# Theses - Applied Mathematics and Theoretical Physics

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Item Open Access Convex Relaxations: Beyond Polynomials, Splitting Methods and Average Case AnalysisFaust, OisinShow more The first part of this thesis concerns the use of semidefinite programming for solving optimisation problems involving non-polynomial and non-semialgebraic functions. We start with the problem of determining the logarithmic Sobolev constant of a finite Markov chain, which can be framed as a nonconvex optimisation problem involving an entropy-like objective function. We demonstrate that it is possible to apply techniques from sum-of-squares programming to these problems. The main obstacle is that sum-of-squares relaxations only work on *polynomial* optimisation problems. We show how to overcome this obstacle via the stepping stone of rational approximations to entropy-like functions, which allow us to formulate sum-of-squares relaxations of entropic functional inequalities including logarithmic Sobolev inequalities. We prove that our semidefinite programming hierarchies converge to the true logarithmic Sobolev constant of the finite Markov chain. This approach extends to other entropic functional inequalities such as modified logarithmic Sobolev inequalities and strong data processing inequalities. We illustrate our semidefinite relaxations on various examples of finite Markov chains. Next, we consider the problem of obtaining accurate semidefinite approximations of the quantum relative entropy, and of more general quantum $f$-divergences. The quantum relative entropy is a jointly convex function of two positive semidefinite Hermitian matrices, but it cannot be expressed as the optimal value a semidefinite program because it is not semialgebraic. Fawzi, Parrilo, and Saunderson used integral representations of the logarithm to define functions which approximate the quantum relative entropy and which have semidefinite representations of modest size. Our focus is on quantifying the dependence of the approximation error for approximations of this form on the size of the semidefinite program. To do this, we are led to study weighted minimax rational approximations to certain operator convex and operator monotone functions. In particular, we prove that the approximation error can decay root-exponentially in the size of the semidefinite approximation, when the approximation is chosen optimally. Our analysis extends beyond the quantum relative entropy to the $\alpha$-quasi-entropies. The second part of the thesis makes contributions to the study of first-order algorithms for convex optimisation and non-convex optimisation. We begin by considering first-order methods for solving linear programs. It is known that Douglas-Rachford splitting when applied to a feasible and bounded linear program eventually enters a region of linear convergence towards a solution. We are concerned with quantifying the rate of linear convergence for random linear programs. We prove that for a quite natural class of random linear programs with $n$ nonnegative variables and $m$ linear constraints, conditional on feasibility of the linear program, on the order of $m(n-m)\log(1/\epsilon)$ iterations are required for an $\epsilon$-accurate solution once the iteration reaches the region of linear convergence. We also consider the problem of certifying infeasibility of a linear program via Dykstra's alternating projections algorithm. We show that as the size $(n,m)$ of the random linear program grows with a fixed ratio $m/n\approx p$, the rate of eventual linear convergence converges in probability to an explicit constant $r(p)$, i.e. it does not deteriorate with the size of the problem. Finally, motivated by an application involving the quantum conditional entropy we consider the difference-of-convex algorithm, which is a method for minimizing (non)convex functions which are expressed as a difference of two convex functions. By interpreting the difference-of-convex algorithm as a Bregman proximal point method, we are able to simplify and strengthen existing non-asymptotic convergence guarantees for the DC algorithm in the nonconvex setting. Under a relative strong convexity assumption we prove a new linear convergence result, and we also present a new DC Polyak-Łojasiewicz condition which ensures a linear convergence rate.Show more Item Open Access Twistor Space and Celestial HolographySeet, SeanShow more This thesis is divided into 4 main parts. The first part is review material, consisting of short descriptions of prerequisite background material, as well as references to good sources that cover the material in more depth. The second part is a discussion of twistor methods for H$^3_\mathbb{C}$ and H$^5_\mathbb{C}$, the complexification of 3 and 5 dimensional hyperbolic space. We discuss the minitwistor space of the bulk and its relation to the ambitwistor space of the boundary. The third part of my thesis is a paper on spectral flow and localisation in AdS$_3$ string theory. The fourth and final part of my thesis is a discussion of off-shell Celestial holography. There is a short appendix consisting of several calculations, intended as supplementary material for the main text.Show more Item Open Access Electron transport, acceleration and loss in Earth's outer radiation beltDaggitt, ThomasShow more Earth's radiation belts are comprised of high energy charged particles trapped by Earth's magnetic field. The outer radiation belt is highly variable, with the trapped electron flux varying by orders of magnitude on timescales of hours. The radiation belts are a hazard for satellite operations as high energy charged particles can damage electrical components. For this reason it is essential to be able to understand, model and predict the processes driving the variability of the outer radiation belt. This is of particular importance during geomagnetically active times, when the variance in the Earth's magnetic field, the plasma wave distribution and the background plasma properties is highest. This thesis investigates the behaviour of relativistic electron flux in the outer radiation belt, and attempts to reproduce rapid storm time flux variations using the British Antarctic Survey radiation belt model (BAS-RBM). A set of events involving rapid drops in electron flux across a range of energies at high $L$ shells near the magnetopause were analysed. The observations were compared to event-specific last closed drift shell models. The results indicate that the magnetic local time at the satellite and its position relative to the last closed drift shell are both important when reproducing observations with radiation belt models. Simulations with the BAS-RBM determined that different last closed drift shell models can significantly alter the reproduction of flux dropouts, but that the current outer boundary conditions and statistical wave models are unable to recreate the inwards extent of flux dropouts in $L^*$. Chorus wave power near the strong diffusion limit was investigated as an alternate driver for electron flux dropouts. Previous analyses have treated strong diffusion primarily as a loss process. Satellite data was used to verify the existence of high chorus wave power leading to strong diffusion in the outer radiation belt. BAS-RBM simulations unexpectedly demonstrated that chorus waves causing strong diffusion led to an increase, not a decrease in the net flux as predicted when treating strong diffusion as a loss process. The results show the existence of a tipping point in chorus wave power between net acceleration and net loss. Due to this, radiation belt models using averaged chorus wave power will misrepresent the effects of chorus when chorus wave power is high. In order to better represent the variation in chorus wave power during geomagnetically active times, a plasma density dependent chorus model was developed for the BAS-RBM. The measurement of the background plasma density was determined to be biased away from low densities. This was corrected for in the chorus model. In combination with an event-specific density model, BAS-RBM simulations using this chorus model are capable of reproducing observations of enhancements of ultra-relativistic electron flux. These simulations confirm that chorus waves are able to directly accelerate electrons to multi-MeV energies with and without including the effects of radial diffusion.Show more Item Open Access Anticipating, Extracting, and Leveraging Information in Clinical Decision-MakingHuyuk, AlihanShow more A principal challenge in machine learning for clinical decision support is the limited availability of high-quality data: Experimental data is hard to acquire and hence scarce, while observational data from the clinic is more abundant but may contain unintended biases. Given that available data is limited either in quantity or quality, it becomes crucial to understand the full extent of its information content - especially in relation to clinical decisions we wish to support. This thesis aims to develop such understanding by studying three modes of handling information in decision-making: (i) anticipating the arrival of future information when making present decisions, (ii) extracting the information encoded in past decisions in an interpretable form, and (iii) leveraging such information to perform other downstream tasks. For anticipating, we consider subpopulation selection in adaptive clinical trials, and formulate this setting as a new type of optimal stopping/switching problem called the optimal commitment problem (OCP). By theoretically analyzing OCP, we discover that, when adapting a trial, decision-makers should factor in the informational value of conducting the trial - beyond just the potential return of its success. For extracting, we tackle the challenge of obtaining transparent representations of an expert's decision-making process purely from demonstrations of their behavior, thereby making the knowledge behind their actions accessible to others. Our efforts establish a new agenda for policy learning focused on understanding - as opposed to merely imitating - human decision-making. We introduce two novel models: interpretable policy learning (Interpole), which explains human actions through decision dynamics and decision boundaries, and lexicographically-ordered reward inference (LORI), which explains human preferences through lexicographically-prioritized objectives. For leveraging, we give two examples of how knowledge extracted from expert demonstrations can inform other downstream tasks: As the first example, we develop inverse contextual bandits (ICB), a method for learning how behavior evolves over time, and show that ICB can help assess the impact of new medical guidelines on actual clinical practice. As the second example, we define the notion of expertise, an information-theoretic measure of how knowledgeable a policy is of its environment, and show that identifying the prominent type of expertise present in a dataset can inform model selection for treatment effect estimation.Show more Item Open Access The cosmological bootstrap and the analytic wavefunctionLee, Mang Hei Gordon; Lee, Mang Hei Gordon [0000-0001-7077-3269]Show more In the past few decades there have been an overabundance of models describing inflation, a period where the universe expands exponentially quickly. This led to the rise of the cosmological bootstrap, which aims at constraining cosmological observables in a model independent way. This is achieved by directly imposing physical principles such as unitarity, locality, symmetry, and analyticity on cosmological correlators. In this thesis we explore the consequences of two such principles: unitarity and analyticity. We show that unitarity implies a set of consistency relations among wavefunction coefficients in perturbation theory, and these relations can be generalized to fields with any mass and integer spin. Unitarity, alongside locality and scale invariance, also implies the vanishing of four-point parity odd correlators at tree level, and we show this is not true at loop level. Analyticity in the S-matrix is linked to causality and serves as the backbone for the S-matrix bootstrap, which provides non-perturbative constraints for scattering. We show that analyticity in the wavefunction is also linked to causality. We study the analytic structure of the wavefunction in detail and demonstrate the relation between singularities in amplitudes and a subset of singularities in the wavefunction. Finally, we write down the dispersion relations of the wavefunction, which serves as a first step towards a non-perturbative bootstrap in cosmology.Show more Item Open Access Mathematical Modelling of Acoustic Diffraction Noise Embracing Diverse Boundary ConditionsHales, Alistair; Hales, Alistair [0000-0003-0445-0247]Show more In this thesis, we develop advanced mathematical tools for modelling acoustic diffraction noise in various contexts and explore improved models for aerofoil-turbulence interaction noise. The thesis is divided into two parts. The first explores variants of half-plane scattering problems and the analytical requirements for their solutions. The second focuses on applying solutions of these scattering problems to leading- and trailing-edge noise models and developing a mathematical framework for the statistical descriptions of turbulence within these models. Part I, "Applying the Wiener–Hopf Technique to Diverse Diffraction Problems in Acoustics", develops a generalised theory to solve diffraction problems with linear boundary conditions prescribed distinctly on each side of a semi-infinite boundary. We choose this framework to reflect physical applications in subsequent parts. Key aspects covered in the technique include the multiplicative splitting of polynomial kernels within Wiener–Hopf equations, adapting edge conditions to more mathematically involved boundary conditions, and solving two-sided diffraction problems with distinct boundary conditions on each side, leading to reworkings of the Wiener–Hopf technique with intriguing results. Part II, "Adapting Leading- and Trailing-Edge Noise Models to Anisotropy and Compliant Plates", focuses on creating a generalised framework to predict aerofoil turbulence interaction noise. This noise arises when turbulent flow scatters off an aerofoil’s sharp leading or trailing edge. Existing analytical models use the Wiener–Hopf technique, approximating turbulence as a single turbulent eddy and scattering it off an edge modelled as a zero-thickness semi-infinite boundary with an infinite span. These solutions enable the construction of a transfer function that relates the incident gust to its far-field scattered pressure. We sum the solutions for all possible eddies using a turbulence spectrum that associates each eddy with its expected energy and length scales. This approach is valid for both leading and trailing-edge noise models. Slight changes in turbulence modelling are addressed carefully throughout this part of the thesis. The scattering component of both models requires theory developed in the first part of the thesis. We verify our leading-edge model with experimental work exploring the interaction of anisotropic turbulence with a rigid leading edge. Then, we repeat the experiment twice more to investigate different types of edges. First, we place a layer of noisereducing foam on either side of the leading edge. Second, we use spanwise-perforated inserts. All experiments show promise for approaching analytical modelling of leading edges using an analytical transfer function based on the solution of the interaction of a gust with a semi-infinite plate with an impedance boundary condition that accounts for steady mean flow effects. Finally, our focus on trailing-edge noise is twofold. First, we predict how changing the trailing edge’s material properties (impedance) can affect the far-field noise. Second, we conduct a theoretical investigation into an analytical pressure spectrum: the TNO–Blake model. We discuss how some analytical simplification can be altered or improved to apply to a broader range of contexts in future studies into either pressure spectra or their use within analytical trailing-edge noise models.Show more Item Open Access Scattering and stability properties of nonlinear wave equations on asymptotically flat spacetimesKadar, IstvanShow more This thesis is the amalgamation of projects focused on the nonlinear behaviour of hyperbolic partial differential equations (PDEs) in asymptotically flat spacetimes. Each project focuses on the behaviour of solutions nearby special ones and are therefore part of a perturbative understanding of these PDEs. Furthermore, all projects are connected by the search for a geometric understanding of the equations and the use of this geometry for the analysis. The first chapter is concerned with the global behaviour of effective field theories (EFT). These are approximate equations that are supposed to model the interaction of light and very heavy particles in the limiting case, where the degrees of freedom connected to the heavy particles are inactive. From a mathematical point of view, these present difficulties, as the equations are higher than second order in time and admit physically non-acceptable run away solutions. The main result is that for a certain coupled system of Klein-Gordon equations in the high mass limit, global solutions as well as the scattering matrix is well approximated by solutions of an EFT. The second chapter is motivated by the weak null condition of Lindblad-Rodnianski, which was introduced to classify systems regarding the stability of their trivial solution. This chapter presents a new heuristic approach to the study of systems of coupled semilinear wave equations in Minkowski space. The non rigorous analysis suggests a condition related to stability for systems of wave equations. The main result of this chapter is the validation of this heuristic for a number of system. In particular we exhibit a system satisfying the weak null condition, but failing our classification which forms singularities in finite time for arbitrary small data. The third and fourth chapters contain results for semilinear wave equations in Minkowski space admitting soliton solutions. In the former, we study the geometric scattering problem in a neighbourhood of timelike infinity around a single soliton and construct solutions admitting a prescribed polynomially decaying radiation field. This study is motivated by the understanding of multi soliton solutions, that is, solutions describing solitons moving away from one another. Indeed, using the technical tools developed in chapter 3, we construct approximate and exact multisoliton solutions for a number of semilinear wave equations in chapter 4. In chapter five we study the scattering problem for a large class of quasilinear system of equations in a neighbourhood of spacelike infinity. This work is based on a collaboration with Leonhard Kehrberger. This study is motivated by the understanding of gravitational radiation for the Einstein equations. We present robust estimates to propagate regularity to different regions of Minkowski space. We use these to classify perturbations that allow for the same estimates to be applied. The perturbations in particular contain the wave equation on Schwarzchild background and therefore allow for the summability of fixed angular mode analysis performed by Kehrberger.Show more Item Open Access Tracing the Cosmos: Probing Cosmology with the Large Scale Structure of the UniverseFarren, GerritShow more In this thesis I will present work on constraining and testing cosmological models with different probes of the large scale structure of the Universe. I will introduce different large scale structure observables and discuss relevant modelling approaches. In particular, I will motivate the use of complementary probes to examine some of the tensions between different datasets widely discussed within cosmology. One such tension is the disagreement between local measurements of the Universe’s expansion rate using the so-called distance ladder and the expansion rate inferred from the Cosmic Microwave Background (CMB). I present work that aims to exploit a novel feature in the distribution of matter in the Universe, the so called matter-radiation equality scale, to obtain independent estimates of the expansion rate. I develop a method to isolate the information from this feature in galaxy surveys, demonstrate its robustness, and present an application to current data as well as forecasts for the next generation of cosmological surveys. Another widely discussed tension is between the amplitude of matter density fluctuations observed in the late universe and predictions from model fits to early universe data. Recent observations have found a slightly smaller abundance of structures than is expected within the standard cosmological model. I obtain independent estimates of the abundance of structures in the late universe using combinations of galaxy surveys and the gravitational lensing of the CMB. Finally, I present the first detection of the three-point correlation between galaxies and CMB lensing. I show that the detection is robust to potential sources of systematic bias and present a comparison to model predictions. This signal can be used to constrain the evolution of structure on smaller scales where non-linear processes are dominant. In summary, I will present methods of probing cosmology with observations of cosmic structure including galaxy surveys and gravitational lensing of the CMB. I will describe new methods to analyse these observations and obtain competitive constraints on cosmological model parameters by working with state of the art data sets. These novel constraints are complementary to other analyses and will contribute to furthering our understanding of the physics of our Universe and shed light on some of the tantalising hints at physics beyond the current standard model of cosmology.Show more Item Open Access Analytic Modelling of Gravitational-Wave Source in General Relativity and Alternative Theories of GravityJain, TamannaShow more The first direct detection of gravitational waves (GWs) by LIGO in 2015 opened up new avenues for probing and better understanding the strong-field dynamics of gravity. The continuous improvement in the sensitivity of LIGO, Virgo and now KAGRA along with the forthcoming next-generation detectors like the Einstein Telescope and LISA, will provide a wealth of GW signals to study strong-field dynamics, and to probe the nature of compact objects. Alternative gravitational theories generally predict dynamics different from GR, therefore, it is also expected that the observations of GWs will shed more light on these theories by constraining their parameters. In addition to the applications in usual parameter estimation to improve the constraints, the analytical modelling of waveform generation considered in our work can be used to build search pipelines for the specific beyond GR model considered here. The detection (searches and parameter estimation) of GWs is based on matching the data observed in the detectors against theoretically predicted waveforms. Hence, highly accurate analytical models are required to construct a bank of waveform templates. The analytical waveform template banks used by LIGO are based on the post-Newtonian (PN) formalism for inspiral and on numerical relativity (NR) for merger and ringdown, or resummation methods such as the effective-one-body (EOB) formalism to describe the complete dynamics. In this thesis, we discuss the construction of analytical waveform models within the EOB formalism for a specific alternative theory of gravity the so-called massless scalar-tensor theory. More specifically, we describe the map of the dynamics of a binary system in scalar-tensor theory into the EOB formalism by building upon the PN expanded Lagrangian of the two-body problem at 3PN order. We discuss the generalization of the 2PN EOB Hamiltonian to 3PN order for the conservative part of the dynamics and then compute the scalar-tensor corrections to EOB potentials for generic orbits. We also compute the corrections in the EOB Hamiltonian due to tidal effects at 3PN order for circular orbits. Following this, we describe the derivation of the angular momentum flux in the scalar-tensor theory for both the scalar part and the tensorial part up to 1PN and 1.5PN order, respectively. We then use these results along with the energy flux to compute generic orbit radiation reaction effects in the EOB formalism up to 1PN order. Finally, we use the 3PN conservative dynamics results in the EOB formalism to derive the scattering angle in the scalar-tensor theories which can be used in future as a check against the scattering angle computed using scattering amplitude techniques. Finally, we implement up to 3PN conservative dynamics results in the EOB based waveform model TEOBResumS. Additionally, we investigate the improvement in the inference of the properties of neutron stars using the information of the observation of electromagnetic counterparts of the GW signal GW170817 and improved fits of the threshold mass. The application of the improved fits to the GW170817 signal implies that a significant part of the parameter space is not supported by the observation of the EM counterpart, implying improved constraints on the neutron star radii and hence the equation-of-state.Show more Item Open Access Geometric Aspects of Supersymmetric Partition FunctionsZhao, BoanShow more We explore the physical interpretations of concepts in (equivariant) K-theory using supersym- metric partition functions. The partition functions are first computed using supersymmetric localisation and then interpreted using suitable concepts in K-theory. We show that quantum field theories predict nontrivial results in K-theory. We focus on supersymmetric quantum field theories in one and three dimensions. Nevertheless, our approach and results can be generalized to other dimensions. We start by discussing the superconformal index of (4,4) superconformal quantum mechanics on hyperkahler cones. The index is computed by coupling the quantum mechanics to a killing vector field. The resulting object has a natural geometric interpretation in terms of the equivariant Euler characteristics of the algebraic differential forms on the resolved space. The superconformal symmetries of the quantum mechanics imply certain symmetries of the Euler characteristics. We also explain how holography predicts an asymptotic behaviour of the index. We then compute the topologically twisted based P 1 index of 3d N = 2 gauge theories. We explain the supersymmetric boundary conditions at the south pole of P 1 . We then prove the existence and uniqueness of the BPS locus compatible with our boundary conditions. A detailed computation of the one-loop determinant around the BPS locus is also provided. The resulting index is interpreted as the K-theoretic Euler characteristic of the corre- sponding based quasimap moduli space. We define the moduli space and construct its tangent-obstruction theory. We show that unmatched fermion modes lie in the tangent- obstruction spaces. We define the virtual structure sheaf using the tangent-obstruction theory and shows that its Euler characteristic is captured by the based P 1 index. In the final chapter we provide a contour integral representation of the based P 1 index by studying the 3d twisted index on a hemisphere. We illustrate the supersymmetric boundary conditions for the hemisphere geometry. We show that the contour integral representation naturally arises as the Coulomb branch localisation of the index. We mention why this presentation is important in the geometry literature.Show more Item Open Access Security Protocols in Relativistic Quantum CryptographyCowperthwaite, GeorgeShow more This thesis explores quantum cryptographic protocols designed to provide secure implementation of three key tasks, through mathematical analysis and qualitative discussion. Each chapter focuses on a distinct cryptographic scheme and explores its utility relative to classical and alternative quantum protocols. In Chapter 2, we examine the concept of singlet testing through discussion of two schemes: the Braunstein-Caves and Random Measurement tests. These schemes allow trusted parties to verify the presence of entangled singlet states through the use of simple quantum tools. Through statistical derivation, we demonstrate that the singlet state consistently exhibits the lowest expected value in both tests. We evaluate the performance of both schemes in a range of important scenarios and discuss the consequences of the results. The Random Measurement test emerges as the preferred choice in several natural scenarios, such that it can be considered as an efficient alternative to CHSH-based schemes under certain conditions. Further merits of both schemes are discussed, both in terms of efficiency and practicality. Chapter 3 introduces the concept of quantum tokens and examines a practical S-money scheme for generating unforgeable quantum tokens. This scheme does not require the utilisation of quantum memories or long-distance quantum transmission, rendering it more practical to implement with current technology. We derive a new unforgeability theorem that bounds the probability of a dishonest user successfully presenting the same token at multiple locations. An ongoing experimental implementation of the scheme is discussed, as well as plausible security guarantees which may result from such a setup. In Chapter 4, we discuss quantum position authentication (QPA) schemes with security derived from physical assumptions. The chapter begins by reviewing the impossibility of unconditionally secure QPA schemes and proposes a scheme based on the assumption that a prover and verifier can share a secret timing schedule. We explore the security principles of a timing-based QPA scheme, including the need to protect synchronised clocks against environmental and adversarial disruptions. We also discuss the expected latencies within such a scheme and the resulting operational impact of such deficiencies. This thesis advances our understanding of quantum cryptographic protocols and their practical applications, by offering new insight into singlet verification schemes, secure quantum tokens, and position authentication protocols in quantum communication.Show more Item Open Access Modelling turbulence and transport of buoyant material in the ocean surface mixed layerDingwall, JenniferShow more The ocean mixed layer (OML) is a significant and dynamically active part of the ocean which plays an important role in climate variability. Here, atmospheric processes such as winds, heat fluxes or density differences drive the generation of small-scale, three-dimensional turbulence and mixing of oceanic waters. These turbulent flows govern the distribution of buoyant materials including oil droplets and microplastics, which have significant implications for marine life and safety. However, turbulent flow structures are often too small to be resolved by global or regional circulation models, and observations at these scales remain limited. The focus of this thesis is to use numerical simulations to improve our understanding of the small-scale, three-dimensional turbulent processes in the OML and examine their role on transporting and accumulating buoyant material. We use high resolution large eddy simulations (LES) and direct numerical simulations (DNS), and model non-inertial, buoyant particles using a combination of buoyant tracers and three-dimensional Lagrangian particles. Surface cooling drives convection, and under this regime persistent convective vortices form which trap and accumulate buoyant particles. We test the resilience of convective vortices under the additional presence of wind, and find that in weak winds, convective vortices survive but are less effective at trapping buoyant material. With sufficiently strong wind forcing, convective vortices are no longer visible, but some clustering occurs in downwelling regions associated with longitudinal wind rolls. Despite their small size, the convective vortices exhibit a bias towards cyclonic vorticity which has not been reported previously. We independently vary the Coriolis acceleration and surface buoyancy flux, and using Lagrangian particles, we find that the large convective vortices develop through the merger of many small unbiased convective vortices. We propose a statistical theory to predict the cyclonic bias of large convective vortices and test the theory using LES results. We apply the theory to typical convective conditions and find that convective vortices in OML are expected to exhibit a bias, but convective vortices in the terrestrial and Martian atmospheres are expected to be largely unbiased. Finally, motivated by accumulation of buoyant material observed at surface fronts in the SUNRISE field campaign in the Gulf of Mexico, we run simulations of a highly idealised front under geostrophic adjustment. By varying the balanced Rossby number, we show that strong fronts develop a three-dimensional instability which generates turbulence near the top and bottom boundaries. We describe the physical mechanisms at play and the energy pathways as the front evolves over time. In the case of the most turbulent dynamics, we additionally model the movement of buoyant particles. Shear instabilities drive turbulence which enhances mixing, and strongly buoyant particles are carried out of the front during the first inertial period, which segregates the particles and leaves a large void in the centre of the front. In contrast, weakly buoyant particles are quickly subducted into the interior, and subsequently move according to the inertial oscillations of the front.Show more Item Open Access Riemannian geometry for inverse problems in cryogenic electron microscopyDiepeveen, Willem; Diepeveen, Willem [0000-0002-4690-9258]Show more This thesis develops theory and algorithms for a Riemannian geometric approach to inverse problems in cryogenic electron microscopy (Cryo-EM). It is divided into two parts, motivated by the sub problem of orientation estimation and that of modelling of protein dynamics. The thesis is concluded with a discussion on how to bring these pieces together to solve the so-called continuous heterogeneous reconstruction problem and a reflection on the general implications and new opportunities that a Riemannian geometry-based approach has for and brings to signal processing and recovery problems. In the first part of the thesis we consider how to use ideas from global optimisation on Riemannian manifolds to regularise the orientation estimation sub problem. Our approach is motivated by the two main challenges in orientation estimation: the high noise levels present in Cryo-EM data and the non-convexity of typical variational problems for estimating orientations. To overcome these challenges jointly, we construct a new regularised global optimisation scheme to solve a variational problem for orientation estimation in a more noise-robust fashion. In the second (and main) part we focus on constructing a Riemannian manifold for protein conformations. In particular, we are interested in constructing Riemannian geometry for protein conformations such that physically realistic protein conformations live in low-dimensional geodesic subspaces. Before constructing a Riemannian manifold, we first consider how curvature causes a discrepancy between data only looking low-dimensional and data actually being low-dimensional, and also consider how to address such curvature effects. Next, we construct a Riemannian manifold of protein conformations with computationally feasible manifold mappings such that realistic protein dynamics data really are low-dimensional under the proposed Riemannian structure, i.e., not suffering from curvature effects. Thirdly, we argue that it could be beneficial to have additional structure on a Riemannian protein geometry to what we have so far. In particular, we consider pullback geometry as a candidate class of Riemannian geometries that comes with the structure of interest and in addition offers a large amount of geometries to choose from. However, instead of directly trying to approximate the target Riemannian geometry, we take a step back and consider how a pullback Riemannian structure affects downstream data processing first and consider how one should go about constructing proper pullback manifolds given a target geometry. In the conclusions we see that the constructed individual parts can be combined into two variational problems for solving the continuous heterogeneous reconstruction problem, one of which being a new strategy for solving general inverse problems under a certain type of learned regulariser. Next, in the light of the findings in this thesis, we also advocate more generally for the development of methodology for processing data under data-driven Riemannian geometry. A key overall takeaway from this thesis is then that over time having a proper account of the Riemannian geometry of our data can change the way we go about handling them in any data analysis pipeline with classically Euclidean data.Show more Item Open Access Magic and Majorana Fermions in Quantum Computing, Topological Matter, and DynamicsMcLauchlan, Campbell; McLauchlan, Campbell [0000-0001-5848-291X]Show more This thesis explores the resource of “magic”, and the use of Majorana fermions, in quantum many-body physics and quantum computing. Along with other quantum resources, magic helps determine the advantage that a quantum circuit can display over classical computation. For the components of the quantum computer itself, Majorana fermions offer promising theoretical advantages, owing to their fermionic nature and noise characteristics. A tool used throughout this thesis is the Pauli-based model of computation (PBC). We extend this model to the regime of fermionic computation, establishing the analogous “fermion-parity-based computation” (FPBC). We develop certain conceptual tools in fermionic computation, including the idea of a “logical Majorana fermion”, which is used throughout this thesis. We find several results that relate to the ease of implementing PBC or FPBC directly, and which relate the magic of (F)PBC to that of the original circuit. We apply these insights to study the dynamics of magic in random quantum systems. We use PBC to identify a phase transition in the magic produced by (and its spread in) random quantum circuits. We relate this transition to previously found phase transitions in entanglement, providing further insight into the purported transition in classical simulability of these systems. We then assess the capabilities of Majorana systems for implementing (F)PBC. We introduce a hardware model with which one can measure arbitrary fermion parities directly, within certain bounds of locality, thus removing an obstacle to (F)PBC present in existing designs. To further investigate noise-reduction methods in large-scale Majorana-based quantum computing, we perform an in-depth investigation of “Majorana surface codes” as intriguing models of both quantum error-correcting codes and fermionic topological quantum matter. We provide a categorisation of anyonic excitations, boundaries and “twist defects” (lattice defects that can store quantum information) in these codes. Finding that twist defects can store logical Majorana fermions, as well as regular logical qubits expected, we introduce methods for computing with all topologically protected degrees of freedom in the code. We introduce new computing techniques. We finally discuss avenues towards improved quantum resource costs, potential implementations and connections to other codes.Show more 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