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Dynamical systems on spectral metric spaces, preprint
"... Abstract. Let (A,H, D) be a spectral triple, namely: A is a separable C∗algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative an ..."
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Abstract. Let (A,H, D) be a spectral triple, namely: A is a separable C∗algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H, D) with additional properties which guarantee that the Connes metric induces the weak∗topology on the state space of A. A “quasiisometric ” ∗automorphism defines a dynamical system. This article gives various answers to the question: Is there a canonical spectral triple based upon the crossed product algebra Aoα Z, characterizing the metric properties of the dynamical system? If α is equivalent to the noncommutative analog of an isometry the answer is yes. Otherwise, the metric bundle construction of Connes and Moscovici is used to replace (A, α) by an equivalent dynamical system that acts isometrically. The difficulties related to the noncompactness of this new system are discussed. Applications, in number theory and coding theory are given at the
Quantum statistical mechanics, Lseries and anabelian geometry, in preparation
"... Abstract. It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C ∗algebra with a one parameter group ..."
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Cited by 12 (8 self)
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Abstract. It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C ∗algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of these associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendieck’s “anabelian ” program, much like the NeukirchUchida theorem characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups. In the second part of the paper, we use these systems to prove the following. If there is a continuous bijection ψ: ˇ G ab K → ˇ G ab L between the character groups (viz., Pontrjagin duals) of the abelianized Galois groups of the two number fields that induces an equality of all corresponding L
NONCOMMUTATIVE MIXMASTER COSMOLOGIES
, 1203
"... Abstract. In this paper we investigate a variant of the classical mixmaster universe model of anisotropic cosmology, where the spatial sections are noncommutative 3tori. We consider ways in which the discrete dynamical system describing the mixmaster dynamics can be extended to act on the noncommut ..."
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Cited by 1 (1 self)
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Abstract. In this paper we investigate a variant of the classical mixmaster universe model of anisotropic cosmology, where the spatial sections are noncommutative 3tori. We consider ways in which the discrete dynamical system describing the mixmaster dynamics can be extended to act on the noncommutative torus moduli, and how the resulting dynamics differs from the classical one, for example, in the appearance of exotic smooth structures. We discuss properties of the spectral action, focussing on how the slowroll inflation potential determined by the spectral action affects the mixmaster dynamics. We relate the model to other recent results on spectral action computation and we identify other physical contexts in which this model may be relevant.
Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology?
, 2014
"... Abstract. We introduce some algebraic geometric models in cosmology related to the “boundaries ” of spacetime: Big Bang, Mixmaster Universe, Penrose’s crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point x. This crea ..."
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Abstract. We introduce some algebraic geometric models in cosmology related to the “boundaries ” of spacetime: Big Bang, Mixmaster Universe, Penrose’s crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point x. This creates a boundary which consists of the projective space of tangent directions to x and possibly of the light cone of x. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose’s idea to see the Big Bang as a sign of crossover from “the end of previous aeon ” of the expanding and cooling Universe to the “beginning of the next aeon ” is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary.
NONCOMMUTATIVE GEOMETRY AND ARITHMETIC
"... Abstract. This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with complex multiplication for imaginary quadratic fields. ..."
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Abstract. This is an overview of recent results aimed at developing a geometry of noncommutative tori with real multiplication, with the purpose of providing a parallel, for real quadratic fields, of the classical theory of elliptic curves with complex multiplication for imaginary quadratic fields. This talk concentrates on two main aspects: the relation of Stark numbers to the geometry of noncommutative tori with real multiplication, and the shadows of modular forms on the noncommutative boundary of modular curves, that is, the moduli space of noncommutative tori. 1.
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, 2008
"... We give an overview of the Integer Quantum Hall Effect. We propose a mathematical framework using NonCommutative Geometry as defined by A. Connes. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localize ..."
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We give an overview of the Integer Quantum Hall Effect. We propose a mathematical framework using NonCommutative Geometry as defined by A. Connes. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.