The aim of this thesis is to create a power line transfer function generator in Matlab. Work on introduction explains the concept of power line and describes the different types of topologies and types of cables that power line are the most widely used today. It is also discussed PLC technology, its structure, characteristics of the transmission channel and interferences. The following are approach the types of the modeling of power line. More details are described generally modeling in the frequency (two-port network model), which is used in the created generator. The next section provides a power line model, whose principle is used in the generator. The following is a description of created the generator and comparison with FTW PLC Simulator. In conclusion, the thesis presents the results of simulations the transfer function, which show the magnitude and phase frequency characteristics of different topologies and parameters.

The theoretical study of this thesis deals with different approaches that can be used to optimize the topology of various structures. Main attributes and principles which they are based on of individual methods are discussed. The first point of the practical part is to design the model for computational analysis of stand and other machining centre parts based on real machine TOS FU. Afterwards, the model is analyzed to determine the dynamic characteristic of the machine. Subsequently, several changes of the stand design are performed in order to improve the dynamic behavior. Eventually, new stand kernel is designed which aims to enhance these dynamic characteristic. This model is analyzed again and the results are compared with the original form of the machining centre.

Le sujet de la topologie dans les oxydes, en particulier à la surface des oxydes de pérovskite comme SrTiO₃, ou à l'interface de LaA1O₃ / SrTiO₃ sera étudié dans cette thèse. Les deux matériaux, à leurs surfaces orientées (001), contiennent un état métallique limité à quelques nanomètres de la surface. De plus, nous montrerons qu'il existe certains croisements de trois bandes autour desquelles des perturbations vont provoquer l'apparition d'un spectre de bandes inversées et gappées. Ceux-ci conduiront à des états de bord topologiques qui peuvent être détectés via la supraconductivité induite comme dans le cas des puits quantiques topologiques ou des nanofils des supraconducteurs-semi-conducteurs. Ensuite, la surface orientée (111) de LaA1O₃ / SrTiO₃ sera étudiée lorsque les mesures de transport de Hall révèlent une transition de un à deux porteurs par dopage électrostatique. Une explication basée sur un modèle de liaisons fortes incluant des corrélations U de Hubbard sera proposée, ce qui donnera lieu à des croisements de bandes entre les sous-bandes favorisant les états topologiques. Enfin, une étude ab-initio de CaTiO₃ sera effectuée pour expliquer l'état métallique qui existe à sa surface (001) orientée et pour prédire le magnétisme dans le système. CaTiO₃ est différent des autres composés étudiés précédemment, en raison de la grande rotation et de l'inclinaison des octaèdres d'oxygène entourant le Ti, ce qui complique les faits. La structure avec et sans lacunes d'oxygène sera étudiée en profondeur pour fournir des détails sur la bande de conduction et leurs caractères orbitaux. / The subject of topology in oxides, in particular at the surfaces of perovskite oxides like SrTiO₃, or at the interface of LaA1O₃/SrTiO₃ will be investigated in this thesis. Both compounds, at their (001) oriented surfaces, contain a metallic state confined to a few nanometers at the surface. In addition, we will show that there exist certain three band crossings around which perturbations will cause an inverted and gapped band spectrum to appear. These will lead to topological edge states which can be detected via induced superconductivity as in the case of topological quantum wells or superconductor-semiconductor nanowires. Next, the (111) oriented surface of LaA1O₃/SrTiO₃ will be studied where Hall transport measurements reveal a one to two carrier transition via electrostatic doping. An explanation based on a tight binding modelling including Hubbard U correlations, will be proposed which will give rise to band crossings between sub-bands promoting topological states. Finally, an ab-initio study of CaTiO₃ will be performed to explain the metallic state which exists at its (001) oriented surface and to predict magnetism in the system. CaTiO₃ is different from the other compounds studied previously, due to the large rotation and tilting of the oxygen octahedra surrounding the Ti, which complicates the picture. The structure with and without oxygen vacancies will be studied in-depth to provide details about the conduction band and their orbital characters.

La molécule d’ADN d’un chromosome décondensé a toujours une taille supérieure au volume de la cellule. Elle doit donc être compactée tout en restant fonctionnelle pour les grandes fonctions cellulaires telles que l’expression des gènes, la réplication et la ségrégation fidèle des chromosomes. Cette compaction fait appel à une structuration finement régulée à différentes échelles chez la bactérie Escherichia coli. A l’échelle moléculaire, le chromosome est maintenu sous une forme surenroulée négative par deux types de surenroulement, l’un « contraint » par la fixation de protéines à l’ADN formant le nucléoïde, et l’autre « libre », diffusible le long du chromosome. A l’échelle sub-cellulaire, le chromosome d'E. coli est composé de quatre régions chromosomiques appelées Macro-Domaines (MD) (Ori, Right, Left, Ter), isolées spatialement et génétiquement, ainsi que de deux régions non structurées (NSR, NSL). La structuration du MD Ter résulte de la liaison de dimères de la protéine MatP sur 22 séquences palindromiques matS de 13 pb. L’absence de MatP entraînant une décondensation relative de l’ADN dans cette région, il est supposé qu’un tétramère de MatP ponte deux sites matS. Mon projet de thèse a consisté à étudier le contrôle de la topologie de l’ADN chromosomique chez la bactérie E. coli et à montrer sa relation avec l’organisation en MD et la formation de la chromatine. J’ai adapté un système rapporteur basé sur la réaction de résolution du transposon gamma delta (Tn1000) qui implique la formation d’une structure d’ADN surenroulée. Ce test a permis de mesurer à la fois le niveau de surenroulement de la molécule d’ADN mais également sa capacité à coulisser, révélant la présence de barrières topologiques. Ces travaux montrent que le niveau de surenroulement diffusible varie localement et suivant les conditions de croissance, impliquant un rôle prépondérant de la réplication, de la transcription et des protéines de fixation à l’ADN. Cependant, le système Res ne suffit pas à déterminer le mécanisme précis par lequel l’ADN est contraint par un déterminant comme MatP ; un système optimal devra combiner les résultats de capture de conformation de chromosome, de Microscopie à Super Resolution et de mesure du surenroulement. / Decondensed DNA molecule of a chromosome is always larger than the volume of the cell. It must therefore be compacted while remaining functional for the major cellular functions such as gene expression, replication and faithful segregation of chromosomes. This compaction uses a tightly regulated structure at different scales in the bacterium Escherichia coli. At the molecular level, the chromosome is maintained under a negative supercoiled form by two types of supercoiling, one "constrained" by the DNA binding protein forming the nucleoid and the other "free" and diffusible along the chromosome. At the subcellular level, the chromosome of E. coli is composed of four chromosomal regions called Macro-Areas (R) (Ori, Right, Left, Ter), spatially and genetically isolated, and two unstructured regions (NSR NSL) .The structure of the MD results from the Ter dimer protein binding of MatP palindromic sequences of 22 bp 13 masts. As the absence of MatP causes relative decondensation of DNA in this region, it is assumed that a tetramer of two MatP bounds two matS sites. My thesis project was to study the control of the chromosomal DNA topology in E. coli and to show its relationship with the organization in MD and chromatin. I’ve adapted a suitable reporter system based on the reaction of resolution of the gamma delta (Tn1000) transposon which involves the formation of a supercoiled DNA structure. This test was used to measure both the level of DNA supercoiling molecule but also its ability to slide, revealing the presence of topological barriers. This work shows that the level of free supercoiling varies locally according to the conditions of growth, implying a major role in the replication, transcription and protein binding to DNA. However, Res system is not sufficient to determine the precise mechanism by which DNA is constrained by a determinant as MatP; an optimal system will combine the results of capture of chromosome conformation of Microscopy Super Resolution and measurement of supercoiling.

Les propriétés mécaniques d’un tricot diffèrent drastiquement de celles du fil dont il est constitué. Par exemple, une étoffe tricotée d’un fil inextensible présente une étonnante propension à la déformabilité. À l’instar des systèmes mécaniques où la géométrie joue un rôle prépondérant, tels les origamis, la réponse mécanique d’un tricot va être déterminée par le chemin imposé au fil. Lors du tricotage, le fil est contraint de se courber et de former des points de croisement suivant un motif répétitif, figeant de cette manière sa topologie. Les trois ingrédients sur lesquels repose la réponse mécanique d’un tricot sont l’élasticité du fil, sa topologie et le frottement aux contacts. Une sélection des nombreux phénomènes qui émergent du couplage entre ces ingrédients fait l’objet de cette thèse. Premièrement, l’intérêt a été porté sur l’élasticité du tricot. En se basant sur une expérience de traction d’un tricot-modèle, une théorie, qui vise à décrire cette réponse mécanique, a été construite en tenant compte de la conservation de la topologie, l’énergie de flexion et l’inextensibilité du fil. Dans un second temps, l’accent est mis sur les fluctuations de la réponse mécanique. Ces fluctuations ont pour origine la friction du fil qui empêche sa répartition dans la maille jusqu’à ce qu’un contact glisse brusquement, déclenchant alors une succession de glissements. La mesure de la réponse en force et du champ de déformations montrent que ces évènements suivent une dynamique d’avalanches. Enfin, l’action de la topologie et de la métrique du tricot sur sa forme tridimensionnelle, ainsi que la transition de configuration spontanée de la structure d’un tricot, ont été examinés. / Knits mechanical properties are fundamentally different from those of its constitutive yarn. For instance, a fabric knitted with an inextensible yarn demonstrates a surprising inclination for deformability. Like mechanical systems where geometry plays a preponderant role, such as origami, the mechanical response of knitted fabrics is governed by the pattern imposed on the yarn. In the process of knitting, the yarn is constrained to bend and to cross itself following a periodic pattern, anchoring its topology. The three factors which determine the mechanical response of a knit are the elasticity of the yarn, its topology, and friction between crossing strands. This thesis explores several phenomena that arise from the interplay of these factors. First, we focused on the elasticity of a knit. Working from experimental data, we developed a theory to decipher the mechanical response of model knits under traction, taking into account the unaltered topology, bending energy, and inextensibility of the yarn. Next, we explored fluctuations in the mechanical response of a knit. Those fluctuations originate from yarn-yarn friction, preventing free yarn redistribution in the stitch until a contact slides and triggers propagative slips. Measures of the force response and deformation fields reveal that those events follow an avalanching dynamic, including a power law distribution of their size. Finally, the impact of topology and metric on knit three-dimensional shapes, along with spontaneous configuration transitions in a knit structure, are studied.

Es handelt sich hierbei um die Dokumentation eines Dateiformats zur Beschreibung dreidimensionaler FEM-Gebiete in Randrepräsentation. Eine interne Datenbasis dient als Verbindung zwischen externem Dateiformat und verschiedenen verarbeitenden Programmen.

Scalar fields are used in many disciplines to represent scalar quantities over some spatial domain. Their versatility and the potential to model a variety of real-world phenomena has made scalar fields a key part of modern data analysis. Examples range from modeling scan results in medical applications (e.g. Magnetic Resonance Imaging or Computer Tomography), measurements and simulations in climate and weather research, or failure criteria in material sciences. But one thing that all applications have in common is that the data is always affected by errors. In measurements, potential error sources include sensor inaccuracies, an unevenly sampled domain, or unknown external influences. In simulations, common sources of error are differences between the model and the simulated phenomenon or numerical inaccuracies. To incorporate these errors into the analysis process, the data model can be extended to include uncertainty. For uncertain scalar fields that means replacing the single value that is given at every location with a value distribution. While in some applications, the influence of uncertainty might be small, there are a lot of cases where variations in the data can have a large impact on the results. A typical example is weather forecasts, where uncertainty is a crucial part of the data analysis. With increasing access to large sensor networks and extensive simulations, the complexity of scalar fields often grows to a point that makes analysis of the raw data unfeasible. In such cases, topological analysis has proven to be a useful tool for reducing scalar fields to their fundamental properties. Scalar field topology studies structures that do not change under transformations like scaling and bending but only depend on the connectivity and relative value differences between the points of the domain. While a lot of research has been done in this area for deterministic scalar fields, the incorporation of uncertainty into topological methods has only gained little attention so far. In this thesis, several methods are introduced that deal with the topological analysis of uncertain scalar fields. The main focus lies on providing fundamental research on the topic and to drive forward a rigorous analysis of the influence of uncertainty on topological properties. One important property that has a strong influence on topological features are stochastic dependencies between different locations in the uncertain scalar field. In the first part of this thesis, we provide a method for extracting regions that show linear dependency, i.e. correlation. Using a combination of point-cloud density estimation, clustering, and scalar field topology, our method extracts a hierarchical clustering. Together with an interactive visualization, the user can explore the correlation information and select and filter the results. A major benefit of our approach is the comprehensive handling of correlation. This also includes global correlation between distant points and inverse correlation, which is often only partially handled by existing methods. The second part of this thesis focuses on the extraction of topological features, such as critical points or hills and valleys of the scalar field. We provide a method for extracting critical points in uncertain scalar fields and track them over multiple realizations. Using a novel approach that operates in the space of all possible realizations, our method can find all critical points deterministically. This not only increases the reliability of the results but also provides complete knowledge that can be used to study the relation and behavior of critical points across different realizations. Through a combination of multiple views, we provide a visualization that can be used to analyze critical points of an uncertain scalar field for real-world data. In the last part, we further extend our analysis to more complex feature types. Based on the well-known contour tree that provides an abstract view on the topology of a deterministic scalar field, we use an approach that is similar to our critical point analysis to extract and track entire regions of the uncertain scalar field. This requires solving a series of new challenges that are associated with tracking features in the multi-dimensional space of all realizations. As our research on the topic falls under the category of fundamental research, there are still some limitations that have to be overcome in the future. However, we provide a full pipeline for extracting topological features that ranges from the data model to the final interactive visualization. We further show the applicability of our methods to synthetic and real-world data. / Skalarfelder sind Funktionen, die jedem Punkt eines Raumes einen skalaren Wert zuweisen. Sie werden in vielen verschiedenen Bereichen zur Analyse von skalaren Messgrößen mit räumlicher Information eingesetzt. Ihre Flexibilität und die Möglichkeit, viele unterschiedliche Phänomene der realen Welt abzubilden, macht Skalarfelder zu einem wichtigen Werkzeug der modernen Datenanalyse. Beispiele reichen von medizinischen Anwendungen (z.B. Magnetresonanztomographie oder Computertomographie) über Messungen und Simulationen in Klima- und Wetterforschung bis hin zu Versagenskriterien in der Materialforschung. Eine Gemeinsamkeit all dieser Anwendungen ist jedoch, dass die erfassten Daten immer von Fehlern beeinflusst werden. Häufige Fehlerquellen in Messungen sind Sensorungenauigkeiten, ein ungleichmäßig abgetasteter Betrachtungsbereich oder unbekannte externe Einflussfaktoren. Aber auch Simulationen sind von Fehlern, wie Modellierungsfehlern oder numerischen Ungenauigkeiten betroffen. Um die Fehlerbetrachtung in die Datenanalyse einfließen lassen zu können, ist eine Erweiterung des zugrunde liegenden Datenmodells auf sogenannte \emph{unsicheren Daten} notwendig. Im Falle unsicherer Skalarfelder wird hierbei statt eines festen skalaren Wertes für jeden Punkt des Definitionsbereiches eine Werteverteilung angegeben, die die Variation der Skalarwerte modelliert. Während in einigen Anwendungen der Einfluss von Unsicherheit vernachlässigbar klein sein kann, gibt es viele Bereiche, in denen Schwankungen in den Daten große Auswirkungen auf die Resultate haben. Ein typisches Beispiel sind hierbei Wettervorhersagen, bei denen die Vertrauenswürdigkeit und mögliche alternative Ausgänge ein wichtiger Bestandteil der Analyse sind. Die ständig steigende Größe verfügbarer Sensornetzwerke und immer komplexere Simulationen machen es zunehmend schwierig, Daten in ihrer rohen Form zu verarbeiten oder zu speichern. Daher ist es wichtig, die verfügbare Datenmenge durch Vorverarbeitung auf für die jeweilige Anwendung relevante Merkmale zu reduzieren. Topologische Analyse hat sich hierbei als nützliches Mittel zur Verarbeitung von Skalarfeldern etabliert. Die Topologie eines Skalarfeldes umfasst all jene Merkmale, die sich unter bestimmten Transformationen, wie Skalierung und Verzerrung des Definitionsbereiches, nicht verändern. Hierzu zählen beispielsweise die Konnektivität des Definitionsbereiches oder auch die Anzahl und Beziehung von Minima und Maxima. Während die Topologie deterministischer Skalarfelder ein gut erforschtes Gebiet ist, gibt es im Bereich der Verarbeitung von Unsicherheit im topologischen Kontext noch viel Forschungspotenzial. In dieser Dissertation werden einige neue Methoden zur topologischen Analyse von unsicheren Skalarfeldern vorgestellt. Der wesentliche Teil dieser Arbeit ist hierbei im Bereich der Grundlagenforschung angesiedelt, da er sich mit der theoretischen und möglichst verlustfreien Verarbeitung von topologischen Strukturen befasst. Eine wichtige Eigenschaft, die einen starken Einfluss auf die Struktur eines unsicheren Skalarfeldes hat, ist die stochastische Abhängigkeit zwischen verschiedenen Punkten. Im ersten Teil dieser Dissertation wird daher ein Verfahren vorgestellt, das das unsichere Skalarfeld auf Regionen mit starker linearer Abhängigkeit, auch \emph{Korrelation} genannt, untersucht. Durch eine Kombination aus hochdimensionaler Punktwolkenanalyse, Clusterbildung und Skalarfeldtopologie extrahiert unsere Methode eine Hierarchie von Clustern, die die Korrelation des unsicheren Skalarfeldes repräsentiert. Zusammen mit einer interaktiven, visuellen Aufbereitung der Daten wird dem Nutzer so ein explorativer Ansatz zur Betrachtung der stochastischen Abhängigkeiten geboten. Anzumerken ist hierbei, dass unser Verfahren auch globale und inverse Korrelation abdeckt, welche in vielen verwandten Arbeiten oft nicht vollständig behandelt werden. Der zweite Teil dieser Dissertation widmet sich der Analyse und Extraktion von topologischen Merkmalen, wie kritischen Punkten oder ganzen Hügeln oder Tälern im Funktionsgraphen des Skalarfeldes. Hierzu wird ein Verfahren zur Berechnung von kritischen Punkten vorgestellt, das diese auch über viele verschiedene Realisierungen des unsicheren Skalarfeldes identifizieren und verfolgen kann. Dies wird durch einen neuen Ansatz ermöglicht, der den Raum aller möglichen Realisierungen nach geometrischen Strukturen untersucht und somit kritische Punkte deterministisch berechnen kann. Dadurch, dass mit diesem Verfahren keine kritischen Punkte ausgelassen werden, steigt nicht nur die Vertrauenswürdigkeit der Resultate, sondern es wird außerdem möglich, Beziehungen zwischen kritischen Punkten zu untersuchen. Zu diesen Beziehungen gehört beispielsweise das Wandern von kritischen Punkten über verschiedene Positionen oder auch die Entstehung von Skalarwerthügeln oder -tälern. Um die Resultate visuell zu untersuchen, stellen wir mehrere verknüpfte Ansichten bereit, die eine Analyse von kritischen Punkten auch in realen Daten ermöglichen. Im letzten Teil dieser Arbeit erweitern wir die Betrachtung der Topologie von kri\-ti\-schen Punkten auf komplexere Strukturen. Basierend auf dem \emph{Konturbaum}, der eine abstrakte Repräsentation der Topologie eines deterministischen Skalarfeldes ermöglicht, untersuchen wir, wie ganze Regionen des Skalarfeldes von Unsicherheit betroffen sind. Dies führt zu einer Reihe von neuen theoretischen und auch praktischen Herausforderungen, wie der stark steigenden Komplexität der notwendigen Berechnungen oder Inkonsistenzen bei der Verfolgung von topologischen Strukturen über mehrere Realisierungen. Auch wenn zur Anwendung unserer Verfahren auf reale Daten aufgrund des großen Möglichkeitsraumes von unsicheren Skalarfeldern noch Einschränkungen notwendig sind, sind viele der theoretischen Erkenntnisse allgemeingültig. Zur Betrachtung der Ergebnisse werden verschiedene Visualisierungen genutzt, um die extrahierten topologischen Strukturen anhand von synthetischen und realen Daten zu zeigen.

Topology, an important topic in physics since several years, is handled as possible solution to many current-state problems in electronics and energy. It could allow to dramatically shrink computational devices or increase their speed without the current problem of heat dissipation, or topological principles can be used to introduce room temperature high-conduction paths within materials. Unfortunately, while many promising materials have been presented yet, the one breakthrough material is still missing. Current style materials are either consisting of toxical elements, obstructing possible use cases, or their electronic structure is too complex to investigate the interplay of all the facets of the electronic structure present in the mateirals. In this thesis, two very promising materials will be thoroughly introduced, namely TaIrTe4 and GaGeTe. Both materials have the potential, to lift one of the shortcomings mentioned. First, TaIrTe4 will be presented. TaIrTe4 is a simplistic Weyl semimetal in terms of its electronic and topological structure - the simplest yet known material. It hosts four Weyl points, the minimum amount of Weyl nodes possible in a non-centrosymmetric material. Predictions state, that these nodes are well separated throughout the Brillouin zone, and are connected by nearly parallel Fermi arcs. The existance of the topological states is proved in this thesis through angle-resolved photoemission spectroscopy (ARPES) and confirmed by spin polarization measurements on these states. GaGeTe is predicted to be a Bi2Se3-style topological insulator, but ARPES data presented shows, that no direct band gap could be observed. Yet, a topological state is still believed to be present. This makes this material interesting in many ways: its elemental composition is less toxic than bismuth and selenium, as well as it is the first realization of such a specific electronic structure. A full discussion of the electronic states close to the Fermi level including the possible existance of topological states is shown in this thesis.

The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.

This thesis deals with Panin and Walter's motivic spectrum BO. This spectrum is constructed using the real and quaternionic Grassmannians RG(r,n) and HGr(r,n) respectively, over schemes were 2 is invertible. We show that the construction of BO does not need the invertibility of 2. We also show that this spectrum is cellular over any base.