arXiv subjects

Alain Connes

Publications and source records attributed to Alain Connes.

At least 37 records · Page 2Linked to original sources

arXiv2021

BC-system, absolute cyclotomy and the quantized calculus

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the "zeta sector" of the latter space as the Scaling Site. The new result that we present concerns the description of the BC-system as the universal Witt ring (i.e. K-theory of endomorphisms) of the "algebraic closure" of the absolute base S. In this way we attain a conceptual meaning of the BC dynamical system at the most basic algebraic level. Furthermore, we define an invariant of Schwartz kernels in 1 dimension and relate the Fourier transform (in 1 dimension) to its role over the algebraic closure of S. We implement this invariant to prove that, when applied to the quantized differential of a function, it provides its Schwarzian derivative. Finally, we survey the roles of the quantized calculus in relation to Weil's positivity, and that of spectral triples in relation to the zeros of the Riemann zeta function.

math.NT↗

arXiv2021

Prolate spheroidal operator and Zeta

In this paper we describe a remarkable new property of the self-adjoint extension W of the prolate spheroidal operator introduced in \cite{college98},\cite{CMbook}. 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. This feature fits with the proof \cite{weilpos} of Weil's positivity at the archimedean place, which uses the compression of the scaling action 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.

math.NT↗

arXiv2021

Tolerance relations and operator systems

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 $d(x,y)< \varepsilon$ 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 $d(x,y)< \varepsilon$ on a path metric measure space.

math.OA↗

arXiv2021

Spectral Triples and Zeta-Cycles

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.

math.NT↗

arXiv2020

Motivic Rhythms

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 $H^1$ of the reduction modulo a prime $p$ of an hyperelliptic curve (defined over $\mathbb Q$). 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.

math.NT↗

arXiv2020

Quasi-inner functions and local factors

We introduce the notion of {\it quasi-inner} function and show that the product $u=ρ_\infty\prod ρ_v$ of $m+1$ ratios of local {$L$-}factors {$ρ_v(z)=γ_v(z)/γ_v(1-z)$} 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 $\Re(z)= \frac 12$ in the following sense. The off diagonal part $u_{21}$ of the matrix of the multiplication by $u$ in the orthogonal decomposition of the Hilbert space $L^2$ of square integrable functions on the critical line into the Hardy space $H^2$ 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 $ρ_v$ is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio $ρ_\infty$ into a product of $m$ quasi-inner functions whose product with each $ρ_v$ retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part $u_{22}$ for the quasi-inner function $u=ρ_\infty$, and when $u(F)=\prod_{v\in F} ρ_v$ 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.

math.NT↗

arXiv2020

Weil positivity and Trace formula, the archimedean place

We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula of the paper "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function". (Selecta Math. 5 (1999), no. 1, 29--106). We explore in great details the simplest case of the single archimedean place. The root of the positivity is 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 cutoff parameter equal to 1. We express the difference between the Weil distribution and the Sonin trace (coming from 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 the difference. All the ingredients and tools used above make sense in the general semi-local case, where Weil positivity implies RH.

math.NT↗

arXiv2020

Spectral truncations in noncommutative geometry and operator systems

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.

math.QA↗

arXiv2020

Segal's Gamma rings and universal arithmetic

Segal's Gamma-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 of Almkvist for the simplest Gamma-ring S. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between Lambda-rings and the Arithmetic site. Then, we concentrate on the Arakelov compactification of Z which acquires a structure sheaf of S-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D, we show how to associate to D a Gamma-space that encodes, in homotopical terms, the Riemann-Roch problem for D.

math.AG↗

arXiv2019

The Scaling Hamiltonian

We first explain the link between the Berry-Keating Hamiltonian and the spectral realization of zeros of the Riemann zeta function of the first author, and why there is no conflict at the semi-classical level between the "absorption" picture of A. Connes and the semiclassical "emission" computations of M. Berry and J. Keating, 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.

math.NT↗

arXiv2019

Noncommutative Geometry, the spectral standpoint

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.

math.QA↗

arXiv2019

Sir Michael Atiyah, a Knight Mathematician A tribute to Michael Atiyah, an inspiration and a friend

Sir Michael Atiyah was considered as one of the world's foremost mathematicians, He is best known for his work in algebraic topology and the co-development of a branch of mathematics called topological K-theory together with the Atiyah-Singer index theorem for which he received Fields Medal (1966). He received also 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 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.

math.HO↗

arXiv2019

On Absolute Algebraic Geometry, the affine case

We develop algebraic geometry for general Segal's Gamma-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under Spec Z). The starting observation is that the category obtained by gluing together the category of commutative rings and that of pointed commutative monoids, that we used in our previous work to define F1-schemes, is naturally a full subcategory of the category of Segal's Gamma-rings. In this paper we develop the affine case of this general algebraic geometry: one distinctive feature is that the spectrum Spec(A) of a Gamma-ring is in general a Grothendieck site rather than a point set endowed with a topology. Two striking features of this new geometry are that it is the natural domain for cyclic homology and for homological algebra, and that new operations, which do not make sense in ordinary algebraic geometry, are here available. For instance, in this new context, the quotient of a ring by a multiplicative subgroup is still a Gamma-ring to which our general theory applies. Thus the adele class space gives rise naturally to a Gamma-ring. Finally, we show that our theory is not a special case of the Töen-Vaqui\' e general theory of algebraic geometry under Spec Z.

math.AG↗

arXiv2019

$\overline{Spec\mathbb Z}$ and the Gromov norm

We define the homology of a simplicial set with coefficients in a Segal's $Γ$-set ($\mathbf S$-module). We show the relevance of this new homology with values in $\mathbf S$-modules by proving that taking as coefficients the $\mathbf S$-modules at the archimedean place over the structure sheaf on $\overline{Spec\mathbb Z}$ introduced in our previous work, 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.

math.AG↗

arXiv2019

On an idea of Michael Atiyah

In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit-Thompson on the solvability of finite groups of odd order, the iterations of the transformation which replaces a representation of a finite group G on a finite dimensional complex vector space E by the difference between the associated representation of G on the sum of exterior powers of E and the trivial representation. We show that G has odd order if and only if the above operation extends to virtual representations and we then express it in terms of the exponential and the Adams operations in the complexified representation ring. We show the relevance of the idea in a concrete example by exhibiting convergence to a non-trivial character.

math.QA↗

arXiv2019

Noncommutative Geometry for Symmetric Non-Self-Adjoint Operators

We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple $(\mathcal{A}, H, D)$ 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 geometry 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 open subsets of $\mathbb{R}^d$, $d \geq 2$.

math.OA↗

arXiv2018

Entropy and the spectral action

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.

hep-th↗

arXiv2018

Around Wilson's theorem

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.

math.NT↗