Theses - Applied Mathematics and Theoretical Physics
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Item Open Access Information and generative deep learning with applications to medical time-seriesEdinburgh, Tom; Edinburgh, Tom [0000-0002-3599-7133]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.Item Embargo Impact cratering with yield-stress fluidsIoannou, GeorgiaImpact 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.Item Open Access Magnetic charges and phase space renormalization of gravityTomova, BilyanaIn 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.Item Open Access Quasinormal Modes of Nearly Extremal Black HolesJoykutty, Jason; Joykutty, Jason [0000-0003-4742-9480]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.Item Open Access The Universe through a magnifying glass: Precision cosmology with CMB lensingQu, Frank JiatianfuThe 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.Item Open Access Deep Learning Approaches for PDE-based Image Analysis and Beyond: From the Total Variation Flow to Medieval Paper AnalysisGroßmann, TamaraPartial 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.Item Open Access Chaos in models of double convectionRucklidge, Alastair MichaelThis 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.Item Open Access On the Factorisation of Matrix Wiener–Hopf Kernels Arising From Acoustic Scattering ProblemsAitken, MungoThe 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.Item Controlled Access The Many Phases of the Surface Code: Coherent Errors and Many-Body LocalisationVenn, FlorianThis 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.Item Open Access On the Relationship between Canonical Quantum Gravity and the Holographic PrincipleAraujo Regado, GoncaloThis 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.Item Open Access Rare events and dynamics in non-equilibrium systemsKikuchi, TakaakiThe 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.Item Open Access Robustness of Fixed Points of Quantum ProcessesSalzmann, RobertThe 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.Item Open Access Human mobility and spatial models for infectious diseaseTang, Maria; Tang, Maria Lan [0000-0002-9671-8302]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.Item Open Access Exploring Non-Minimality in New Physics Beyond the Standard ModelBanks, HannahThe 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.Item Open Access Contact and coalescence of viscous dropsBeaty, Edward; Beaty, Edward [0000-0001-6995-8645]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.Item Open Access Advances in Reinforcement Learning for Decision SupportJarrett, DanielOn the level of decision support, most algorithmic problems encountered in machine learning are instances of pure prediction or pure automation tasks. This dissertation takes a holistic view of decision support, and begins by identifying four important problem classes that lie between the two extremes: exploration, mediation, interpretation, and generation---specifically with an eye towards the role of reinforcement learning in helping humans 'close the loop' in sequential decision-making. In particular, we focus on the problems of: exploring new environments without guidance, interpreting observed behavior from data, generating synthetic time series, and mediating between humans and machines. For each of these, we proffer novel mathematical formalisms, propose algorithmic solutions, and present empirical illustration of their utility. In the first instance, we refine our notion of curiosity-driven exploration to separate epistemic knowledge from aleatoric variation in hindsight, propose an algorithmic framework that yields a simple and scalable generalization of curiosity that is robust to all types of stochasticity, and demonstrate state-of-the-art results in a popular benchmark. In the second instance, we formalize a unifying perspective on inverse decision modeling that generalizes existing work on imitation learning and reward learning while opening up a broader class of research problems in behavior representation, and instantiate an example for learning interpretable representations of boundedly rational decision-making. In the third instance, we propose a probabilistic generative model of time-series data that optimizes a local transition policy by reinforcement from a global energy model learned by contrastive estimation, draw a rich analogy between synthetic generation and sequential imitation, and verify that it yields useful samples on real-world datasets. In the fourth instance, we formalize the sequential problem of online decision mediation with abstentive feedback, propose an effective solution that seeks to trade off immediate loss terms against future improvements in generalization error, and illustrate its efficacy relative to applicable benchmark algorithms on a variety of metrics. Like so, this dissertation contributes and advances a broader perspective on machine learning for augmenting decision-making processes.Item Embargo Radiative Transitions in CharmoniumDelaney, JamesIn this thesis, we analyse radiative transitions in charmonium using Lattice QCD, to provide insight into the internal structure and photocoupling of charmonium states. Distillation is used to allow the efficient computation of a high number of correlation functions, leading to the construction of highly optimized operators ideal as an input for the calculation of matrix elements. We explore techniques which leverage data over many time slices to increase statistics as well as a robust method of determining multiple form factors with generic kinematics. From these matrix elements, form factors are determined over a wide range of *Q2* and extrapolations are performed to the zero virtuality point. Quantities computed are in reasonable agreement with other lattice studies and experimental values, where available. First determinations of radiative partial widths of exotic charmonia using dynamical quarks are calculated.Item Open Access Inference and Entropy in Free-Surface FlowsYoung, BenjaminIn spite of the ubiquity of free-surface flows across both nature and industry, theoretical and numerical modelling remains challenging. Continuum flows, whether Newtonian or Non-Newtonian, are governed by the Navier–Stokes equations, whilst the particle-particle interactions within a granular flow can be modelled by simple collision mechanics. However, the computational effort required to solve these equations can often render numerical modelling intractable as modellers must either resolve all scales of the velocity field or model every particle-particle interaction. This issue is compounded by the fact that in many cases, modellers are not interested in the fine detail of the flow, but instead wish to study the bulk features of the flow. These bulk features include: the development of a free-surface due to topographic changes or the spatial distribution of the coarse-grained stress field within a granular flow, to name the two examples explored in this thesis. A common approach to model bulk features, without resolving the smallest scale features, is to perform a filtering operation on the governing Navier–Stokes equations or equations of collision mechanics. Unfortunately, the resulting equations often rely on a heuristic closure model to capture the interaction between the sub-filter scale features and the bulk features. In this thesis we explore two methods for deriving closure laws for filtered models – information entropy/energy dissipation maximisation and rheological inference – and apply these to two free-surface flows. In our first problem, we theoretically and numerically examine the interaction between a Blasius boundary layer and free-surface within a shallow Newtonian fluid. Depth-averaging the Navier Stokes equations yields an infinite system of equations that describes how the shape-factors (depth-wise moments of the stream-wise velocity profile) of the flow evolve. We apply an entropy maximisation method that: (i) produces a first-order accurate closure model for our equations; (ii) predicts analytical, steady-state solutions to the free-surface flow and (iii) provides valuable, new insight into the relationship between the rheology and information entropy of a flow. In our second problem, we experimentally and numerically investigate the stress/deformation-rate relationship (granular rheology) in the statistically steady flow of a refractive-index-matched, granular suspension within a rotating drum. Coarse-graining the governing equations yields the Cauchy momentum equations, which describe the relationship between coarse-grained velocity, pressure and stress. We develop and apply an inference algorithm to estimate the latent coarse-grained stress and pressure fields from granular velocimetry data and examine the relationship between inferred stress and shear rate within the context of existing rheology models. We demonstrate that our inferred data follows many of the previously observed stress/shear-rate trends and discuss where our data deviates away from theory. Finally, we posit how our method could be used to develop and validate more accurate models of granular rheology.Item Open Access Cauchy Slice Holography and the Semiclassical ApproximationShafi, Mohammed Rifath KhanWe investigate a new approach to holography in asymptotically AdS spacetimes, in which time rather than space is the emergent dimension. By making a sufficiently large $T^2$-deformation of a Euclidean CFT, we define a holographic theory that lives on Cauchy slices of the Lorentzian bulk. (More generally, for an arbitrary Hamiltonian constraint equation that closes, we show how to obtain it by an irrelevant deformation from a CFT with suitable anomalies.) The partition function of this theory defines a natural map between the bulk canonical quantum gravity theory Hilbert space, and the Hilbert space of the usual (undeformed) boundary CFT. We argue for the equivalence of the ADM and CFT Hamiltonians. We also explain how bulk unitarity emerges naturally, even though the boundary theory is not reflection-positive. This allows us to reformulate the holographic principle in the language of Wheeler-DeWitt canonical quantum gravity. Along the way, we outline a procedure for obtaining a bulk Hilbert space from the gravitational path integral with Dirichlet boundary conditions. Following previous conjectures, we postulate that this finite-cutoff gravitational path integral agrees with the $T^2$-deformed theory living on an arbitrary boundary manifold---at least near the semiclassical regime. However, the $T^2$-deformed theory may be easier to UV complete, in which case it would be natural to take it as the definition of nonperturbative quantum gravity. We then explore the semiclassical approximation to canonical quantum gravity and how a classical background emerges from the Wheeler-DeWitt (WDW) states. By employing the Wigner functional analysis, we derive the backreacted Einstein-Hamilton-Jacobi equation as an approximation to the WDW equation, along with the requisite validity conditions. We then apply this understanding to both AdS/CFT and dS/CFT correspondences, to explain how the bulk is encoded in the correlation functions of the $T^2$-deformed theory. We then explain an appropriate description for scenarios in which gravity behaves quantum mechanically in certain regions of spacetime and explain its relation to subregion holography. We derive the validity conditions for gravity to be semiclassical near any co-dimension 1 time-like surface and employ these conditions to explore the black hole information paradox. Our analysis suggests that for evaporating black holes, there might be a violation of semiclassical gravity in the near-horizon region close to the Page time, although this is contingent upon certain assumptions. This also provides insights into the fate of information trapped within evaporating black holes. We then explore this issue from the perspectives of both external and infalling observers. We then explain how to employ this new approach to study the retrieval of information from evaporating black holes, presenting a comprehensive approach to tackle this complex issue in quantum gravity.Item Open Access Neural Network Training and Inversion with a Bregman Learning FrameworkWang, XiaoyuDeep Neural Networks (DNNs) are powerful computing systems that have revolutionised a wide range of research domains and have achieved remarkable success in various realworld applications over the past decade. Despite their significant recent advancements, training DNNs still remains a challenging task due to the non-convex and (potentially) non-smooth nature of the objective function. Back-propagation in combination with gradient-based minimisation approaches has been the predominant strategy for training DNNs for decades. Yet the popular error backpropagation algorithm is susceptible to potential drawbacks and limitations, for example its non-parallelisablity and biological implausibility, and vanishing or exploding gradients issues, etc. Inverting DNNs to infer likely inputs of the system from given outputs, is the other side of the same coin. Early ideas of DNNs inversion trace back to the 1990s, but research interests in more generic network inversion problems have been rekindled and primarily driven due to the rapid advancements in generative modelling in recent years. While several approaches for the inversion of DNNs have been proposed, the stability of the inversion is an often neglected crucial aspect. The neural network inversion problem is ill-posed as the solution does not depend continuously on the input datum hence can be highly sensitive to perturbations. The core theme of this thesis is at the training of DNNs. Built up on distributed optimisation approaches, this work contributes to both the learning problems and the inversion problems of DNNs. In particular, we propose a lifted Bregman learning framework that goes beyond the classical back-propagation approach, and aims to address unresolved and overlooked issues in training and the inversion of DNNs. More specifically, we propose a family of loss (penalty) functions that are based on a tailored Bregman distance. We provide detailed mathematical analysis on the derived Bregman learning framework and propose a whole range of deterministic and stochastic optimisation strategies to enable solving the learning problem. Bringing techniques and tools from Inverse Problems and Regularisation Theory, we provide theoretical guarantees as well as computational optimisation strategies for the stable, model-based inversion of neural networks.