In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}} {{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}} {{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.

Ce mémoire a deux objectifs principaux. Premièrement de développer et interpréter les groupes de cohomologie de Hochschild de basse dimension et deuxièmement de borner la dimension cohomologique des k-algèbres par dessous; montrant que presque aucune k-algèbre commutative est quasi-libre. / The aim of this master’s thesis is two-fold. Firstly to develop and interpret the low dimensional Hochschild cohomology of a k-algebra and secondly to establish a lower bound for the Hochschild cohomological dimension of a k-algebra; showing that nearly no commutative k-algebra is quasi-free.

Diese Arbeit befasst sich mit Quantenfeldtheorien auf nicht-kommutativen Räumen. Solche Modelle treten im Zusammenhang mit der Stringtheorie und mit der Quantengravitation auf. Ihre nicht-störungstheoretische Behandlung ist üblicherweise schwierig. Hier untersuchen wir jedoch drei nicht-kommutative Quantenfeldtheorien nicht-perturbativ, indem wir die Wirkungsfunktionale in eine äquivalente Matrixformulierung übersetzen. In der Matrixdarstellung kann die jeweilige Theorie dann numerisch behandelt werden. Als erstes betrachten wir ein regularisiertes skalares Modell auf der nicht-kommutativen Ebene und untersuchen den Kontinuumslimes bei festgehaltener Nicht-Kommutativität. Dies wird auch als Doppelskalierungslimes bezeichnet. Insbesondere untersuchen wir das Verhalten der gestreiften Phase. Wir finden keinerlei Hinweise auf die Existenz dieser Phase im Doppelskalierungslimes. Im Anschluss daran betrachten wir eine vier-dimensionale U(1) Eichtheorie. Hierbei sind zwei der räumlichen Richtungen nicht-kommutativ. Wir untersuchen sowohl die Phasenstruktur als auch den Doppelskalierungslimes. Es stellt sich heraus, dass neben den Phasen starker und schwacher Kopplung eine weitere Phase existiert, die gebrochene Phase. Dann bestätigen wir die Existenz eines endlichen Doppelskalierungslimes, und damit die Renormierbarkeit der Theorie. Weiterhin untersuchen wir die Dispersionsrelation des Photons. In der Phase mit schwacher Kopplung stimmen unsere Ergebnisse mit störungstheoretischen Berechnungen überein, die eine Infrarot-Instabilität vorhersagen. Andererseits finden wir in der gebrochenen Phase die Dispersionsrelation, die einem masselosen Teilchen entspricht. Als dritte Theorie betrachten wir ein einfaches, in seiner Kontinuumsform supersymmetrisches Modell, welches auf der "Fuzzy Sphere" formuliert wird. Hier wechselwirken neutrale skalare Bosonen mit Majorana-Fermionen. Wir untersuchen die Phasenstruktur dieses Modells, wobei wir drei unterschiedliche Phasen finden. / This work deals with three quantum field theories on spaces with noncommuting position operators. Noncommutative models occur in the study of string theories and quantum gravity. They usually elude treatment beyond the perturbative level. Due to the technique of dimensional reduction, however, we are able to investigate these theories nonperturbatively. This entails translating the action functionals into a matrix language, which is suitable for numerical simulations. First we explore a scalar model on a noncommutative plane. We investigate the continuum limit at fixed noncommutativity, which is known as the double scaling limit. Here we focus especially on the fate of the striped phase, a phase peculiar to the noncommutative version of the regularized scalar model. We find no evidence for its existence in the double scaling limit. Next we examine the U(1) gauge theory on a four-dimensional spacetime, where two spatial directions are noncommutative. We examine the phase structure and find a new phase with a spontaneously broken translation symmetry. In addition we demonstrate the existence of a finite double scaling limit which confirms the renormalizability of the theory. Furthermore we investigate the dispersion relation of the photon. In the weak coupling phase our results are consistent with an infrared instability predicted by perturbation theory. If the translational symmetry is broken, however, we find a dispersion relation corresponding to a massless particle. Finally, we investigate a supersymmetric theory on the fuzzy sphere, which features scalar neutral bosons and Majorana fermions. The supersymmetry is exact in the limit of infinitely large matrices. We investigate the phase structure of the model and find three distinct phases. Summarizing, we study noncommutative field theories beyond perturbation theory. Moreover, we simulate a supersymmetric theory on the fuzzy sphere, which might provide an alternative to attempted lattice formulations.

Dado um C$^*$-sistema dinâmico $(A, G, \\alpha)$ define-se um homomorfismo, denominado de caráter de Chern-Connes, que leva elementos de $K_0(A) \\oplus K_1(A)$, grupos de K-teoria da C$^*$-álgebra $A$, em $H_{\\mathbb}^*(G)$, anel da cohomologia real de deRham do grupo de Lie $G$. Utilizando essa definição, nós calculamos explicitamente esse homomorfismo para os exemplos $(\\overline{\\Psi_^0(S^1)}, S^1, \\alpha)$ e $(\\overline{\\Psi_^0(S^2)}, SO(3), \\alpha)$, onde $\\overline{\\Psi_^0(M)}$ denota a C$^*$-álgebra gerada pelos operadores pseudodiferenciais clássicos de ordem zero da variedade $M$ e $\\alpha$ a ação de conjugação pela representação regular (translações). / Given a C$^*$-dynamical system $(A, G, \\alpha)$ one defines a homomorphism, called the Chern-Connes character, that take an element in $K_0(A) \\oplus K_1(A)$, the K-theory groups of the C$^*$-algebra $A$, and maps it into $H_{\\mathbb}^*(G)$, the real deRham cohomology ring of $G$. We explictly compute this homomorphism for the examples $(\\overline{\\Psi_^0(S^1)}, S^1, \\alpha)$ and $(\\overline{\\Psi_^0(S^2)}, SO(3), \\alpha)$, where $\\overline{\\Psi_^0(M)}$ denotes the C$^*$-álgebra gene\\-rated by the classical pseudodifferential operators of zero order in the manifold $M$ and $\\alpha$ the action of conjugation by the regular representation (translations).

Dado um C$^*$-sistema dinâmico $(A, G, \\alpha)$ define-se um homomorfismo, denominado de caráter de Chern-Connes, que leva elementos de $K_0(A) \\oplus K_1(A)$, grupos de K-teoria da C$^*$-álgebra $A$, em $H_{\\mathbb}^*(G)$, anel da cohomologia real de deRham do grupo de Lie $G$. Utilizando essa definição, nós calculamos explicitamente esse homomorfismo para os exemplos $(\\overline{\\Psi_^0(S^1)}, S^1, \\alpha)$ e $(\\overline{\\Psi_^0(S^2)}, SO(3), \\alpha)$, onde $\\overline{\\Psi_^0(M)}$ denota a C$^*$-álgebra gerada pelos operadores pseudodiferenciais clássicos de ordem zero da variedade $M$ e $\\alpha$ a ação de conjugação pela representação regular (translações). / Given a C$^*$-dynamical system $(A, G, \\alpha)$ one defines a homomorphism, called the Chern-Connes character, that take an element in $K_0(A) \\oplus K_1(A)$, the K-theory groups of the C$^*$-algebra $A$, and maps it into $H_{\\mathbb}^*(G)$, the real deRham cohomology ring of $G$. We explictly compute this homomorphism for the examples $(\\overline{\\Psi_^0(S^1)}, S^1, \\alpha)$ and $(\\overline{\\Psi_^0(S^2)}, SO(3), \\alpha)$, where $\\overline{\\Psi_^0(M)}$ denotes the C$^*$-álgebra gene\\-rated by the classical pseudodifferential operators of zero order in the manifold $M$ and $\\alpha$ the action of conjugation by the regular representation (translations).

Cette thèse montre en quoi la quantification de Berezin--Toeplitz peut être incorporée dans le cadre de la géométrie non commutative.Tout d'abord, nous présentons les principales notions abordées : les opérateurs de Toeplitz (classiques et généralisés), les quantifications géométrique et par déformation, ainsi que quelques outils de la géométrie non commutative.La première étape de ces travaux a été de construire des triplets spectraux (A,H,D) utilisant des algèbres d'opérateurs de Toeplitz sur les espaces de Hardy et Bergman pondérés relatifs à des ouverts Omega de Cn à bord régulier et strictement pseudoconvexes, ainsi que sur l'espace de Fock sur Cn. Nous montrons que les espaces non commutatifs induits sont réguliers et possèdent la même dimension que le domaine complexe sous-jacent. Différents opérateurs D sont aussi présentés. Le premier est l'opérateur de Dirac usuel sur L2(Rn) ramené sur le domaine par transport unitaire, d'autres sont formés à partir de l'opérateur d'extension harmonique de Poisson ou de la dérivée normale complexe sur le bord de Omega.Dans un deuxième temps, nous présentons un triplet spectral naturel de dimension n+1 construit à partir du produit star de la quantification de Berezin--Toeplitz. Les éléments de l'algèbre correspondent à des suites d'opérateurs de Toeplitz dont chacun des termes agit sur un espace de Bergman pondéré. Plus généralement, nous posons des conditions pour lesquelles une somme infinie de triplets spectraux forme de nouveau un triplet spectral, et nous en donnons un exemple. / The results of this thesis show links between the Berezin--Toeplitz quantization and noncommutative geometry.We first give an overview of the three different domains we handle: the theory of Toeplitz operators (classical and generalized), the geometric and deformation quantizations and the principal tools we use in noncommutative geometry.The first step of the study consists in giving examples of spectral triples (A,H,D) involving algebras of Toeplitz operators acting on the Hardy and weighted Bergman spaces over a smoothly bounded strictly pseudoconvex domain Omega of Cn, and also on the Fock space over Cn. It is shown that resulting noncommutative spaces are regular and of the same dimension as the complex domain. We also give and compare different classes of operator D, first by transporting the usual Dirac operator on L2(Rn) via unitaries, and then by considering the Poisson extension operator or the complex normal derivative on the boundary.Secondly, we show how the Berezin--Toeplitz star product over Omega naturally induces a spectral triple of dimension n+1 whose construction involves sequences of Toeplitz operators over weighted Bergman spaces. This result led us to study more generally to what extent a family of spectral triples can be integrated to form another spectral triple. We also provide an example of such triple.

The irrational rotation algebra A_θ is known to be Poincaré self-dual in the KK-theoretic sense. The spectral triple representing the required K-homology fundamental class was constructed by Connes out of the Dolbeault operator on the 2-torus, but so far, there has not been an explicit description of the dual element. We geometrically construct, for certain elements g of the modular group, a finitely generated projective module L_g over A_θ ⊗ A_θ out of a pair of transverse Kronecker flows on the 2-torus. For upper triangular g, we find an unbounded cycle representing the dual of said module under Kasparov product with Connes' class, and prove that this cycle is invertible in KK(A_θ,A_θ), allowing us to 'untwist' L_g to an unbounded cycle representing the unit for the self-duality of A_θ. / Graduate

Indiana University-Purdue University Indianapolis (IUPUI) / I study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.

It is argued that the geometric dual of a noncommutative operator algebra represents a notion of quantum state space which differs from existing notions by representing observables as maps from states to outcomes rather than from states to distributions on outcomes. A program of solving for an explicitly geometric manifestation of quantum state space by adapting the spectral presheaf, a construction meant to analyze contextuality in quantum mechanics, to derive simple reconstructions of noncommutative topological tools from their topological prototypes is presented. We associate to each unital C*-algebra A a geometric object--a diagram of topological spaces representing quotient spaces of the noncommutative space underlying A—meant to serve the role of a generalized Gel'fand spectrum. After showing that any functor F from compact Hausdorff spaces to a suitable target category C can be applied directly to these geometric objects to automatically yield an extension F^∼ which acts on all unital C*-algebras, we compare a novel formulation of the operator K₀ functor to the extension K^∼ of the topological K-functor. We then conjecture that the extension of the functor assigning a topological space its topological lattice assigns a unital C*-algebra the topological lattice of its primary ideal spectrum and prove the von Neumann algebraic analogue of this conjecture.

Le théorème de l'indice d'Atiyah et Singer, démontré en 1963, est un résultat qui a permis de relier des thématiques mathématiques variées, allant des équations aux dérivées partielles a la topologie et la géométrie différentielle. Plus précisément, il fait le lien entre la dimension de l'espace des solutions d'une équation aux dérivées partielles elliptique et des invariants topologiques du type (co)homologie, et a des applications importantes, regroupant plusieurs théorèmes majeurs venant de divers domaines (géométrie algébrique, topologie différentielle, analyse fonctionnelle). D'un autre cote, les fonctions zêta associées à des opérateurs pseudo différentiels sur une variété riemannienne close contiennent dans leurs propriétés analytiques des informations intéressantes. On peut par exemple retrouver dans les résidus le théorème de Weyl sur l asymptotique du nombre de valeurs propres d'un laplacien, et en particulier le volume de la variété. En se plaçant dans le cadre de la géométrie différentielle non commutative développée par Connes, on peut pousser cette idée plus loin. Plus précisément, on peut obtenir, en combinant des techniques de renormalisation zêta avec la propriété d'excision en cohomologie cyclique, des théorèmes d'indice dans l'esprit de celui d'Atiyah-Singer. L'intérêt de ce point de vue réside dans sa généralisation possible à des situations géométriques plus délicates. La présente thèse établit des résultats dans cette direction / The index theorem of Atiyah and Singer, discovered in 1963, is a striking result which relates many different fields in mathematics going from the analysis of partial differential equations to differential topology and geometry. To be more precise, this theorem relates the dimension of the space of some elliptic partial differential equations and topological invariants coming from (co)homology theories, and has important applications. Many major results from different fields (algebraic topology, differential topology, functional analysis) may be seen as corollaries of this result, or obtained from techniques developed in the framework of index theory. On another side, zeta functions associated to pseudodifferential operators on a closed Riemannian manifold contain in their analytic properties many interesting informations. For instance, the Weyl theorem on the asymptotic number of eigenvalues of a Laplacian may be recovered within the residues of the zeta function. This gives in particular the volume of the manifold, which is a geometric data. Using the framework of noncommutative geometry developed by Connes, this idea may be pushed further, yielding index theorems in the spirit of the one of Atiyah Singer. The interest in this viewpoint is to be suitable for more delicate geometrical situations. The present thesis establishes results in this direction