This paper presents a study on the optimization of floating offshore wind farms. The aim of this study is mainly to create a tool that can help determine the most profitable layout option for the floating offshore wind power company Windeed. This report contains an overview of wind power theory including wind roses, wake losses, fatigue loading, and construction theory. The methodology used for modeling and optimization is the programming language Python together with additional tools such as TopFarm and PyWake. Challenges in the process of designing the layout of a floating wind farm are discussed and the two mathematical models, Bastankhah Gaussian and NOJ, are compared for their ability to recreate wake effects. In the results, the discoveries of the study and the tool created for the company are presented. It was found that allowing freely placed turbines rather than placing the turbines in a strict hexagonal pattern, with shared anchors, gave notably higher annual power production for examined wind farms. Although the levelized cost of energy of the farms with hexagonal patterns were lower for some of the investigated scenarios. Some of the key factors that need to be considered when choosing layout for an off shore wind farm, as well as the potential improvements of the tool are also highlighted in the discussion chapter. Overall, this study provides valuable insights into the design and optimization of floating offshore wind farms.

This report aims to give an insight to the path integral formalism in quantum mechanics. After explaining the kernel's construction, some of its properties and ways to compute it, we see how it relates to the Schrödinger picture. Moreover, we see how its representation can change if it is defined in the space, momentum, time or energy space. Finally, we derive Born's expansion with the kernel showing how this formalism helps to understand perturbation theory and thus scattering. The path integral formalism is then used in quantum field theory with proofs and examples of simple correlation functions. Furthermore, supersymmetry in zero and one dimension are studied with use of the localization principle and the Witten index.

Quantum field theories are very good at describing the world around us but use complicated computations that cannot always be solved exactly. Introducing conformal symmetry to quantum field theory can reduce this complexity and allow for quite simple calculation in the best case. This report aims to describe the critical part of the Ising model in 2 dimensions using conformal field theory while assuming only some knowledge of quantum mechanics and complex analysis from the reader. This is done by using the book Conformal Field Theory as the source for information about conformal field theory. / Kvantfältteorier är mycket bra på att beskriva verkligheten runt om oss men de använder sig av avancerade beräkningar som inte alltid kan lösas exakt. Genom att ge systemet konform symmetri så kan dessa avancerade beräkningar förenklas och bli ganska enkla i de bästa fallen. Målet med denna rapport är att beskriva hur en modell som kallas för "Ising model" kan beskrivas i sitt kritiska tillstånd med hjälp utav konform fältteori. Läsaren antas kunna kvantmekanik samt komplex analys. Informationen om konform fältteori hämtas ifrån boken Conformal Field Theory

Motivated by the successes in the field of deep learning, the scientific community has been increasingly interested in neural networks that are able to reason about physics. As neural networks are universal approximators, they could in theory learn representations that are more efficient than traditional methods whenever improvements are theoretically possible. This thesis, done in collaboration with Algoryx, serves both as a review of the current research in this area and as an experimental investigation of a subset of the proposed methods. We focus on how useful these methods are as \textit{learnable simulators} of mechanical systems that are possibly constrained and multiscale. The experimental investigation considers low-dimensional problems with training data generated by either custom numerical integration or by use of the physics engine AGX Dynamics. A good learnable simulator should express some important properties such as being stable, accurate, generalizable, and fast. Importantly, a generalizable simulator must be able to represent reconfigurable environments, requiring a model known as a graph neural network (GNN). The experimental results show that black-box neural networks are limited to approximate physics in the states it has been trained on. The results also suggest that traditional message-passing GNNs have a limited ability to represent more challenging multiscale systems. This is currently the most widely used method to realize GNNs and thus raises concern as there is not much to be gained by investing time into a method with fundamental limitations.

The Bose-Hubbard Model, a tight binding model within solid state theory can be solved exactly using a number theoretical approach. From this approach, in the two sited Bose-Hubbard model, the hopping term in the model takes the form of a Pauli x matrix. The hopping term can be interpreted as a two energy level system or a dimer with k+1 particles. The statistical properties of this dimer is investigated assuming Boltzmann distribution. The partition function and particle density on each site in the dimer is calculated for spin 1/2 system. The entropy and average energy is also calculated. The particle density is calculated and plotted as a function of temperature for the spin 1/2,1,3/2 and 2 system. At low temperature the particles are more likely to be found in the lower energy site and at high temperatures the particles is equally distributed at the both sites. / Bose-Hubbard modellen är en tight binding modell inom fasta tillstånd- ets fysik som kan lösas exakt genom att använda en talteoretisk lösning- smetod. Genom att göra detta med bara två interagerande platser i modellen tar hoppingtermen i modellen formen av en Pauli x-matris. Hopping modellen kan tolkas som ett system med två energinivåer eller en så kallad dimer med k+1 partiklar. Dimerens statistiska egenskaper undersöks utifrån Boltzmannfördelningen. Partitionsfunktionen och partikeldensiteten på varje plats i dimeren beräkn- as för ett spin 1/2 system, tillsammans med entropin och medelenergin. Vidare är partikeldensiteten  beräknad och plottad som funktion av temperaturen för spinn 1/2, 1, 3/2 och 2 system. Vid låg temperatur befinner sig partiklarna i dimeren i den lägre energinivån och vid hög temperatur är partiklarna jämt fördelade i de två energinivåerna

This is a project with the ambition of demonstrating the possibilities and applications of the quantum approximation optimization algorithm (QAOA). Throughout the paper discussions on the theoretical background and fundamentals of the algorithm will be done by examining the relevant nomenclature. Then a set of possible application problems will be considered where it will be discussed why this specific algorithm is of interest for each individual problem. In the fourth section these problems will concretely be tested via simulations of the QAOA and lastly an analysis of the outcomes will be done.

The focus of this thesis is to introduce the path integral and some of its applications. One interpretation of quantum mechanics is that a microscopic system which moves from an initial- to a final state moves through each possible intermediate state. The path integral uses the principle of least action to sum over all such intermediate states to find the evolution of a quantum mechanical system. We compare the path integral approach to that of the Schrödinger equation and show that the two give an equivalent description of quantum mechanics. To demonstrate the usefulness of the path integral, we introduce low-dimensional quantum field theory (QFT). In particular, we discuss Feynman diagrams. The idea behind Feynman diagrams is to sum over all possible weak interactions between fields to evaluate the properties of a system through the path integral. We also carry out a computation of a low energy effective action in a 0-dimensional model. The result of the computation shows that there is free energy also in a vacuum. Finally, we briefly generalize some of the previous discussion to 1-dimensional QFT. To give an example of a practical application, we give a qualitative discussion of how the path integral can be applied to statistical mechanics to predict the behaviour of superfluids. / Målet med den här rapporten är att introducera konceptet vägintegral och några av dess applikationer. En tolkning av kvantmekanik är att ett mikroskopiskt system som går från ett initialt- till ett slutgiltigt tillstånd kommer att passera genom alla möjliga mellanliggande tillstånd. Vägintegralen använder sig av principen om minsta verkan för att summera över alla sådana mellanliggande tillstånd för att hitta utvecklingen hos ett system. Vi kommer att jämföra vägintegralen med Schrödingers ekvation och visa att de två ger en ekvivalent beskrivning av kvantmekaniken. För att demonstrera vägintegralens användbarhet introducerar vi lågdimensionell kvantfältteori. Vi diskuterar speciellt Feynmandiagram. Idén bakom Feynmandiagram är att summera över alla möjliga svaga interaktioner mellan fält för att utvärdera fysikaliska egenskaper hos system med hjälp av vägintegraler. Vi kommer också att utvärdera en effektiv verkan i 0-dimensionell kvantfältteori. Resultatet visar att det finns fri energi även i ett vakuum. Slutligen generaliserar vi delar av vår tidigare diskussion till 1-dimensionell kvantfältteori. Som ett exempel på praktiska applikationer för vi en kvalitativ diskussion kring hur vägintegraler kan användas inom statistisk mekanik för att förutsäga egenskaper hos superfluider.

Gromov-Witten invariants play a crucial role in symplectic- and enumerative Geometry as well as topological String Theory. Essentially, theseinvariants are a count of (pseudo)holomorphic curves of a given genus,going through n-marked points on a symplectic manifold. In the last fewdecades, this has been a huge research topic for both physicists as well asmathematicians, and breakthroughs in calculation techniques have beenmade using Mirror Symmetry. We investigate and explicitly calculateclosed genus zero Gromov-Witten invariants of toric Calabi-Yau threefolds, namely O(−3) → P2 and the resolved conifold. This will be doneby using localization techniques, mirror symmetry and the so called diskpartition function.

Superconformal field theories (SCFTs) are an important class of quantum field theories. These SCFTs have been a significant component in exploring and comprehending the fundamental framework of quantum field theories. In the vast realm of quantum field theories, integrability plays a crucial role, providing powerful analytic tools that allow us to solve certain physical quantities exactly. In this thesis, we focus on the representation theory of the algebraic structure in six-dimensional (6D) SCFTs and investigate the intricate interplay between 6D SCFTs and integrability. To begin, we delve into the fundamental concepts of representation theory, establishing a solid foundation for our subsequent analysis. The discussion then will move on to all possible generators in the SCFTs, explaining how they are realized in terms of bosonic and fermionic oscillators. Finally, we investigate spin chains and their application in 6D SCFTs. We demonstrate that symmetry arguments derived from representation theory are not sufficient to establish the integrability of the spin chains in 6D SCFTs. This conclusion does not imply the absence of integrable systems within 6D SCFTs; rather, it suggest there are other potential methods available e.g., correlation functions, to explore the appearance of integrable systems in 6D SCFTs. / Superkonforma fältteorier (SCFTs) är en viktig klass av kvantfältteorier. Dessa SCFTs utgör en viktig komponent för att utforska och förstå det fundamentala ramverket för kvantfältteorin. Inom det stora riket av kvantfältteori spelar integrabilitet en avgörande roll, vilket tillhandahåller kraftfulla analytiska verktyg som gör att vi kan lösa vissa fysiska storheter exakt. I denna avhandling fokuserar vi på representationsteorin av den algebraiska strukturen i sexdimensionella (6D) SCFTs och undersöker det intrikat samspelet mellan 6D SCFTs och integrabilitet. Till att börja med kommer vi att fördjupa oss i de grundläggande begreppen inom representationsteori och skapa en gedigen grund för vår efterföljande analys.  Diskussionen kommer sedan att gå vidare till alla möjliga generatorer i SCFTs, och förklarar hur de realiseras i termer av bosoniska och fermioniska oscillatorer. Slutligen kommer spinnkedjor och dess tillämpningar i 6D SCFTs att undersökas. Vi kommer visa att symmetriargument som härleds från representationsteori inte är  tillräckliga för att fastställa integrerbarhet av spinnkedjor i 6D SCFTs. Denna slutsats innebär inte att integrerbara system inte existerar inom 6D SCFTs, utan föreslår att det finns andra potentiella metoder, till exempel korrelationsfunktioner, för att utforska existensen av integrerbara system i 6D SCFTs.

Nanoelectronics is a field undergoing rapid development, meaning knowledge of the materials and methods used in nano-scale systems is a driving force in the industry. Silicon is a well known material in nanoelectronics commonly used as a semiconductor and is therefore a good representative for nanomaterials in general. In this thesis work the effects of the helium and neon ions with the energies 100 keV and 200 keV respectively on surface contaminants and the bulk material of nanometer thick silicon membranes are being studied. Beyond interactions based on different incident ions, the effects are studied inboth the geometries of transmission and backscattering, giving information about the immediate effects on the surface, as well as bulk effects. Using Medium Energy Ion Scattering (MEIS), the positively charged particles on the surfaces and in the bulk of the materials, which are either sputtered or desorbed, can be detected. While the ions are different, the energies in this work are chosen to be such, that the electronic stopping power is the same, while the nuclear stopping of neon is vastly higher. From this work, it is concluded that both ions have the same qualitative effects on the membrane contaminants, consisting of hydrocarbons, which are desorbed electronically. Furthermore, neon has the effect of destructively sputtering the bulk material. A synergistic effect of electronic and nuclear deposition was also found, as quantitatively, more hydrocarbons per incident Ne+ ion were desorbed than per incident He+ ion. The change in effect based on different geometries can to a large extent be attributed to the energy loss of the ions in the material. The one change between geometries which can not be explained by energy loss in the material, is a 50% under representation of desorbed hydrogen ions in transmission geometryfor He+ incident ion. It is also concluded that the method used has the potential to be a viable, non-destructive and scalable cleaning and measuring method for contaminations on nano-scale materials, such as 2 dimensional materials. / Nanoelektronik är ett område under snabb utveckling, vilket gör kunskap kring de material som bygger upp elektroniken drivande i utvecklingen av teknologin. Kisel är ett välkänt material inom nanoteknologin då det är välanvänt som halvledare, vilket gör det till en bra representant för en mängd andra nanomaterial. I detta examensarbete undersöks de effekter som 100 keV helium- och 200 keV neonjoner har på ytföroreningar och bulkmaterial av kiselmembran i nanometertjocklek. Utöver olika jonslag undersöks även skillnaderna mellan membranets geometri i transmission och backåtspridning. Detta ger information, både om de direkta effekterna som jonerna har på materialets bestrålade yta, samt effekterna på den motsatta ytan. Genom användningen av ”Medium Energy Ion Scattering” (MEIS) kan de positivt laddade partiklarna på ytan och i bulkmaterialet, som genom elektronisk och nukleär växelverkan bryts läs, utforskas. Partiklarnas energier har valts så att den elektroniska bromsförmågan är densamma för de två partiklarna, medan den nukleära bromsförmågan skiljer sig kraftigt. Från detta arbete kan slutsatsen dras att båda joner har samma kvalitativa effekt på kiselmembranets föroreningar i form av kolväten. Dock har neon en betydlig nukleär, destruktiv effekt på bulkmaterialet. En synergistisk effekt har också kvantitativt observerats, där de elektroniska effekterna på kolväten är högre för neonjoner. De förändringar som påvisades i de olika geometrierna kan i stort förklaras av energiförlust av jonerna under passagen genom membranet. Den geometribaserade differens som inte följer förklaringen energiförlust är desorberat väte, vilket är 50 % mindre i transmissionsgeometri för heliumjoner. Den andra slutsatsen som kan dras från detta arbete är att den använda metoden har potentialen att utvecklas till en icke-destruktiv och skalbar metod för rengöring och mätning av föroreningar på nanomembran så som 2 dimensionella material.