# An introduction to NCG

8 septembre 2021 | Conférences

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8 septembre 2021 | Conférences

- 29 septembre 2021
- 27 septembre 2021
- 8 septembre 2021

Prolate spheroidal operator and Zeta

Alain Connes, Henri Moscovici

À paraître.

@article{Connes2021bb,

title = {Prolate spheroidal operator and Zeta},

author = {Alain Connes and Henri Moscovici },

url = {https://alainconnes.org/wp-content/uploads/draft4.pdf},

year = {2021},

date = {2021-12-13},

urldate = {2021-12-13},

abstract = {In this paper we describe a remarkable new property of the self-adjoint extension of the prolate spheroidal operator. The restriction of this operator to the interval J whose characteristic function commutes with it is well known, has discrete positive spectrum and is well understood. What we have discovered is that the restriction of W to the complement of J admits (besides a replica of the above positive spectrum) negative eigenvalues whose ultraviolet behavior reproduce that of the squares of zeros of the Riemann zeta function. Furthermore, their corresponding eigenfunctions belong to the Sonin space. As a byproduct we construct an isospectral family of Dirac operators whose spectra have the same ultraviolet behavior as the zeros of the Riemann zeta function.},

keywords = {},

pubstate = {forthcoming},

tppubtype = {article}

}

In this paper we describe a remarkable new property of the self-adjoint extension of the prolate spheroidal operator. The restriction of this operator to the interval J whose characteristic function commutes with it is well known, has discrete positive spectrum and is well understood. What we have discovered is that the restriction of W to the complement of J admits (besides a replica of the above positive spectrum) negative eigenvalues whose ultraviolet behavior reproduce that of the squares of zeros of the Riemann zeta function. Furthermore, their corresponding eigenfunctions belong to the Sonin space. As a byproduct we construct an isospectral family of Dirac operators whose spectra have the same ultraviolet behavior as the zeros of the Riemann zeta function.

Tolerance relations and operator systems

Alain Connes, Walter D. van Suijlekom

À paraître.

@article{Connes2021bb,

title = {Tolerance relations and operator systems},

author = {Alain Connes and Walter D. van Suijlekom },

url = {https://alainconnes.org/wp-content/uploads/2111.02903-2.pdf},

year = {2021},

date = {2021-12-01},

abstract = {We extend the scope of noncommutative geometry by generalizing the construction of the noncommutative algebra of a quotient space to situations in which one is no longer dealing with an equivalence relation. For these so-called tolerance relations, passing to the associated equivalence relation looses crucial information as is clear from the examples such as coarse graining in physics or the relation d(x,y)<ε on a metric space. Fortunately, thanks to the formalism of operator systems such an extension is possible and provides new invariants, such as the C*-envelope and the propagation number. After a thorough investigation of the structure of the (non-unital) operator systems associated to tolerance relations, we analyze the corresponding state spaces. In particular, we determine the pure state space associated to the operator system for the relation d(x,y)<ε on a path metric measure space.},

keywords = {},

pubstate = {forthcoming},

tppubtype = {article}

}

We extend the scope of noncommutative geometry by generalizing the construction of the noncommutative algebra of a quotient space to situations in which one is no longer dealing with an equivalence relation. For these so-called tolerance relations, passing to the associated equivalence relation looses crucial information as is clear from the examples such as coarse graining in physics or the relation d(x,y)<ε on a metric space. Fortunately, thanks to the formalism of operator systems such an extension is possible and provides new invariants, such as the C*-envelope and the propagation number. After a thorough investigation of the structure of the (non-unital) operator systems associated to tolerance relations, we analyze the corresponding state spaces. In particular, we determine the pure state space associated to the operator system for the relation d(x,y)<ε on a path metric measure space.

Spectral Triples and Zeta-Cycles

Alain Connes, Caterina Consani

Enseignement Mathématique, À paraître.

@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.