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11 janvier 1985 | Cours

- 11 février 2024
- 29 septembre 2021
- 27 septembre 2021

Noncommutativity and physics: a non-technical review

Ali Chamseddine, Alain Connes, Walter van Suijlekom

Eur. Phys. J. Spec. Top., vol. 232, p. 3581–3588, 2023.

@article{Chamseddine2023,

title = {Noncommutativity and physics: a non-technical review},

author = {Ali Chamseddine and Alain Connes and Walter van Suijlekom},

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

year = {2023},

date = {2023-09-30},

urldate = {2023-09-30},

journal = {Eur. Phys. J. Spec. Top.},

volume = {232},

pages = {3581--3588},

abstract = {Abstract We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the coexistence of discrete and continuous variables. The spectral approach to geometry is then explained to encompass two natural ingredients: the line element and the algebra. The relation between these two is dictated by so-called higher Heisenberg relations, from which both spin geometry and non-abelian gauge theory emerges. Our exposition indicates some of the applications in physics, including Pati-Salam unification beyond the Standard Model, the criticality of dimension 4 , second quantization and entropy.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

Abstract We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the coexistence of discrete and continuous variables. The spectral approach to geometry is then explained to encompass two natural ingredients: the line element and the algebra. The relation between these two is dictated by so-called higher Heisenberg relations, from which both spin geometry and non-abelian gauge theory emerges. Our exposition indicates some of the applications in physics, including Pati-Salam unification beyond the Standard Model, the criticality of dimension 4 , second quantization and entropy.

Alain Connes, Caterina Consani

Bull. Sci. Math. , vol. 187, 2023.

@article{Connes2023b,

title = {Riemann-Roch for Spec Z},

author = {Alain Connes and Caterina Consani},

url = {https://alainconnes.org/wp-content/uploads/Riemann-Roch-for-Spec-Z_2023.pdf},

year = {2023},

date = {2023-08-01},

urldate = {2023-08-01},

journal = {Bull. Sci. Math. },

volume = {187},

abstract = {We prove a Riemann-Roch theorem of an entirely novel nature for divisors on the Arakelov compactification of the algebraic spectrum of the integers. This result relies on the introduction of three key concepts: the cohomologies (attached to a divisor), their integer dimension, and Serre duality. These notions directly extend their classical counterparts for function fields. The Riemann-Roch formula equates the (integer valued) Euler characteristic of a divisor with a slight modification of the traditional expression in terms of the sum of the degree of the divisor and the logarithm of 2 . Both the definitions of the cohomologies and of their dimensions rely on a universal arithmetic theory over the sphere spectrum that we had previously introduced using Segal's Gamma rings. By adopting this new perspective we can parallel Weil's adelic proof of the Riemann-Roch formula for function fields including the use of Pontryagin duality.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

We prove a Riemann-Roch theorem of an entirely novel nature for divisors on the Arakelov compactification of the algebraic spectrum of the integers. This result relies on the introduction of three key concepts: the cohomologies (attached to a divisor), their integer dimension, and Serre duality. These notions directly extend their classical counterparts for function fields. The Riemann-Roch formula equates the (integer valued) Euler characteristic of a divisor with a slight modification of the traditional expression in terms of the sum of the degree of the divisor and the logarithm of 2 . Both the definitions of the cohomologies and of their dimensions rely on a universal arithmetic theory over the sphere spectrum that we had previously introduced using Segal's Gamma rings. By adopting this new perspective we can parallel Weil's adelic proof of the Riemann-Roch formula for function fields including the use of Pontryagin duality.

Hochschild homology, trace map and zeta-cycles

Alain Connes, Caterina Consani

Proc. Sympos. Pure Math , vol. 105, p. 83–101, 2023.

@article{Connes2023c,

title = { Hochschild homology, trace map and zeta-cycles},

author = {Alain Connes and Caterina Consani},

editor = { },

url = {https://alainconnes.org/wp-content/uploads/Hochschild-homology-trace-map-and-ζ-cycles-2023.pdf},

year = {2023},

date = {2023-04-11},

urldate = {2023-04-11},

journal = {Proc. Sympos. Pure Math },

volume = {105},

pages = {83--101},

abstract = {ABSTRACT. In this paper we consider two spectral realizations of the zeros of the Riemann zeta function. The first one involves all non-trivial (i.e. non-real) zeros and is expressed in terms of a Laplacian intimately related to the prolate wave operator. The second spectral realization affects only the critical zeros and it is cast in terms of sheaf cohomology. The novelty is that the base space is the Scaling Site playing the role of the parameter space for the $zeta$-cycles and encoding their stability by coverings.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

ABSTRACT. In this paper we consider two spectral realizations of the zeros of the Riemann zeta function. The first one involves all non-trivial (i.e. non-real) zeros and is expressed in terms of a Laplacian intimately related to the prolate wave operator. The second spectral realization affects only the critical zeros and it is cast in terms of sheaf cohomology. The novelty is that the base space is the Scaling Site playing the role of the parameter space for the $zeta$-cycles and encoding their stability by coverings.