# Theses - Applied Mathematics and Theoretical Physics

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Item Open Access Dualities and Categorical Structures from 2D UpPasquarella, VeronicaShow more String Theory is the most promising candidate unifying theory of fundamental interactions so far; however, the Standard Model (SM) still features many open questions. The present work aims at providing a step further towards reconciling the two, analysing part of the richness that underlying mathematical structures and dualities are able to provide in, both, gravitating systems and Quantum Field Theories (QFTs) alike. In doing so, our approach will be of the top-down kind. In particular, we will be relying upon the key tools of holographic duality and categorical algebraic geometry. The use of the former is justified by the lack of a non-perturbative formulation of String Theory, whereas the latter is dictated by the great advancement there has been in the past decades in studying algebraic varieties associated to moduli spaces, specifically Higgs and Coulomb branches. A fundamental step towards studying string theory vacua, and, ultimately their stability, is that of understanding the underlying mathematical structure of the QFT resulting from its dimensional reduction on Calabi-Yau (CY) manifolds, the latter being complex manifolds admitting a category theory description. In particular, the work of Kapustin, Rozansky and Saulina (KRS) has shown how this can be achieved in terms of a 3D TFT equipped with a 2-categorical structure. Our analysis develops in two main directions, namely on the gravitational, and supersym metric quiver gauge theory side. In both cases, our treatment focuses on lower-dimensional structures necessitating extensions and generalisations of well-established dualities and correspondences, specifically, holographic duality, homological mirror symmetry, and 3D mirror symmetry. As we shall see, the common ground in between the two paths taken in this treatment is the role played by amplitutdes in studying fundamental interactions and the properties of the vacuum structure, as well as the role played by dualities in understanding analytic results.Show more Item Open Access Dynamics of Chiral FermionsOnder, KaanShow more This thesis studies the dynamics of a variety of two and four dimensional quantum field theories containing Weyl fermions respecting some chiral symmetry. There are severe challenges in lattice regularising such theories and chiral gauge theories, where a non-anomolous chiral symmetry is gauged, can display interesting strong coupling dynamics such as confinement without chiral symmetry breaking. We explore such gauge dynamics in two and four dimensions using a variety of different techniques. Furthermore, we use tensor network methods to study chiral fermions on the lattice in two dimensions. We start by studying the dynamics of chiral $SU(N)$ gauge theories in four dimensions. These contain Weyl fermions in the symmetric or anti-symmetric representation of the gauge group, together with further fermions in the fundamental and anti-fundamental. We revisit an old proposal of Bars and Yankielowicz who match the ‘t Hooft anomalies of this theory to free fermions. We show that there are novel and, in some cases, quite powerful constraints on the dynamics in the large $N$ limit. In addition, we study these $SU(N)$ theories with an extra Weyl fermion transforming in the adjoint representation. Here we show that all $21$ ‘t Hooft anomalies for global symmetries are matched with those of a Spin$(8)$ gauge theory. This suggests a non-supersymmetric extension of the duality of Pouliot and Strassler. We then discuss some non-supersymmetric dualities with vector-like matter content for $SO(N)$ and $Sp(N)$ gauge theories and the constraints imposed by Weingarten inequalities. We then move on to study the dynamics of analogous chiral gauge theories in two dimensions which also contain Weyl fermions in the symmetric, antisymmetric, and fundamental representations. A consistent infrared limit of these theories consists of certain coset conformal field theories. There is also a free-fermion phase which shares the same central charge and ’t Hooft anomalies but does not coincide with the coset models. We show that these two theories sit on a conformal manifold of infrared theories and are related by a current-current deformation. We further consider extensions of these theories by adding Dirac fermions and comment on possible renormalization group flows. Finally, we present matrix product state simulations of the 3450 lattice chiral fermion model in two dimensions. We consider a lattice setup introduced by Wang and Wen which realises two left and two right-moving fermions on one edge of a thin Chern insulator. The partner mirror fermions are localised on the opposing edge. We turn on symmetry preserving six-fermion gapping interactions on one edge of the Chern insulator to gap the doublers whilst preserving an anomaly free $U(1)$ chiral symmetry and leaving the opposing edge gapless. This provides a candidate lattice regularisation of chiral fermions. We present numerical results for the entanglement entropy scaling to extract the central charge and study excited states by using a quasi-particle ansatz. We observe a BKT transition at six-fermion coupling strength $g=7$ and a central charge $c=2$ scaling regime at $g=25$. We conclude by discussing subtleties of working with symmetric MPS at a fixed charge density.Show more Item Open Access Dynamics of super-absorbent hydrogelsWebber, Joseph; Webber, Joseph [0000-0002-0739-9574]Show more This thesis explores the behaviour of hydrogels, a broad class of materials comprising a hydrophilic polymer scaffold surrounded by adsorbed water molecules, potentially comprising over 99% water by volume. In general, hydrogels are soft, elastic, porous materials that can swell or dry to a significant degree by imbibing or expelling water. Any modelling of their behaviour must take into account the interplay between elasticity, osmotic effects arising from the attraction of water to the polymer, the pressure-driven flow through their porous structure and conservation of water and polymer. Owing to the large swelling or drying strains seen in super-absorbent gels, linear theories fail to predict the dynamics seen in experiments, so we introduce a new `linear-elastic-nonlinear-swelling' theory that linearises with respect to small deviatoric shearing strains but allows for nonlinearity in the isotropic strains that result from volumetric change. This theory is founded on three material parameters describing any gel (a shear modulus, an osmotic modulus and a permeability), all of which depend on the local polymer fraction and are macroscopically measurable, agnostic of the particular model used to describe the microscopic structure of the gel. In effect, modelling a gel in this manner is the same as treating a hydrogel swollen to any degree as its own distinct linear-elastic material. Swelling and drying are driven by the accumulation or expulsion of water within the matrix, with flows driven by gradients in pore pressure, and these gradients can be deduced by a momentum balance between pore pressures, osmotic pressures and elastic stresses. Given these theoretical foundations, we can solve a number of gel swelling and drying problems, using the continuum-mechanical foundations introduced here to describe the physical processes describing the transient state as water flows through the matrix, and the dependence of the gel's behaviour on its material properties. This theory underlines the importance of deviatoric stresses in understanding the dynamics of hydrogels, showing how the dynamics of three-dimensional swelling is qualitatively different from simple one-dimensional models, and underlining a distinct difference between the dynamics of gels and other colloidal materials where such stresses do not arise. Furthermore, it is seen how differential swelling introduces shear stresses and sets the shape of hydrogels, forming curved interfaces and wrinkled surfaces. It is also shown how our framework can be used to understand interfacial instabilities at the swelling front, with the patterns resulting from a complex interplay between elasticity and osmotic effects. Separating out the contributions of these two driving processes results in a rich range of phenomena exhibited at different stages during the swelling process, and can be used to explain the formation, development and healing of patterns seen in experiments. Finally, two extensions to this modelling are illustrated, underlining the utility of our poroelastic approach. First, the freezing of hydrogels is discussed, which results in phase separation behaviour as water is driven out of the polymer matrix to form pure ice and a partially-dried hydrogel from which water has been expelled. Second, we incorporate surface tension effects at the interface between gels and water, an effect that can not only modify the behaviour discussed in earlier chapters, but also gives rise to novel qualitative phenomena including the bulk transport of interstitial fluid and the suppression of instabilities.Show more Item Open Access Parton Distributions in Beyond the Standard Model TheoriesMoore, James; Moore, James [0000-0002-0066-0362]Show more Parton distributions are a key ingredient of precise predictions for collider experiments. They are usually determined from fits to experimental data under the assumption that the Standard Model (SM) of particle physics is complete; however, this can bias studies of beyond the Standard Model (BSM) physics if these SM-like PDFs are used in these analyses. It is important to quantify the extent to which this occurs, in order to avoid making incorrect conclusions about BSM physics. We begin in Chapter 1 with a review of perturbative quantum chromodynamics (QCD) and parton distribution functions (PDFs), providing a definition of the PDFs at next- to-leading order in QCD perturbation theory. At the end of the Chapter, in Sect. 1.4, we introduce the main problem that this thesis aims to address in a variety of special cases, namely the simultaneous extraction of PDFs together with other theory parameters (specifically BSM theories). In Chapters 2, 3 and 4, we describe the interplay between PDFs and the parameters of various BSM models. In more detail, in Chapter 2, we perform an approximate simultaneous extraction of PDFs together with the parameters of a dark photon model; in particular, we use projected high-luminosity LHC (HL-LHC) data to investigate the sensitivity of the HL-LHC to our particular class of light, leptophobic dark photons. Subsequently, in Chapter 3, we introduce the Standard Model Effective Field Theory (SMEFT), and carry out a simultaneous determination of PDFs together with two parameters drawn from the SMEFT; we show that at the HL-LHC, there will be significant interplay between extraction of PDFs and SMEFT parameters. In Chapter 4, we perform a much more comprehensive analysis of the PDF-SMEFT interplay in the top sector, using a new efficient methodology, SIMUnet. Importantly in Sect. 4.7, we also comment on the efficacy of the Monte Carlo replica method for error propagation, which forms the heart of the uncertainty calculation in both the NNPDF and SIMUnet methodologies. In the second half of this thesis, we focus on future issues in PDF fitting, related to the work presented in the previous chapters. In Chapter 5, we explore how New Physics in the data might be inadvertently ‘fitted away’ into the PDFs, if the data is treated as SM-like. We also recommend strategies for disentangling PDFs and BSM effects. Finally, in Chapter 6, we discuss the Monte Carlo replica method used in many of the previous chapters, and discuss the need for its replacement in future PDF and BSM fits.Show more Item Open Access Modelling the propagation of subglacial floodsTobin, SophieShow more Subglacial flooding, in which large volumes of water are suddenly released beneath a glacier, is a process which has the potential to significantly modify the dynamics of the overlying ice. The routing of the water beneath the glacier and the extent of its incorporation into existing drainage networks determines the response of the ice. As a result, modelling of subglacial flooding is both necessary for understanding the detailed dynamics of glaciers responding to meltwater and also a useful test case for investigating the properties of the contact between glacial ice and the bed on which it sits. A number of studies have looked at the initial axisymmetric spreading of subglacial floodwater by considering the coupling between the flow of the water and the elastic deformation of the ice. Other studies have examined the movement of the water downstream, but without modelling the detailed, potentially elasticity-controlled dynamics at the flood front. In this thesis I combine these two approaches in order to model the processes which determine the propagation speeds of subglacial floods and their impact on the overlying ice. In chapter 1 I discuss subglacial flooding in the broader context of subglacial drainage systems and review previous modelling approaches. In chapter 2 develop a model for flood propagation beneath glaciers by considering the behaviour of a blister of water trapped between a rigid sloping base and an elastic sheet. I use an asymptotic analysis to show that, by removing a jump in curvature otherwise present at the upslope edge, the presence of a sloping base results in a new, nearly-translating regime in which the body of the blister moves at an approximately constant speed, leaving behind a thin layer of fluid. In chapter 3 I compare this model to GPS observations of six different subglacial flooding events. The observed uplifts are compared to those predicted by the model and processes at the front of the blister which could regulate flood propagation speeds are discussed. Linking observations of ice acceleration to the hydraulic jacking of the ice caused by subglacial floods requires combining both viscous and elastic deformation, so in chapter 4 I investigate the impacts of viscoelasticity on ice dynamics. To explore potential effects, I investigate the impact of viscoelastic bending on the movement of grounding lines. I then develop a model for viscoelastic bending and stretching of ice which I discuss in the context of subglacial flooding. In chapter 5 I conclude and discuss future directions for this work.Show more Item Open Access Information and generative deep learning with applications to medical time-seriesEdinburgh, Tom; Edinburgh, Tom [0000-0002-3599-7133]Show more Physiological time-series data are a valuable but under-utilised resource in intensive care medicine. These data are highly-structured and contain a wealth of information about the patient state, but can be very high-dimensional and difficult to interpret. Understanding temporal relationships between time-series variables is crucial for many important tasks, in particular identifying patient phenotypes within large heterogeneous cohorts, and predicting and explaining physiological changes to a patient over time. There are wide- ranging complexities involved in learning such insights from longitudinal data, including a lack of a universal accepted framework for understanding causal influence in time-series, issues with poor quality data segments that bias downstream tasks, and important privacy concerns around releasing sensitive personal data. These challenges are by no means unique to this clinical application, and there are significant domain-agnostic elements within this thesis that have a broad scope to any research area that is centred around time-series monitoring (e.g. climate science, mathematical finance, signal processing). In the first half of this thesis, I focused firstly on information and causal influence in time- series data and then on flexible time-series modelling and hierarchical model comparison using Bayesian methods. To aid these tasks, I reviewed and developed new statistical methodology, particularly using integrated likelihoods for model evidence estimation. Together, this provided a framework for evaluating trajectories of the information contained within and between physiological variables, and allowed a comparison between patient cohorts that showed evidence of impaired physiological regulation in Covid-19 patients. The second half of this thesis introduced generative deep learning models as a tool to address some of the key difficulties in clinical time-series data, including artefact detection, imputation and synthetic dataset generation. The latter is especially important in the future of critical care research, because of the inherent challenges in publishing clinical datasets. However, I showed that that there are many obstacles that must be addressed before large-scale synthetic datasets can be utilised fully, including preserving complex relationships between physiological time-series variables within the synthetic data.Show more Item Embargo Impact cratering with yield-stress fluidsIoannou, GeorgiaShow more Impact cratering is the process where a moving object hits a deformable target, causing material to be ejected away from the impact point, at least most of the times, and opening a crater on the target surface. This process has been studied extensively to understand the dynamics of planetary impact cratering and other similar natural or industrial processes. Most of the relevant experimental works involve water and granular media, and very few yield-stress fluids like soft materials. Except for the common experiment of a water drop impacting a water pool, in most works the impactor is a solid, non-deformable sphere. However, in the relevant geological process, and in most other applications, both target and impactor are deformable. Also, the material used in lab experiments to mimic the relevant geological process must have rheological properties that allow for it to hold a shape at the end of the process so that the resulted crater does not vanish. Ideal experimental materials are the yield-stress fluids, which behave as solids when low stresses are applied, but deform as fluids when the applied stresses exceed a threshold value. In this work, we conduct impact cratering experiments with a yield-stress fluid as both target and impactor. We explore many aspects of the time-dependent features of this highly transient process by recording the dynamics with high-speed cameras. The transient features we study are the transient cavity (air-gel interface) dimensions and shape, the spreading of the drop material upon impact, and the duration of the cavity growth. The dynamics of this transient process are considered using an energy balance. We find that only a small percentage of the impactor kinetic energy is converted into potential energy of the cavity, unlike Newtonian fluids. Here, most of the impactor kinetic energy is converted into elastic energy stored in the material. A particle tracking method is employed to visualise the response of the target material upon impact. Interestingly, the cavity does not grow radially as a hemi-sphere, like in Newtonian fluids, but growth is faster in horizontal than in vertical direction. Additionally, growth in vertical direction ceases before that in the horizontal direction. After the crater is formed, the target material undergoes a damped oscillation for a time period 50 times greater than the duration of cavity growth. We explore the dependence of the period of oscillation on material properties and examine whether the material oscillates in phase everywhere in the target. Our study of the transient features expands to the ejecta sheet that emerges from the target, which is primarily material expelled from the point of impact. We perform a qualitative study of sheet shapes, categorising the ejecta into regimes according to the instabilities that arise at the edge of the sheet. These regimes are determined by a single dimensionless number that compares the inertial stresses to the dissipative stresses of the flow. Additionally, we study the dimensions and shape of the ejecta sheet and how these quantities evolve with time and compare our findings with the ejecta emerging from water and granular impact cratering. When the transient part of the process finishes, a final crater that has a static shape in time is formed on the surface of the target. Using laser profilometry, we acquire the three-dimensional shape of the crater formed from which we categorise the different morphological regimes and examine how the final dimensions of the crater are related to its transient conformation. Moreover, we compare the size and shape of our craters with those reported in the literature when the target is a granular bed or a planetary body. We augment our experimental study of impact cratering with simulations that imitate the laboratory experiments. For the simulations we use OpenFOAM, an open-source software package, and investigate various constitutive models for non-Newtonian fluids. Only the cavity growth stage is studied, when the flow is presumed to be stable and axisymmetric. The size and shape of the transient cavity for the different models are compared with each other and with the experimental results. We conclude with a summary of our findings and a discussion of future directions of research.Show more Item Open Access Magnetic charges and phase space renormalization of gravityTomova, BilyanaShow more In the first part of this thesis we perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions – those that are analytic near I+ – admit a non-trivial action of the generalised Bondi-Metzner-van der Burg- Sachs (GBMS) group which contains infinite-dimensional supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff×Weyl transformations of the celestial S4. Using the covariant phase space formalism and a new technique which we present in this thesis (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are local and covariant. The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. In the second part of this thesis, we study the dual charges of N = 1 supergravity in asymptotically flat spacetime. The action considered is the usual supergravity action with a topological contribution. This is the Nieh-Yan term and the magnetic term of the free Rarita-Schwinger field. Through methods of the covariant phase space formalism we construct the charges conjugate to supersymmetry, diffeomorphism and Lorentz transformations. The additional term in the action will lead to new, non-vanishing contributions to these charges. The magnetic diffeomorphism charges are equivalent to the ones previously found for gravity, while the dual supersymmetric charges are new and do not appear for the free Rarita-Schwinger field. We find that the asymptotic symmetry group for supergravity can only include global conformal transformations on the celestial sphere.Show more Item Open Access Quasinormal Modes of Nearly Extremal Black HolesJoykutty, Jason; Joykutty, Jason [0000-0003-4742-9480]Show more Quasinormal modes are the gravitational wave analogue to the overtones heard after striking a bell; like many physical systems, black holes emit radiation as a response to perturbations. After a dynamical event, for example a black hole merger, the system is expected to relax to a stationary black hole solution. After sufficient time, the system can be treated as a perturbation to this stationary solution in what is called the ringdown phase. The observed gravitational wave signal is dominated by the ringing associated with these solutions to the linear perturbation equations in this period of the evolution. Each quasinormal mode is characterised by a complex frequency which encodes its behaviour in time: the imaginary part determines its oscillation and the real part its exponential decay. In light of the observation of gravitational wave signals in the past few years, quasinormal modes are important from an astronomical perspective. By comparing the observed gravitational wave signal from some dynamical event with the predictions provided by computing quasinormal frequencies, one can compare the fit given by general relativity against some modified theory of gravity and test which is a better model for these phenomena. This black hole spectroscopy could also be used to deduce the parameters of an astrophysical object from the gravitational wave signal. As horizons become extremal, various computations (from a range of authors including Detweiler, Hod and Zimmerman) have shown that in many cases, there exists a sequence of frequencies which become purely oscillatory in the limit and which cluster on a line in the complex plane. These zero-damped modes are typically the most slowly decaying resonances of the equation and hence are key to understanding stability. In the case of a positive cosmological constant, they are closely tied to the Strong Cosmic Censorship Conjecture: if the spectral gap is too large, the modes don't decay slowly enough to destabilise the Cauchy horizon. From the large variety of examples in the literature of nearly extremal black holes with zero-damped modes, it is natural to conjecture that this phenomenon is generic. This thesis explores mathematically rigorous results that can be obtained toward resolving this question. In particular, we shall review the literature on quasinormal modes (focussing on zero-damped modes), discuss the mathematical definition of these objects and the idea of co-modes or dual resonant states: solutions to the adjoint problem which can make identifying the frequencies easier. Finally, we shall use this framework and Gohberg-Sigal theory to prove existence results for zero-damped modes: firstly in the case of wave equations with potentials which decay sufficiently rapidly, then for a large class of static, spherically symmetric black hole spacetimes. There are also partial results toward resolving the question for the Kerr-de Sitter spacetime.Show more Item Open Access The Universe through a magnifying glass: Precision cosmology with CMB lensingQu, Frank JiatianfuShow more The cosmic microwave background provides a unique back-light for illuminating the growth of structures in our universe. Measuring the arcminute-scale lensing deflections experienced by the CMB photons as they travel from the last scattering surface to our telescopes enables the mapping of the matter distribution to very high redshifts. This lensing signal provides a clean window for constraining fundamental physics, such as the sum of neutrino masses, structure growth, and the nature of dark energy. Among other applications, precisely measuring this lensing signal will enable robust tests of the standard cosmological model via the comparison of high-precision measurements of structure growth at late times with predictions. This thesis explores several uses and applications of CMB lensing. On the theory side, we construct novel statistical methods to measure lensing, such as using the shear estimator on the full sky to be more robust to extragalactic foregrounds in Chapter 4. Chapter 3 exploits the synergy between CMB lensing and other probes of structure growth. It discusses the method of combining CMB lensing map with appropriately scaled large-scale structure tracers to construct a high-redshift mass map and leverage more robust inference of cosmological parameters and sample variance cancellation. On the data side, the main section of the thesis focuses on CMB lensing measurements obtained from data release 6 of the Atacama Cosmology Telescope. This work provides a state-of-the-art lensing power spectrum measurement at a significance of $43\sigma$ and an associated signal-dominated lensing mass map that enable a host of cosmological and astrophysical science goals. This lensing measurement, largely independent of measurements from *Planck* or galaxy survey data, provides a novel avenue to obtain information about large-scale growth and new insight into potential tensions in structure formation. The thesis also discusses novel methods to tackle key systematics affecting precision ground-based CMB lensing. These include using cross-correlation-based lensing estimators robust to noise modelling and a repertoire of foreground mitigation techniques for suppressing the contamination from extragalactic foregrounds. Two hundred null tests accompany the analysis to ensure the measurement is free from unmitigated systematic effects. The lensing analysis and pipeline used here provide a foundation for high-resolution, ground-based lensing measurements covering a significant portion of the sky. This framework will be used for ongoing analyses of ACT data incorporating day-time observations from 2017-2022 and night-time data recorded in 2022. Moreover, the analysis presented here paves the way for upcoming surveys like the Simons Observatory.Show more Item Open Access Deep Learning Approaches for PDE-based Image Analysis and Beyond: From the Total Variation Flow to Medieval Paper AnalysisGroßmann, TamaraShow more Partial differential equations (PDEs) play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences, as well as in engineering and computer science. Image analysis is a prime example for a field where PDEs have triggered many innovations. One such PDE that has gained considerable attention in the last few years is the total variation (TV) flow. The TV flow generates a scale-space representation of an image based on the TV functional. This gradient flow has desirable features for images such as sharp edges and enables spectral, scale, and texture analysis. Based on the solution to the TV flow, a non-linear spectral decomposition can be derived. Due to its ability to extract spectral components corresponding to objects of different size and contrast, such decompositions enable filtering, feature transfer, image fusion and other applications. However, obtaining the spectral TV decomposition involves solving multiple non-smooth optimisation problems to solve the governing PDE - the TV flow - and is therefore computationally highly intensive. In the first part of the thesis, we present a supervised neural network approximation of the spectral TV decomposition which significantly speeds up its numerical solution. We report up to four orders of magnitude speedup in processing of mega-pixel size images, compared to classical GPU implementations of spectral TV. Our proposed network, the TVspecNET, is able to implicitly learn the underlying PDE and, despite being entirely data-driven, inherits equivariances of the model-based transform. To the best of our knowledge, this is the first approach towards learning a non-linear spectral decomposition of images. The TVspecNET, however, is designed as a supervised learning approach and in that relies on ground truth data. It is additionally constrained to produce fixed spectral bands of the image. We therefore extend the work to learn the TV flow solution in the third part of the thesis. Learning the solution to PDEs has been a rapidly growing area at the intersection of machine learning and PDEs. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. So-called physics-informed neural networks (PINNs) and their variants have shown to be able to successfully approximate a large range of PDEs. However, before the advent of deep learning, many classical numerical methods had been developed to approximate PDE solutions on a discrete level. The finite element method (FEM), for instance, is one standard methodology to do so. So far, PINNs and FEM have mainly been studied in isolation of each other. In the second part of the thesis, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and non-linear PDEs: the Poisson equation in 1D, 2D, and 3D, the Allen-Cahn equation in 1D, and the semilinear Schrödinger equation in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, PINNs have not been able to outperform FEM in our study. In some experiments, they were faster at evaluating the solved PDE. In the third part of the thesis, we consider the deep learning approximation of the TV flow solution. Compared to the TVspecNET that learns the entire spectral TV decomposition pipeline, this unsupervised approach is inspired by the PINN framework and is more flexible in terms of scale representation and does not require ground truth data. Computing the TV flow is challenging because the subdifferential of TV is not a singleton unless the image has no constant regions. Numerical methods amount to either modifying the gradient of the image in constant regions to make sure that the subdifferential is single-valued or an implicit scheme which requires solving multiple non-smooth optimisation problems. The first option includes FEM approaches, however, due to the gradient modifications this introduces artefacts. The second option is the classical approach to solve the TV flow. Even with state-of-the-art convex optimisation techniques, this is often prohibitively expensive and strongly motivates the use of alternative, faster approaches. Inspired by and extending the framework of PINNs, we propose the TVflowNET, an unsupervised neural network approach to approximate the solution of the TV flow given an initial image and a time instance. We require no ground truth data but rather make use of the PDE for optimisation of the network parameters. We circumvent the challenges related to the subdifferential by additionally learning the related diffusivity term. We significantly speed up the computation time and show that the TVflowNET approximates the TV flow solution with high fidelity for different image sizes and image types. Additionally, we give a full comparison for different network architecture designs as well as training regimes to highlight the fidelity of our approach. The last part of the thesis concerns the application of the spectral TV decomposition to medieval paper analysis. Medieval paper, a handmade product, is made with a mould which leaves an indelible imprint on the sheet of paper. This imprint includes chain lines, laid lines and watermarks which are often visible on the sheet. Extracting these features allows the identification of paper stock and gives information about chronology, localisation and movement of manuscripts and people. Most computational work for feature extraction of paper analysis has so far focused on radiography or transmitted light images. While these imaging methods provide clear visualisation for the features of interest, they are expensive and time consuming in their acquisition and not feasible for smaller institutions. However, reflected light images of medieval paper manuscripts are abundant and possibly cheaper in their acquisition. We propose algorithms to detect and extract the laid and chain lines from reflected light images. We tackle the main drawback of reflected light images, that is, the low contrast attenuation of chain and laid lines and intensity jumps due to noise and degradation, by employing the spectral TV decomposition and develop methods for subsequent chain and laid line extraction. Our results clearly demonstrate the feasibility of using reflected light images in paper analysis. This work enables the feature extraction for paper manuscripts that have otherwise not been analysed due to a lack of appropriate images. We also open the door for paper stock identification at scale.Show more Item Open Access Chaos in models of double convectionRucklidge, Alastair MichaelShow more This dissertation concentrates on the derivation and analysis of low-order sets of ordinary differential equations (ODEs) that accurately describe the behaviour of a fluid in convective motion. A second-order set of ODEs is presented and analysed, and then related to a particular double convection problem (compressible convection in a vertical magnetic field); the low-order model proves to be useful in interpreting the behaviour of the full system. Equations describing several types of double convection (convection in a magnetic field, convection in a rotating layer of fluid and convection in a solute gradient) are reduced to low-order sets of ODEs that are asymptotically exact descriptions of the partial differential equations (PDEs) from which they were derived. The ODE model for incompressible convection in a vertical magnetic field is analysed in detail, and a rich variety of periodic orbits and chaotic behaviour is found. A numerical study of the full set of PDEs for this case confirms that the low-order model provides an asymptotically correct description of the full problem; in particular, the PDEs have the chaotic solutions predicted by the low-order model.Show more Item Open Access On the Factorisation of Matrix Wiener–Hopf Kernels Arising From Acoustic Scattering ProblemsAitken, MungoShow more The research undertaken in this thesis is in the broad area of diffraction theory. We consider three separate and distinct problems of acoustic scattering with rectangular geometries, which have a common underlying mathematical structure. The geometries are: the infinite wedge, the waveguide with a barrier, and the semi-infinite plate of finite thickness. It turns out that these problems may be formulated as matrix Wiener–Hopf problems with the special property that their matrix kernels $\mathsf K$ may be formulated as $\mathsf K = \mathsf M^{-1} \mathsf J \mathsf M$, where $\mathsf J^2 = \mathsf I$, the identity matrix. This special property makes the problems amenable to factorisation which enables an exact solution to be derived, at least in theory. In practice, in two of the cases, we end up with an infinite system of equations which must be truncated to allow for practical computation of coefficients. However, these coefficients are rapidly convergent aided by the use of a novel technique termed the `corner singularity method', in which the integration contour of an integral is shifted upwards in the complex plane to pick up a contribution from the infinite 'tail'. This work has applications in industrial and marine acoustics, and bears promise of fruitful extension to elastodynamics and other areas of wave theory.Show more Item Controlled Access The Many Phases of the Surface Code: Coherent Errors and Many-Body LocalisationVenn, FlorianShow more This thesis investigates the far-from-ground-state physics of the surface code, in particular its quantum error correction applications and formulations. We contribute to this field via two lines of research: we study the behaviour of the surface code under coherent errors, which create superpositions of excited states, and we probe topological many body localization (MBL) which protects topological order for all eigenstates. In the first strand, we develop an interpretation of the error correction threshold for coherent error rotations as a phase transition. For this, we first generalize a numerical method for the simulation of coherent errors in surface codes on square lattices to work with surface codes on general planar graphs. This method is based on a mapping to a free fermion model which allows calculating the expectation values using fermion linear optics. Using this method, we show that the connectivity of the graph can shift the error correcting performance between resilience against *X*- and *Z*-rotations. Building on this work, we further explore the relationship between coherent errors in surface codes and free fermion models. We develop a formalism to map the surface code under coherent errors to a complex Ising model and from there to a Majorana fermion scattering model. We analyze its conductivity and find that for rotations below the error correction threshold the resulting model is an insulator, and it becomes a metal above the threshold. By estimating the position of this phase transition, we obtain the achievable error correction threshold for coherent errors. The second line of research is focused on the disordered and perturbed toric code. We implement a recently proposed method that numerically approximates the local integrals of motion that are present in (topological) MBL phases using sets of stabilizers that are dressed by optimized quantum circuits. First, we apply this method to the disordered Kitaev chain as a benchmark. Then, we proceed by adapting it to the toric code. We show how it can be used to distinguish topological and trivial MBL and how it can be combined with exact diagonalization to obtain an approximate phase diagram.Show more Item Open Access On the Relationship between Canonical Quantum Gravity and the Holographic PrincipleAraujo Regado, GoncaloShow more This thesis explores the connection between two approaches to the problem of quantum gravity. On the one hand, we have the canonical approach which imposes the gauge constraints on the physical states. This leads to the notoriously hard problem of solving the Wheeler-deWitt (WdW) equation. On the other hand, we have the holographic principle, which defines the gravitational path integral in terms of the partition function of a non-gravitational CFT living on the boundary, leading to the flourishing field of the AdS/CFT correspondence. The connection between the two becomes clear after a reformulation of the holographic principle in which the emergent dimension is time instead of space. For that we need to consider Euclidean field theories living on a slice of space. They are defined starting from the usual type of holographic CFTs followed by a special type of deformation called the $T^2$ deformation. Such partition functions solve the WdW equation, thus providing canonical quantum states of the bulk theory. The deformation flow is uniquely fixed by the bulk gauge constraints and it has several exotic properties. This formulation extends the AdS/CFT framework naturally to other quantum gravity scenarios. We explain the what, how and why of the $T^2$ deformation in quantum gravity by studying general solutions to the WdW equation. This leads naturally to an explicit map between field theory states living on the boundary of space and quantum gravity states living on the bulk of space. This is a manifestation of the holographic principle, hiding inside the WdW equation. We also propose a reconstruction of the boundary state from bulk data. We conjecture about an isomorphism between the quantum gravity and field theory Hilbert spaces. The dynamics of the boundary state with respect to boundary time is shown to induce a time evolution of the quantum gravity state. We discuss, at several points in the thesis, how the bulk theory manages to keep being unitary, despite the lack of unitarity of the deformed field theory. Along the way, we also propose a more general version of the holographic principle in the language of equating bulk and boundary path integrals. We discuss at length the application of this formalism to quantum cosmology. This requires us to consider complexified deformations. Crucially, we are forced to consider superpositions of field theory branches in order to describe the bulk. This leads to several discussions about the structure of quantum gravity and its hypothetical UV completion. In particular, we discuss the phenomenon of spontaneous CPT breaking for the UV completion of the $T^2$-deformed theory along its RG flow. The partition function is computed explicitly in minisuperspace, touching base with previously known solutions to the WdW equation applied to this restricted toy model. We then go on to conjecture that the choice of lapse contour in the gravitational path integral is intimately related to the superposition of field theory branches and, therefore, to the different UV completions for the holographic dual. All these features point in the direction of the long-standing conjecture that there is a unique quantum state of the universe.Show more Item Open Access Rare events and dynamics in non-equilibrium systemsKikuchi, TakaakiShow more The matter of this thesis is divided in two parts, both of which are substantially different from the other, but nevertheless belong to disciplines that lie within the purview soft matter physics. In the first part, we study the infinite-dimensional probability space of stochastic differential equations. In particular, we study the transition path ensemble (TPE), the set of transition paths between meta-stable states of Ito diffusions. In the limit of vanishing diffusivity, the Freidlin-Wentzell action characterises the asymptotics of the path-probability distribution over the TPE. We develop spectral Ritz methods to efficiently find minimisers of this action, and to construct quasipotentials of steady-state distributions, and we test our algorithm on a number of benchmark systems. To study the TPE in the finite temperature regime, we develop an MCMC algorithm to sample the infinite-dimensional space of transition paths, which we call the *teleporter MCMC*. The algorithm was designed to efficiently sample the TPEs of Ito diffusions with multiple competing transition channels, avoiding the issue of slow-mixing common to MCMC schemes. We concluded this part of the thesis by applying our MCMC method to study the temperature-dependence of the TPE. Using two model systems, we show that the dominant transition channel does not in general coincide with the most probable path of the path distribution, even in a low-to-intermediate temperature regime. In the second part of this thesis we develop a general theory of the geometric mechanics of a broad class of microstructured continuum systems. Specifically, we consider systems with configuration spaces that are either Lie groups, or homogeneous spaces. We demonstrate that this theory, which we call a generalised geometric Cosserat theory (GGCT), can be seen as a unifying framework with which to study classical Cosserat systems, and numerous non-classical variations. As a paradigmatic example we first study the Cosserat rod model, we identify its configuration space as a curve in $SE(3)$, the Lie group of translations and rotations on Euclidean space, and use the Lie algebra-Lie group correspondence to relate its configuration to curves in the Lie algebra. Using the Euler-Poincaré theorem we then proceeded to formulate the dynamics of the Cosserat rod on the dual Lie algebra. The resulting kinodynamical - kinematic and dynamic - theory of the Cosserat rod is defined completely on the trivialisation of the tangent bundle of $SE(3)$, the Lie algebra $\mathfrak{se(3)}$. We then constructed the GGCT by extrapolating these above steps to systems with generalised configuration spaces. In the final chapter of this thesis, we constructed geometric numerical integrators designed to preserve the qualitative features of the system geometry.Show more Item Open Access Robustness of Fixed Points of Quantum ProcessesSalzmann, RobertShow more The thesis combines two independent lines of research, both of which lie in the general area of the theory of robustness of fixed points (or invariant states) of quantum processes. In the first part of the thesis, we address the following question: Given a quantum channel and a quantum state which is almost a fixed point of the channel, can we find a new channel and a new state, which are respectively close to the original ones, such that they satisfy an exact fixed point equation? This question can be asked under many interesting constraints in which the original channel and state are assumed to have certain structures which the new channel and state are supposed to satisfy as well. We answer this question in the affirmative under fairly general assumptions on afore-mentioned structures through a compactness argument. We then concentrate on specific structures of states and channels and establish explicit bounds on the approximation errors between the original- and new states and channels respectively. We find a particularly desirable form of these approximation errors for a variety of interesting examples. These include the structure of general quantum states and general quantum channels, unitary channels, mixed unitary channels and unital channels, as well as the structure of classical states and classical channels. On the other hand, for the setup of bipartite quantum systems for which the considered channels are demanded to act locally, we are able to lower bound the possible approximation errors. Here, we show that these approximation errors necessarily scale in terms of the dimension of the quantum system in an undesirable manner. We apply our results to the robustness question of quantum Markov chains (QMC) and establish the following: For a tripartite quantum state we show the existence of a dimension-dependent upper bound on the distance to the set of QMCs, which decays as the conditional mutual information of the state vanishes. In the second part of the thesis we prove the so-called quantum Zeno- and strong damping limits for infinite-dimensional open quantum systems. In the former case, which we refer to as the quantum Zeno regime, the dynamics of the open quantum system is governed by a quantum dynamical semigroup, which is repeatedly and frequently interrupted by the action of a quantum operation. The quantum operation is considered to be mixing, in the sense that if applied multiple times it converges to its fixed point space. We then analyse the effective dynamics of the overall process in the limit of the application frequency of the quantum operation going to infinity. The strong damping regime can be considered as a continuous variant of the quantum Zeno regime. Here, the discrete and frequent action of the quantum operation is replaced by an additional term in the generator of the dynamical semigroup, whose individual dynamics is mixing, in the sense that it converges to its fixed point space in the infinite time limit. We analyse the overall dynamics in the limit of infinite interaction strength. All previous proofs of quantum Zeno limits in the literature relied on an assumption given by a certain spectral condition. We give a full characterisation of quantum operations which are mixing in the uniform topology under this assumption. Then, using a novel perturbation technique, we are able to go beyond this assumption and prove quantum Zeno- and strong damping limits in an unified way, if the mixing happens in the strong sense, i.e. pointwise for a given state. Here, we see that the effective processes converge to the fixed point spaces, on which they are governed by an effective quantum Zeno dynamics. The result is quantitative and gives a bound on the speed of convergence of the quantum Zeno- and strong damping limits, given a bound on the speed of convergence of the mixing process.Show more Item Open Access Human mobility and spatial models for infectious diseaseTang, Maria; Tang, Maria Lan [0000-0002-9671-8302]Show more Human mobility is an important determinant for the spatial spread of human infectious diseases such as influenza but obtaining human mobility datasets has historically been difficult. This thesis investigates two ways to represent human mobility in spatial metapopulation models for the spread of influenza in the US and UK – using gravity models with data-based distance metrics and using survey mobility data from the BBC Pandemic project and the 2011 UK census. Our metapopulation models describe the spread of influenza on a network of geographically segregated subpopulations that make up the whole population. Interactions between subpopulations are characterised by the human mobility proxies, while homogeneous mixing is assumed within subpopulations. The choice of subpopulations can therefore potentially have a large influence on the model output, and so this thesis also considers how this choice of spatial scale for the aggregation of the human mobility data and for the model can affect the epidemic dynamics produced. Chapter 2 investigates the use of data-based distance metrics in a gravity model framework fit to influenza spread in the US. Given that people do not move via straight lines, we consider driving distance by road and driving time as alternative distance metrics to great-circle distance. Gravity models are fit to outbreak onset dates in the US for the 2009 A/H1N1pdm influenza pandemic and the 2003/04 and 2007/08 influenza seasons, derived from influenza-like-illness medical claims timeseries at the scale of 3-digit ZIP codes (ZIPs). Driving distance and time are found to give better gravity model fits than great-circle distance to this data and simulations highlight spatial differences in the spread predicted by the different distance metrics. Chapter 3 explores the effect that spatial scale of the data and model has on the results in the previous chapter and considers two spatial scales in addition to ZIPs: sectional centre facilities (SCFs) and states. We compare the results from using different scales for obtaining outbreak onset dates from the influenza-like-illness timeseries, model fitting to the outbreak onset dates, and simulating from the model parameters. The better modelling performance of driving distance and driving time compared to great-circle distance persisted at the SCF level but not at the state level. Chapter 4 describes the England mobility data from the BBC Pandemic citizen science project that recorded location data of participants via a mobile phone app in 2017-2018. Compared to the most widely used open-source England human mobility data in the last decade, the 2011 census commuter workflow matrices, the BBC location data is more recent and records the movement of a wider range of people and trips but is relatively sparser. To compare the two datasets, we aggregate the BBC data into origin-destination matrices and fit competing destination models, an extension of the gravity model, to both BBC and census mobility data at three spatial scales: local authority districts (LADs), upper tier local authorities (UTLAs) and regions. Model preference was similar between datasets and scales, but parameter estimates differed. Chapter 5 uses the fitted mobility matrices in the previous chapter in a compartmental metapopulation model for influenza disease spread in England to compare simulated output from using the BBC and census mobility datasets. The resulting simulated epidemic dynamics are evaluated at the three scales (LADs, UTLAs, regions). Additionally, Chapter 6 presents a retrospective analysis of another source of survey data – for coughs, colds, and influenza-like illness in the University of Cambridge from 2007-2008. This self-reported data from university students and staff is one of the most detailed datasets of infectious respiratory disease in UK universities pre-COVID-19. Although a simple survey that comes with biases, it provides insights into risk factors for infectious disease in the relatively closed environment of a university and suggests ways in which future surveys could be carried out.Show more Item Open Access Exploring Non-Minimality in New Physics Beyond the Standard ModelBanks, HannahShow more The need to extend the Standard Model of particle physics is now well established with a multitude of observations heralding the existence of new physics beyond the realms of our present understanding. A plethora of new theoretical possibilities have been proposed to this end, each with vastly different microphysical realisations and in turn, phenomenological signatures. The notion of minimality has traditionally been appealed to as a guiding force in the organisation of our experimental explorations of this space to date, with a handful of simple benchmark scenarios receiving the lion's share of attention. With all dedicated searches for new physics as-yet returning null results however, it is becoming increasingly apparent that a more thorough survey of the diverse landscape of prospective theoretical models is required. This thesis considers a number of different ways in which we might introduce complexity into our searches for new physics beyond the Standard Model in order to probe previously unchartered theoretical territory. We begin in the arena of flavour physics where we re-interpret LHC search data to place exclusion bounds on a specific extension of the Standard Model which, in order to address both the hierarchy of the fermion masses and anomalies observed in meson decay processes, is non-trivial in its flavour structure. The latter part of this thesis then focuses on new physics relating to the dark sector. We begin by developing an entirely general analysis framework with which to structure searches for scalar operator `fifth forces' that may arise between Standard Model particles due to the exchange of new light states. By encapsulating the phenomenology of an extremely broad range of theoretical possibilities in terms of a single real, positive-definite spectral density function, we demonstrate that this approach enables exotic scenarios which go beyond the simplest possibility of tree-level scalar exchange to be considered with ease. We also show how this prescription provides the scaffolding to probe speculative violations of quantum field theoretic principles such as unitarity, causality and locality. Continuing along the lines of generalising searches for new light physics, we next apply ourselves to the phenomenon of neutrino oscillations. Here, we introduce a new, flexible language in which a diverse range of new physics effects on neutrino propagation, such as the existence of additional light neutrino species, are described by a single spectral function. We further demonstrate that the relevant phenomenology of a host of complex theoretical models can be conveniently approximated by way of a simple mass spectrum which comprises three `broadened' states. By allowing for a model-independent analysis of neutrino oscillation data, we illustrate how this phenomenological ansatz enables deviations from the canonical three-neutrino scenario to be probed in a systematic and general fashion. We finally turn to a specific possible manifestation of complexity in the dark sector - namely the formation of exotic compact objects. Provided such structures form binary systems, they may generate unique, identifiable signals at near future gravitational wave observatories sensitive to sub-Hz frequencies. We show that studying the gravitational wave background generated by the mergers of such objects may not only provide an indication of their existence but offer a unique opportunity to probe their properties and in turn, the dark sector states from which they are composed.Show more Item Open Access Contact and coalescence of viscous dropsBeaty, Edward; Beaty, Edward [0000-0001-6995-8645]Show more When two fluid drops come into contact, surface tension quickly pulls the drops together into a single larger drop. This coalescence process is an example of a singular flow resulting from a topological transition, in this case between the disjoint and connected drops. In this thesis, we consider theoretically the flow in two viscous drops with inertia either side of this topological transition. The deformation of drops prior to contact sets the initial shape of the drops for the subsequent coalescence. We consider a mechanism for contact between nearby, effectively stationary drops: when such drops are in sufficiently close proximity, van der Waals attraction between the drops overcomes surface tension and deforms the surfaces into contact. We solve for the viscous dynamics of this deformation both with and without inertia and find in each case a self-similar solution. The self-similar surface evolution determines the initial surface profile, and therefore the strength of the singularity in surface curvature, for the subsequent coalescence. At sufficiently early times, coalescence is described geometrically by a small fluid bridge over which the touching drops are in contact. At the edge of the fluid bridge the surface is tightly curved. The corresponding surface tension drives an expansion of the fluid bridge and consequently coalescence. We solve for the dynamics of viscous coalescence for general initial surface profiles set by a contact process. The strength of the singularity in surface curvature at contact determines the initial rate of coalescence at leading order. For drops with both viscosity and inertia, the early-time coalescence dynamics transitions between several regimes that are determined by the relative strengths of viscosity and inertia on the different length scales of the problem. We identify regimes in which the momentum imparted on the fluid by surface tension is confined to a viscous wake over the fluid bridge. Entrainment into the wake alters the drop profile ahead of the fluid bridge and subsequently alters the rate of coalescence.Show more