# Parcours d’un Mathématicien (Savoirs Pour Tous)

December 25th, 2017 | Lectures

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December 25th, 2017 | Lectures

- November 23rd, 2020
- February 27th, 2019
- December 22nd, 2018

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. |

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. |

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. |