp. 105–114, Amer. Math. Soc., Providence, RI, 2001. |

164. | American Mathematical Society, Providence, RI, 2001, ISBN: 0-8218-2614-X. BookAbstract |

163. | From local perturbation theory to Hopf and Lie algebras of Feynman graphs 56 (1), pp. 3–15, 2001, ISSN: 0377-9017, (EuroConférence Moshé Flato 2000, Part I (Dijon)). |

162. | Noncommutative manifolds, the instanton algebra and isospectral deformations Comm. Math. Phys., 221 (1), pp. 141–159, 2001, ISSN: 0010-3616. Journal ArticleAbstract |

161. | Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The ?-function, diffeomorphisms and the renormalization group Comm. Math. Phys., 216 (1), pp. 215–241, 2001, ISSN: 0010-3616. |

160. | Odile Jacob, Paris, 2000, ISBN: 9780821826140. BookAbstract |

159. | A lecture on noncommutative geometry (Special Issue), pp. 31–64, 2000, ISSN: 1120-6330, (Mathematics towards the third millennium (Rome, 1999)). Incollection |

158. | Noncommutative geometry year 2000 (Special Volume, Part II), pp. 481–559, 2000, ISSN: 1016-443X, (GAFA 2000 (Tel Aviv, 1999)). IncollectionAbstract |

157. | Cyclic cohomology and Hopf algebra symmetry Conférence Moshé Flato 1999, Vol. I (Dijon), 21 , pp. 121–147, Kluwer Acad. Publ., Dordrecht, 2000. Incollection |

156. | Geometric ?-theory for Lie groups and foliations Enseign. Math. (2), 46 (1-2), pp. 3–42, 2000, ISSN: 0013-8584. Journal Article |

155. | A short survey of noncommutative geometry J. Math. Phys., 41 (6), pp. 3832–3866, 2000, ISSN: 0022-2488. |

154. | Noncommutative geometry and the Riemann zeta function Mathematics: frontiers and perspectives, pp. 35–54, Amer. Math. Soc., Providence, RI, 2000. Incollection |

153. | Comm. Math. Phys., 210 (1), pp. 249–273, 2000, ISSN: 0010-3616. Journal ArticleAbstract |

152. | Hopf algebras, renormalization and noncommutative geometry Quantum field theory: perspective and prospective (Les Houches, 1998), 530 , pp. 59–108, Kluwer Acad. Publ., Dordrecht, 1999. IncollectionAbstract |

151. | Renormalization in quantum field theory and the Riemann-Hilbert problem J. High Energy Phys., (9), pp. Paper 24, 8, 1999, ISSN: 1126-6708. |

150. | Cyclic cohomology and Hopf algebras 48 (1), pp. 97–108, 1999, ISSN: 0377-9017, (Moshé Flato (1937--1998)). |

149. | Lessons from quantum field theory: Hopf algebras and spacetime geometries 48 (1), pp. 85–96, 1999, ISSN: 0377-9017, (Moshé Flato (1937--1998)). IncollectionAbstract |

148. | Hypoelliptic operators, Hopf algebras and cyclic cohomology Algebraic ?-theory and its applications (Trieste, 1997), pp. 164–205, World Sci. Publ., River Edge, NJ, 1999. Incollection |

147. | Trace formula on the adèle class space and Weil positivity Current developments in mathematics, 1997 (Cambridge, MA), pp. 5–64, Int. Press, Boston, MA, 1999. Incollection |

146. | Trace formula in noncommutative geometry and the zeros of the Riemann zeta function Selecta Math. (N.S.), 5 (1), pp. 29–106, 1999, ISSN: 1022-1824. Journal ArticleAbstract |

145. | A new proof of Morley's theorem Les relations entre les mathématiques et la physique théorique, pp. 43–46, Inst. Hautes Études Sci., Bures-sur-Yvette, 1998. IncollectionAbstract |

144. | Hopf algebras, renormalization and noncommutative geometry Comm. Math. Phys., 199 (1), pp. 203–242, 1998, ISSN: 0010-3616. |

143. | Hopf algebras, cyclic cohomology and the transverse index theorem Comm. Math. Phys., 198 (1), pp. 199–246, 1998, ISSN: 0010-3616. |

142. | Noncommutative differential geometry and the structure of space-time The geometric universe (Oxford, 1996), pp. 49–80, Oxford Univ. Press, Oxford, 1998. Incollection |

141. | Noncommutative geometry: the spectral aspect Symétries quantiques (Les Ħouches, 1995), pp. 643–686, North-Holland, Amsterdam, 1998. Incollection |

140. | Noncommutative geometry and matrix theory: compactification on tori J. High Energy Phys., (2), pp. Paper 3, 35, 1998, ISSN: 1126-6708. Journal ArticleAbstract |

139. | Brisure de symétrie spontanée et géométrie du point de vue spectral Fields Medallists' lectures, 5 , pp. 340–371, World Sci. Publ., River Edge, NJ, 1997. |

138. | Noncommutative differential geometry and the structure of space time Quantum fields and quantum space time (Cargèse, 1996), 364 , pp. 45–72, Plenum, New York, 1997. Incollection |

137. | Noncommutative differential geometry and the structure of space time Operator algebras and quantum field theory (Rome, 1996), pp. 330–358, Int. Press, Cambridge, MA, 1997. |

136. | Brisure de symétrie spontanée et géométrie du point de vue spectral J. Geom. Phys., 23 (3-4), pp. 206–234, 1997, ISSN: 0393-0440. |

135. | Trace formula in noncommutative geometry and the zeros of the Riemann zeta function Journées "Équations aux Dérivées Partielles" (Saint-Jean-de-Monts, 1997), pp. Exp. No. IV, 28, École Polytech., Palaiseau, 1997. IncollectionAbstract |

134. | Noncommutative differential geometry and the structure of space time Deformation theory and symplectic geometry (Ascona, 1996), 20 , pp. 1–33, Kluwer Acad. Publ., Dordrecht, 1997. Incollection |

133. | Noncommutative differential geometry and the structure of space time Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), 17 , pp. 17–42, Amer. Math. Soc., Providence, RI, 1997. |

132. | Brisure de symétrie spontanée et géométrie du point de vue spectral (241), pp. Exp. No. 816, 5, 313–349, 1997, ISSN: 0303-1179, (Séminaire Bourbaki, Vol. 1995/96). IncollectionAbstract |

131. | Comm. Math. Phys., 186 (3), pp. 731–750, 1997, ISSN: 0010-3616. Journal ArticleAbstract |

130. | Aspherical gravitational monopoles Nuclear Phys. B, 490 (1-2), pp. 391–431, 1997, ISSN: 0550-3213. Journal ArticleAbstract |

129. | Matrix Vieta theorem revisited Lett. Math. Phys., 39 (4), pp. 349–353, 1997, ISSN: 0377-9017. Journal ArticleAbstract |

128. | Gravity coupled with matter and the foundation of non-commutative geometry Comm. Math. Phys., 182 (1), pp. 155–176, 1996, ISSN: 0010-3616. Journal ArticleAbstract |

127. | Formule de trace en géométrie non-commutative et hypothèse de Riemann C. R. Acad. Sci. Paris Sér. I Math., 323 (12), pp. 1231–1236, 1996, ISSN: 0764-4442. Journal Article |

126. | Universal formula for noncommutative geometry actions: unification of gravity and the standard model Phys. Rev. Lett., 77 (24), pp. 4868–4871, 1996, ISSN: 0031-9007. |

125. | Polarized modules and Fredholm modules Mat. Fiz. Anal. Geom., 2 (1), pp. 15–24, 1995, ISSN: 1027-1767. Journal Article |

124. | Non-commutative geometry and physics Gravitation et quantifications (Les Ħouches, 1992), pp. 805–950, North-Holland, Amsterdam, 1995. Incollection |

123. | Round table: physics and mathematics XIth International Congress of Mathematical Physics (Paris, 1994), pp. 691–705, Int. Press, Cambridge, MA, 1995. Incollection |

122. | Quantized calculus and applications XIth International Congress of Mathematical Physics (Paris, 1994), pp. 15–36, Int. Press, Cambridge, MA, 1995. Incollection |

121. | Selecta Math. (N.S.), 1 (3), pp. 411–457, 1995, ISSN: 1022-1824. Journal ArticleAbstract |

120. | Noncommutative geometry and reality J. Math. Phys., 36 (11), pp. 6194–6231, 1995, ISSN: 0022-2488. Journal ArticleAbstract |

119. | Geometry from the spectral point of view Lett. Math. Phys., 34 (3), pp. 203–238, 1995, ISSN: 0377-9017. |

118. | The local index formula in noncommutative geometry Geom. Funct. Anal., 5 (2), pp. 174–243, 1995, ISSN: 1016-443X. Journal ArticleAbstract |

117. | Conversations on mind, matter, and mathematics Princeton University Press, Princeton, NJ, 1995, ISBN: 0-691-08759-8. BookAbstract |

116. | Academic Press, Inc., San Diego, CA, 1994, ISBN: 0-12-185860-X. BookAbstract |

115. | Classical Quantum Gravity, 11 (12), pp. 2899–2917, 1994, ISSN: 0264-9381. Journal ArticleAbstract |

114. | Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes Topology, 33 (4), pp. 663–681, 1994, ISSN: 0040-9383. Journal ArticleAbstract |

113. | Classifying space for proper actions and ?-theory of group ?*-algebras ?*-algebras: 1943--1993 (San Antonio, TX, 1993), 167 , pp. 240–291, Amer. Math. Soc., Providence, RI, 1994. |

112. | Interprétation géométrique du modèle standard de la physique des particules et structure fine de l'espace-temps C. R. Acad. Sci. Paris Sér. Gén. Vie Sci., 10 (3), pp. 223–234, 1993, ISSN: 0762-0969. Journal Article |

111. | Formules locales pour les classes de Pontrjagin topologiques C. R. Acad. Sci. Paris Sér. I Math., 317 (5), pp. 521–526, 1993, ISSN: 0764-4442. Journal Article |

110. | Transgression and the Chern character of finite-dimensional ?-cycles Comm. Math. Phys., 155 (1), pp. 103–122, 1993, ISSN: 0010-3616. |

109. | Editorial (dedication of this issue to Huzihiro Araki) Comm. Math. Phys., 155 (1), pp. 1–2, 1993, ISSN: 0010-3616. |

108. | Group cohomology with Lipschitz control and higher signatures Geom. Funct. Anal., 3 (1), pp. 1–78, 1993, ISSN: 1016-443X. |

107. | Géométrie non commutative et physique quantique Mathématiques quantiques, 1992 , pp. 20, Soc. Math. France, Paris, 1992. Incollection |

106. | Mathématiques quantiques. Géométrie non commutative et physique quantique. Sur l'équation de Schrödinger. Journée annuelle SMF, 23 mai 1992, 1992 , pp. 52, 1992. Journal Article |

105. | Noncommutative geometry In the forest of symbols (Finnish), pp. 244–263, Art House, Helsinki, 1992. Incollection |

104. | Round-table discussion Mathematical research today and tomorrow (Barcelona, 1991), 1525 , pp. 87–108, Springer, Berlin, 1992. |

103. | Noncommutative geometry Mathematical research today and tomorrow (Barcelona, 1991), 1525 , pp. 40–58, Springer, Berlin, 1992. |

102. | The metric aspect of noncommutative geometry New symmetry principles in quantum field theory (Carg`ese, 1991), 295 , pp. 53–93, Plenum, New York, 1992. Incollection |

101. | Produits eulériens et facteurs de type III C. R. Acad. Sci. Paris Sér. I Math., 315 (3), pp. 279–284, 1992, ISSN: 0764-4442. Journal Article |

100. | Closed star products and cyclic cohomology Lett. Math. Phys., 24 (1), pp. 1–12, 1992, ISSN: 0377-9017. |

99. | Sur la nature de la réalité mathématique Elem. Math., 47 (1), pp. 19–26, 1992, ISSN: 0013-6018. Journal Article |

98. | On the Chern character of ? summable Fredholm modules Comm. Math. Phys., 139 (1), pp. 171–181, 1991, ISSN: 0010-3616. |

97. | Caractères des représentations ?-sommables des groupes discrets C. R. Acad. Sci. Paris Sér. I Math., 312 (9), pp. 661–666, 1991, ISSN: 0764-4442. Journal Article |

96. | Particle models and noncommutative geometry Recent advances in field theory (Annecy-le-Vieux, 1990), 18B , pp. 29–47 (1991), 1990, ISSN: 0920-5632. |

95. | Essay on physics and noncommutative geometry The interface of mathematics and particle physics (Oxford, 1988), 24 , pp. 9–48, Oxford Univ. Press, New York, 1990. Incollection |

94. | InterEditions, Paris, 1990, ISBN: 2-7296-0284-4. BookAbstract |

93. | Introduction à la géométrie non-commutative The legacy of John von Neumann (Hempstead, NY, 1988), 50 , pp. 91–118, Amer. Math. Soc., Providence, RI, 1990. |

92. | Cyclic cohomology, the Novikov conjecture and hyperbolic groups Topology, 29 (3), pp. 345–388, 1990, ISSN: 0040-9383. Journal ArticleAbstract |

91. | Déformations, morphismes asymptotiques et ?-théorie bivariante C. R. Acad. Sci. Paris Sér. I Math., 311 (2), pp. 101–106, 1990, ISSN: 0764-4442. Journal Article |

90. | Conjecture de Novikov et fibrés presque plats C. R. Acad. Sci. Paris Sér. I Math., 310 (5), pp. 273–277, 1990, ISSN: 0764-4442. Journal Article |

89. | Odile Jacob, Paris, 1989, ISBN: 9782738100733. BookAbstract |

88. | Compact metric spaces, Fredholm modules, and hyperfiniteness Ergodic Theory Dynam. Systems, 9 (2), pp. 207–220, 1989, ISSN: 0143-3857. |

87. | Embedding of U(1)-current algebras in noncommutative algebras of classical statistical mechanics Comm. Math. Phys., 121 (3), pp. 507–525, 1989, ISSN: 0010-3616. |

86. | Hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks Pacific J. Math., 137 (2), pp. 225–243, 1989, ISSN: 0030-8730. |

85. | Noncommutative geometry Nonperturbative quantum field theory (Carg`ese, 1987), 185 , pp. 33–69, Plenum, New York, 1988. Incollection |

84. | Trace de Dixmier, modules de Fredholm et géométrie riemannienne 5B , pp. 65–70, 1988, ISSN: 0920-5632, (Conformal field theories and related topics (Annecy-le-Vieux, 1988)). |

83. | ?-theory for discrete groups Operator algebras and applications, Vol. 1, 135 , pp. 1–20, Cambridge Univ. Press, Cambridge, 1988. |

82. | Caractere multiplicatif d'un module de Fredholm ?-Theory, 2 (3), pp. 431–463, 1988, ISSN: 0920-3036. Journal ArticleAbstract |

81. | Conjecture de Novikov et groupes hyperboliques C. R. Acad. Sci. Paris Sér. I Math., 307 (9), pp. 475–480, 1988, ISSN: 0249-6291. Journal Article |

80. | Cyclic cohomology and noncommutative differential geometry Géométrie différentielle (Paris, 1986), 33 , pp. 33–50, Hermann, Paris, 1988. Incollection |

79. | Entire cyclic cohomology of Banach algebras and characters of ?-summable Fredholm modules ?-Theory, 1 (6), pp. 519–548, 1988, ISSN: 0920-3036. |

78. | The action functional in noncommutative geometry Comm. Math. Phys., 117 (4), pp. 673–683, 1988, ISSN: 0010-3616. |

77. | Quasi homomorphismes, cohomologie cyclique et positivité Comm. Math. Phys., 114 (3), pp. 515–526, 1988, ISSN: 0010-3616. |

76. | Chern character for discrete groups A fête of topology, pp. 163–232, Academic Press, Boston, MA, 1988. |

75. | Cyclic cohomology and noncommutative differential geometry Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pp. 879–889, Amer. Math. Soc., Providence, RI, 1987. |

74. | Dynamical Entropy of ?* Algebras and von Neumann Algebras Comm. Math. Phys., 112 (4), pp. 691–719, 1987, ISSN: 0010-3616. Journal ArticleAbstract |

73. | Yang-Mills for noncommutative two-tori Operator algebras and mathematical physics (Iowa City, Iowa, 1985), 62 , pp. 237–266, Amer. Math. Soc., Providence, RI, 1987. |

72. | Transgression du caractère de Chern et cohomologie cyclique C. R. Acad. Sci. Paris Sér. I Math., 303 (18), pp. 913–918, 1986, ISSN: 0249-6291. Journal Article |

71. | Cyclic cohomology and the transverse fundamental class of a foliation Geometric methods in operator algebras (Kyoto, 1983), 123 , pp. 52–144, Longman Sci. Tech., Harlow, 1986. IncollectionAbstract |

70. | Indice des sous facteurs, algèbres de Hecke et théorie des nœuds (d'après Vaughan Jones) (133-134), pp. 289–308, 1986, ISSN: 0303-1179, (Seminar Bourbaki, Vol. 1984/85). Incollection |

69. | Leafwise homotopy equivalence and rational Pontrjagin classes Foliations (Tokyo, 1983), 5 , pp. 1–14, North-Holland, Amsterdam, 1985. |

68. | Noncommutative differential geometry Inst. Hautes Études Sci. Publ. Math., (62), pp. 257–360, 1985, ISSN: 0073-8301. Journal ArticleAbstract |

67. | Diameters of state spaces of type III factors Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), 1132 , pp. 91–116, Springer, Berlin, 1985. |

66. | Entropie de Kolmogoroff-Sinai et mécanique statistique quantique C. R. Acad. Sci. Paris Sér. I Math., 301 (1), pp. 1–6, 1985, ISSN: 0249-6291. Journal Article |

65. | Introduction to noncommutative differential geometry Workshop Bonn 1984 (Bonn, 1984), 1111 , pp. 3–16, Springer, Berlin, 1985. |

64. | Approximately transitive flows and ITPFI factors Ergodic Theory Dynam. Systems, 5 (2), pp. 203–236, 1985, ISSN: 0143-3857. |

63. | Factors of type III₁, property ?' J. Operator Theory, 14 (1), pp. 189–211, 1985, ISSN: 0379-4024. Journal Article |

62. | Property T for von Neumann algebras Bull. London Math. Soc., 17 (1), pp. 57–62, 1985, ISSN: 0024-6093. |

61. | The longitudinal index theorem for foliations Publ. Res. Inst. Math. Sci., 20 (6), pp. 1139–1183, 1984, ISSN: 0034-5318. Journal ArticleAbstract |

60. | Caractère multiplicatif d'un module de Fredholm C. R. Acad. Sci. Paris Sér. I Math., 299 (19), pp. 963–968, 1984, ISSN: 0249-6291. Journal Article |

59. | A connection between the classical and the quantum mechanical entropies Operator algebras and group representations, Vol. I (Neptun, 1980), 17 , pp. 113–123, Pitman, Boston, MA, 1984. Incollection |

58. | Cohomologie cyclique et foncteurs Ext C. R. Acad. Sci. Paris Sér. I Math., 296 (23), pp. 953–958, 1983, ISSN: 0249-6291. Journal ArticleAbstract |

57. | A survey of foliations and operator algebras Operator algebras and applications, Part I (Kingston, Ont., 1980), 38 , pp. 521–628, Amer. Math. Soc., Providence, R.I., 1982. IncollectionAbstract |

56. | ?²-index theory on homogeneous spaces and discrete series representations Operator algebras and applications, Part I (Kingston, Ont., 1980), 38 , pp. 419–433, Amer. Math. Soc., Providence, R.I., 1982. |

55. | Classification des facteurs Operator algebras and applications, Part 2 (Kingston, Ont., 1980), 38 , pp. 43–109, Amer. Math. Soc., Providence, R.I., 1982. Incollection |

54. | The ?²-index theorem for homogeneous spaces of Lie groups Ann. of Math. (2), 115 (2), pp. 291–330, 1982, ISSN: 0003-486X. |

53. | A II₁ factor with two nonconjugate Cartan subalgebras Bull. Amer. Math. Soc. (N.S.), 6 (2), pp. 211–212, 1982, ISSN: 0273-0979. |

52. | An amenable equivalence relation is generated by a single transformation Ergodic Theory Dynam. Systems, 1 (4), pp. 431–450 (1982), 1981, ISSN: 0143-3857. |

51. | Feuilletages et algèbres d'opérateurs Bourbaki Seminar, Vol. 1979/80, 842 , pp. 139–155, Springer, Berlin-New York, 1981. Incollection |

50. | Théorème de l'indice pour les feuilletages C. R. Acad. Sci. Paris Sér. I Math., 292 (18), pp. 871–876, 1981, ISSN: 0249-6291. Journal Article |

49. | An analogue of the Thom isomorphism for crossed products of a ?*-algebra by an action of ℝ Adv. in Math., 39 (1), pp. 31–55, 1981, ISSN: 0001-8708. |

48. | Property T and asymptotically invariant sequences Israel J. Math., 37 (3), pp. 209–210, 1980, ISSN: 0021-2172. |

47. | A factor of type II₁ with countable fundamental group J. Operator Theory, 4 (1), pp. 151–153, 1980, ISSN: 0379-4024. Journal Article |

46. | A construction of approximately finite-dimensional non-ITPFI factors Canad. Math. Bull., 23 (2), pp. 227–230, 1980, ISSN: 0008-4395. |

45. | ?*-algèbres et géométrie différentielle C. R. Acad. Sci. Paris Sér. A-B, 290 (13), pp. A599–A604, 1980, ISSN: 0151-0509. Journal ArticleAbstract |

44. | Von Neumann algebras Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 97–109, Acad. Sci. Fennica, Helsinki, 1980. Inproceedings |

43. | On the spatial theory of von Neumann algebras J. Functional Analysis, 35 (2), pp. 153–164, 1980, ISSN: 0022-1236. |

42. | On the equivalence between injectivity and semidiscreteness for operator algebras Algèbres d'opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), pp. 107–112, CNRS, Paris, 1979. Inproceedings |

41. | Sur la théorie non commutative de l'intégration Algèbres d'opérateurs (Sém., Les Plans-sur-Bex, 1978), 725 , pp. 19–143, Springer, Berlin, 1979. IncollectionAbstract |

40. | The ?²-index theorem for homogeneous spaces Bull. Amer. Math. Soc. (N.S.), 1 (4), pp. 688–690, 1979, ISSN: 0273-0979. |

39. | The von Neumann algebra of a foliation Mathematical problems in theoretical physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977), pp. 145–151, Springer, Berlin-New York, 1978. Inproceedings |

38. | Errata: "The flow of weights on factors of type III" Tohoku Math. J. (2), 30 (4), pp. 653–655, 1978, ISSN: 0040-8735. |

37. | On the cohomology of operator algebras J. Functional Analysis, 28 (2), pp. 248–253, 1978. |

36. | Homogeneity of the state space of factors of type III₁ J. Functional Analysis, 28 (2), pp. 187–196, 1978. |

35. | The flow of weights on factors of type III Tohoku Math. J. (2), 29 (4), pp. 473–575, 1977, ISSN: 0040-8735. |

34. | Periodic automorphisms of the hyperfinite factor of type II₁ Acta Sci. Math. (Szeged), 39 (1-2), pp. 39–66, 1977, ISSN: 0001-6969. Journal ArticleAbstract |

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