The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*-algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*-algebras respectively. Of particular interest are the crossed product C*-algebras; we derive important structure results about them and compute their K-theory.

No description available.

In this work, we examine how neutrino and collider experiments can each and together put constraints on new physics more stringently than ever. Constraints arise in three ways. First, possible new theoretical frameworks are reviewed and analyzed for the compatibility with collider experiments. We study alternate theories such as the superconnection formalism and non-commutative geometry (NCG) and show how these can be put to test, if any collider excess were to show up. In this case, we use the previous diboson and diphoton statistical excess as examples to do the analysis. Second, we parametrize low energy new physics in the neutrino sector in terms of non-standard interactions (NSI), which are constrained by past and proposed future neutrino experiments. As an example, we show the capability of resolving such NSI with the OscSNS, a detector proposed for Oak Ridge National Lab and derive interesting new constraints on NSI at very low energy (≲ 50 MeV). Apart from this, in order to better understand the NSI matter effect in long baseline experiments such as the future DUNE experiment, we derive a new compact formula to describe the effect analytically, which provides a clear physical picture of our understanding of the NSI matter effect compared to numerical computations. Last, we discuss the possibility of combining neutrino and collider data to get a better understanding of where the new physics is hidden. In particular, we study a model that produces sizable NSI to show how they can be constrained by past collider data, which covers a distinct region of the model parameter space from the DUNE experiment. In combining the two, we show that neutrino experiments are complementary to collider searches in ruling out models such as the ones that utilize a light mediator particle. More general procedures in constructing such models relevant to neutrino experiments are also described. / Ph. D.

An analogue of the Riemannian structure of a manifold is created for an ultrametric Cantor set using the techniques of Noncommutative Geometry. In particular, a spectral triple is created that can recover much of the fractal geometry of the original Cantor set. It is shown that this spectral triple can recover the metric, the upper box dimension, and in certain cases the Hausdorff measure. The analogy with Riemannian geometry is then taken further and an analogue of the Laplace-Beltrami operator is created for an ultrametric Cantor set. The Laplacian then allows to create an analogue of Brownian motion generated by this Laplacian. All these tools are then applied to the triadic Cantor set. Other examples of ultrametric Cantor sets are then presented: attractors of self-similar iterated function systems, attractors of cookie cutter systems, and the transversal of an aperiodic, repetitive Delone set of finite type. In particular, the example of the transversal of the Fibonacci tiling is studied.

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.

No description available.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Nesta tese estudamos a mecânica quântica não-comutativa na situação não-relativística. Nesse contexto, a expansão-1/N é introduzida e aplicada para alguns potenciais de interesse, como o do oscilador anarmônico e do potencial Coulombiano. A convergência da série é então discutida. Propomos uma versão modificada do potencial Coulombiano nãocomutativo, o qual fornece uma expansão 1/N bem comportada. A seguir, introduzimos um novo conjunto de relações de comutação no espaço-tempo não-comutativo satisfazendo uma álgebra de Heisenberg deformada. A equação de Pauli modificada é usada para o cálculo de correções para a energia, com o uso de teoria da perturbação, no contexto da não-comutatividade dependente do spin. / In this thesis we study non-commutative quantum mechanics in nonrelativistic situation. In this context, the 1/N-expansion is introduced and applied to some potentials of interest as the anharmonic oscillator and the Coulomb potential. The convergence of the serie is discussed. We proposed a modied version of the noncommutative Coulombian potential which provides a well-behaved 1/N expansion. Subsequently, we introduce a new set of noncommutative space-time commutation relations which satisfy a spin dependent nonstandard Heisenberg algebra. Modied Pauli equation is used to calculate corrections to the energy by the use of perturbation theory in the noncommutativity spin-dependent context.

Noncommutative Riemannian geometry is an area that has seen intense activity over the past 25 years. Despite this, noncommutative complex geometry is only now beginning to receive serious attention. The theory of quantum groups provides a large family of very interesting potential examples, namely the quantum flag manifolds. Thus far, only the irreducible quantum flag manifolds have been investigated as noncommutative complex spaces. In a series of papers, Heckenberger and Kolb showed that for each of these spaces, there exists a q-deformed Dolbeault double complex. In this thesis a comprehensive framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully at quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. It is shown that when applied to the quantum projective spaces, this theory reproduces the q-Dolbeault double complexes of Heckenberger and Kolb. Furthermore, the framework is used to q-deform results from Borel{Bott{ Weil theory, and to produce the beginnings of a theory of noncommutative Kahler geometry.

The primary objects of study in this thesis are noncommutative surfaces; that is, noncommutative noetherian domains of GK dimension 2. Frequently these rings will also be singular, in the sense that they have infinite global dimension. Very little is known about singularities of noncommutative rings, particularly those which are not finite over their centre. In this thesis, we are able to give a precise description of the singularities of a few families of examples. In many examples, we lay the foundations of noncommutative singularity theory by giving a precise description of the singularities of the fundamental examples of noncommutative surfaces. We draw comparisons with the fundamental examples of commutative surface singularities, called Kleinian singularities, which arise from the action of a finite subgroup of SL(2; k) acting on a polynomial ring. The main tool we use to study the singularities of noncommutative surfaces is the singularity category, first introduced by Buchweitz in [Buc86]. This takes a (possibly noncommutative) ring R and produces a triangulated category Dsg(R) which provides a measure of "how singular" R is. Roughly speaking, the size of this category reflects how bad the singularity is; in particular, Dsg(R) is trivial if and only if R has finite global dimension. In [CBH98], Crawley-Boevey-Holland introduced a family of noncommutative rings which can be thought of as deformations of the coordinate ring of a Kleinian singularity. We give a precise description of the singularity categories of these deformations, and show that their singularities can be thought of as unions of (commutative) Kleinian singularities. In particular, our results show that deforming a singularity in this setting makes it no worse. Another family of noncommutative surfaces were introduced by Rogalski-Sierra-Stafford in [RSS15b]. The authors showed that these rings share a number of ring-theoretic properties with deformations of type A Kleinian singularities. We apply our techniques to show that the "least singular" example has an A1 singularity, and conjecture that other examples exhibit similar behaviour. In [CKWZ16a], Chan-Kirkman-Walton-Zhang gave a definition for a quantum version of Kleinian singularities. These require the data of a two-dimensional AS regular algebra A and a finite group G acting on A with trivial homological determinant. We extend a number of results in [CBH98] to the setting of quantum Kleinian singularities. More precisely, we show that one can construct deformations of the skew group rings A#G and the invariant rings AG, and then determine some of their ring-theoretic properties. These results allow us to give a precise description of the singularity categories of quantum Kleinian singularities, which often have very different behaviour to their non-quantum analogues.