In this thesis, we consider q-deformations of multiplicative Hypertoric varieties, where q∈𝕂^x for 𝕂 an algebraically closed field of characteristic 0. We construct an algebra D_q of q-difference operators as a Heisenberg double in a braided monoidal category. We then focus on the case where q is specialized to a root of unity. In this setting, we use D_q to construct an Azumaya algebra on an l-twist of the multiplicative Hypertoric variety, before showing that this algebra splits over the fibers of both the moment and resolution maps. Finally, we sketch a derived localization theorem for these Azumaya algebras.

Le sujet principal de ce mémoire est le comportement asymptotique de la géométrie et topologie des variétés localement symétriques Gamma\ X quand le volume tend vers l’infini. Notre premier résultat porte sur la convergence Benjamini-Schramm des 2 ou 3-variétés hyperboliques arithmétiques. Une suite d'espaces localement symétriques (Gamma_n\ X) converge Benjamini-Schramm vers l'espace symétrique X si pour chaque R>0 la limite de \Vol((\Gamma\X)_{<R})/Vol(\Gamma\bs X). On montre qu'il existe une constante réelle C=C_R satisfaisant la propriété suivante: pour chaque réseau arithmétique de congruence Gamma de \PGL(2,R) ou PGL(2,C) sans torsion on a Vol ((Gamma\ X)_{<R})<= C_R \ Vol (Gamma\ X)^0.986. Il n'y a qu'un nombre fini de réseaux arithmétiques de covolume borné par une constante donc ce résultat implique la convergence Benjamini-Schramm pour des variétés arithmétiques de congruence. On donne aussi une version de (\ref{AbsFr1}) un peu plus faible qui reste vraie pour des réseaux arithmétiques qui ne sont pas de congruence. Les majorations de volume de la partie $R$-mince sont déduites d'une version forte de la propriété de la multiplicité limite satisfaite par les réseaux arithmétiques de PGL(2,R) et PGL(2,C). En utilisant nos résultats on confirme la conjecture de Gelander pour des 3-variétés arithmétiques hyperboliques: pour chaque telle variété M on construit un complexe simplicial N homotope à M dont le nombre des simplexes est O(Vol(M)) et le degré des nœuds est uniformément borné par une constante absolue. Dans la deuxième partie on s'intéresse aux espaces localement symétriques Gamma\X où X est de rang supérieur ou égal à 2. Notre résultat principal affirme que la dimension du premier groupe d'homologie à coefficients dans F_2 (corps avec 2 éléments) est sous-linéaire en le volume. Ce résultat est à comparer avec des travaux de Calegari et Emerton sur la cohomologie mod-p dans les tours p-adiques des 3-variétés et les résultats d'Abert, Gelander et Nikolov sur le rang des sous-groupes d'un réseau de rang supérieur à angles droits. Le point fort de notre approche est qu'il n'y a pas besoin de travailler dans une seule classe de commensurabilité. La troisième partie est indépendante des deux premières. Elle porte sur une extension du théorème de Kesten. Le théorème de Kesten affirme que si Gamma est un groupe engendré par un ensemble fini symétrique S, N est un sous-groupe normal de Gamma alors N est moyennable si et seulement si les rayons spectraux du graphe de Cayley Cay(Gamma,S) et du graphe de Scheier Sch(Gamma/N,S) coïncident. En utilisant les techniques de Abert, Glasner et Virag on généralise le theorème de Kesten aux N-uniformément récurrents. / The main theme of this work is the study of geometry and topology of locally symmetric spaces Gamma\ X as ther volume Vol(\Gamma\ X) tends to infinity. Our first main result concerns the Benjamini-Schramm convergence for arithmetic hyperbolic 2 or 3-manifolds. A sequence of locally symmetric spaces (Gamma_n\ X) converges Benjamini-Schramm to X if and only if for every radius R>0 the limit Vol((Gamma\ X)_{<R}/Vol (Gamma\ X) as n goes to infinity is 0, where (\Gamma\X)_{<R} stands for the R-thin part of Gamma\ X. We prove that there exists a positive constant C=C_R with the following property: for every torsion free, uniform, congruence arithmetic lattice Gamma in PGL(2,R) or PGL(2,C) Vol ((Gamma\ X)_{<R})<= C Vol (Gamma\X))^0.986. There is only finitely many arithmetic lattices of covolume bounded by a constant so the result above implies the Benjamini-Schramm convergence for any sequence of congruence arithmetic hyperbolic 3-manifolds. We also prove a similar but slightly weaker inequality for non-congruence subgroups. Our results are deduced form a strong form of the limit multiplicity property that holds for arithmetic lattices in PGL(2,R) of PGL(2,C). As an application of our bounds we confirm Gelander's conjecture on the triangulations of arithmetic hyperbolic 3-manifolds: we show that every arithmetic hyperbolic 3-manifold M admits a triangulation with O(Vol(M)) simplices and degrees of vertices bounded uniformly by an absolute constant. Next, we move to the setting of higher rank locally symmetric spaces. Let M_n=Gamma_n\ X be a sequence of pairwise distinct locally symmetric spaces modeled after a higher rank symmetric space X. We show that the dimension of the first homology group with coefficients in F_2 is sublinear in volume. This can be compared with the results of Calegari and Emerton on mod-p homology growth in p-adic analytic towers of 3-manifolds as well as the results of Abert, Gelander and Nikolov on the rank gradient of right-angled lattices in higher rank Lie groups.The main strength of our theorem is that we do not need to assume that the manifolds in question are commensurable. Our third result is independent of the first two. Kesten theorem asserts that if Gamma is group generated by a finite symmetric set S and N is a normal subgroup of Gamma then N is amenable if and only if the spectral radii of the Cayley graphs Cay(Gamma, S) and the Schreier graph Sch(Gamma/N,S) are equal. Building on the work of Abert, Glasner and Virag we extend Kesten's theorem to uniformly recurrent subgroups.

Pour toute courbe projective lisse C et théorie homologique orientée de Borel-Moore libre A, on construit un produit associatif de type Hall sur les A-groupes du champ de modules des faisceaux de Higgs de torsion sur C.On montre que l'algèbre AHa0C qu'on obtient admet une présentation de battage naturelle, qui est fidèle dans le cas où A est l'homologie de Borel-Moore usuelle.On introduit de plus les espaces de modules des triplets stables M(d,n), fortement inspirés par les variétés de carquois de Nakajima.Ces espaces de modules sont des variétés lisses symplectiques, et admettent une autre caractérisation comme les espaces de modules de faisceaux sans torsion stables encadrés sur P(T*C)$.De plus, on munit leurs A-groupes avec une action de AHa0C, qui généralise les opérateurs de modification ponctuelle de Nakajima sur l'homologie des schémas de Hilbert de T*C. / For a smooth projective curve C and a free oriented Borel-Moore homology theory A, we construct a Hall-like associative product on the A-theory of the moduli stack of Higgs torsion sheaves on C.We show that the resulting algebra AHa0C admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel-Moore homology groups.We also introduce moduli spaces of stable triples M(d,n), heavily inspired by Nakajima quiver varieties.These moduli spaces are shown to be smooth symplectic varieties, which admit another characterization as moduli of framed stable torsion-free sheaves on P(T*C).Moreover, we equip their A-theory with an AHa0C-action, which generalizes Nakajima's raising operators on the homology of Hilbert schemes of points on T*C.

The intention of this thesis is to investigate whether there are consistent narrative patterns of images produced by female photojournalists under the Instagram hashtag #8mfotografascolombia in the context of the feminist mobilization of March 8, 2021, that took place in Colombia. I aim to establish if indeed these new communicative strategies in expansion respond to narratives widely used by traditional photojournalism, or if they operate under a different set of dynamics. Under the lens of Representation Theory I want to study how hegemonic depiction and absent stories form photojournalism are configuring counter narratives on social media platforms. On the other hand the perspective of Feminist Media Theory will provide understanding and context about the processes of production, circulation and absent feminine gaze within the media. The subsequent analysis shows that in fact narratives are being configured opposed to the structures of large media organizations in which the female gaze produces not only aesthetically different results, but the photographic process is intrinsically linked to performative actions, the recognition of subjects and away of the logic of spectacle and violence of the big media, but also outside of what the Instagram algorithm privileges.

Let E/F be a quadratic extension of p-adic fields. The local Langlands correspondence establishes a bijection between n-dimensional Frobenius semisimple representations of the Weil-Deligne group of E and smooth, irreducible representations of GL(n, E). We reinterpret this bijection in the setting of the Weil restriction of scalars Res(GL(n), E/F), and show that the Asai L-function and epsilon factor on the analytic side match up with the expected Artin L-function and epsilon factor on the Galois side.

In this thesis, we will explore the structure of Terwilliger algebras over several different types of finite groups. We will begin by discussing what a Schur ring is, as well as providing many different results and examples of them. Following our discussion on Schur rings, we will move onto discussing association schemes as well as their properties. In particular, we will show every Schur ring gives rise to an association scheme. We will then define a Terwilliger algebra for any finite set, as well as discuss basic properties that hold for all Terwilliger algebras. After specializing to the case of Terwilliger algebras resulting from the orbits of a group, we will explore bounds of the dimension of such a Terwilliger algebra. We will also discuss the Wedderburn decomposition of a Terwilliger algebra resulting from the conjugacy classes of a group for any finite abelian group and any dihedral group.

Let $G$ be a locally compact group (usually a reductive algebraic group over an algebraic number field $F$). The main aim of the theory of Automorphic Forms is to understand the right regular representation of the group $G$ on the space $L^{2}(\Gamma \ G)$ for certain \emph{nice} closed subgroups $\Gamma$. Usually, $\Gamma$ is taken to be a lattice or even an arithmetic subgroup. In the case of uniform lattices, the space $L^{2}(\Gamma \ G)$ decomposes into a direct sum of irreducible unitary representations of the group $G$ with each such representation $\pi$ occurring with a \emph{finite} multiplicity $m(\Gamma, \pi)$. It seems quite difficult to obtain an explicit formula for this multiplicity; however, the limiting behaviour of these numbers in case of certain \emph{nice} sequences of subgroups $(\Gamma_{n})_{n}$ seems more tractable. We study this problem in the global set-up where $G$ is the group of adelic points of a reductive group defined over the field of rational numbers and the relevant subgroups are the maximal compact open subgroups of $G$. As is natural and traditional, we use the Arthur trace formula to analyse the multiplicities. In particular, we expand the geometric side to obtain the information about the spectral side---which is made up from the multiplicities $m(\Gamma, \pi)$. The geometric side has a contributions from various conjugacy classes, most notably from the unipotent conjugacy class. It is this \emph{unipotent} contribution that is the subject of Part III of this thesis. We estimate the contribution in terms of level of the maximal compact open subgroup and make conclusions about the limiting behaviour. Part IV is then concerned with the spectral side of the trace formula where we show (under certain conditions) that the trace of the discrete part of the regular representation is the only term that survives in the limit.

The Lorentz group is a symmetry group on Minkowski space, and as such is central to studying the geometry of this and related spaces. The group therefore shows up also from physical considerations, such as trying to formulate quantum physics in anti-de Sitter space. In this thesis, the Lorentz group in 2+1 dimensions and its representations are investigated, and comparisons are made to the analogous rotation group. Firstly, all unitary irreducible representations are found and classified. Then, those representations are realised as the square-integrable, analytic functions on the unit circle and the unit disk, which turn out to correspond to the projective lightcone and the hyperbolic plane, respectively. Also, a way to realise a particular class of representations on 1+1-dimensional anti-de Sitter space is shown. / Lorentzgruppen är en symmetrigrupp på Minkowski-rum, och är således central för att studera geometrin i detta och relaterade rum. Gruppen dyker också därför upp från fysikaliska frågeställningar, såsom att försöka formulera kvantfysik i anti-de Sitter-rum. Denna uppsats undersöker Lorentzgruppen i 2+1 dimensioner och dess representationer, och jämför med den analoga rotationsgruppen. Först konstrueras och klassificeras alla unitära irreducibla representationer. Sedan realiseras dessa representationer som de analytiska funktioner på enhetscirkeln och enhetsskivan vars belopp i kvadrat är integrerbara. Det visar sig att denna cirkel respektive skiva svarar mot den projektiva ljuskonen respektive det hyperboliska planet. Dessutom visas att en särskild klass av representationer blir relevanta för att formulera kvantfysik i 1+1-dimensionellt anti-de Sitter-rum.

In this dissertation, we explore the nature of Schur rings over finite cyclic groups, both algebraically and combinatorially. We provide a survey of many fundamental properties and constructions of Schur rings over arbitrary finite groups. After specializing to the case of cyclic groups, we provide an extensive treatment of the idempotents of Schur rings and a description for the complete set of primitive idempotents. We also use Galois theory to provide a classification theorem of Schur rings over cyclic groups similar to a theorem of Leung and Man and use this classification to provide a formula for the number of Schur rings over cyclic p-groups.

Textbooks are one of the most common educational materials used at Swedish upper secondary schools. Diversity and inclusion have become increasingly important topics in Swedish society, and it is important to ensure that they are reflected in learning materials such as textbooks. The purpose of this research project was therefore to, through an intersectional theoretical approach, achieve increased knowledge about which constructions of social identities and roles as professional workers that are conveyed through teaching materials in three social studies textbooks. This study was governed by two research questions which are as follows: What constructions of social identities are conveyed through the books' images and text? and What constructions of professionals are conveyed through the books' images and text? These questions in the study were explored through a theoretical framework that presents representation in three perspectives, namely the age, gender, and ethnicity. Qualitative analysis method was applied to analyze both visual and textual content in this study. With regards to the visual aspect of the study, an iconographic analysis tool, which looks at the icons or symbols in images to determine the images original meaning or intent was used. The written text was analyzed with the help of Ammerts Typology which examined the establishing, explaining, reflecting/analyzing and normative presentation types of the surrounding text. The study shows that the textbooks' images and text partly lack aspects of inclusion and diversity. This result indicates that a possible revision existing textbook is needed, which may entail creating new materials that are more inclusive. It takes continuous effort and commitment to ensure that all social identities and occupations are represented in educational materials.