# Noncommutative Geometry, The spectral aspect (Harvard CMSA Math Science Literature)

23 November 2020 | Lectures

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23 November 2020 | Lectures

- September 29th, 2021
- September 27th, 2021
- September 8th, 2021

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. |

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. |

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. |