Mathematics, School of
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Preconditioned iterative methods for optimal control problems with timedependent PDEs as constraints
(The University of Edinburgh, 20220627)In this work, we study fast and robust solvers for optimal control problems with Partial Differential Equations (PDEs) as constraints. Speci cally, we devise preconditioned iterative methods for timedependent PDEconstrained ... 
Bayesian analysis of nonstationary extremes
(The University of Edinburgh, 20220627)Extreme value theory (EVT) provides theoretical foundation to assess the probability of rare and extreme events. This thesis provides novel Bayesian semiparametric models for capturing the behaviour of extreme events and ... 
Role of structure in oscillatory integral estimates
(The University of Edinburgh, 20220622)Although oscillatory integrals intrinsically depend only on measuretheoretic properties of the phase and amplitude functions, in practice most techniques for deriving estimates invoke structural information either implicitly ... 
Stochastic dynamics and partitioned algorithms for model parameterization in deep learning
(The University of Edinburgh, 20220616)In this thesis, we study model parameterization for deep learning applications. Part of the mathematical foundation for this work lies in stochastic differential equations and their constrained counterparts. We will study ... 
Local and global analysis of geometric partial differential equations and their application to curvature flow problems
(The University of Edinburgh, 20220615)“An analytical approach to many problems in geometry leads to the study of partial differential equations.” (A. V. Pogorelov, Foreword to The Minkowski Multidimensional Problem) This thesis concerns itself with three ... 
Uncertainty quantification in seismic interferometry
(The University of Edinburgh, 20220613)It is a wellestablished principle that crosscorrelating wavefield observations at different receiver locations yields new responses that, under certain conditions, provide a useful estimate of the Green's function between ... 
Multijoints and multilinear duality
(The University of Edinburgh, 20220525)The joints problem is related to geometric questions at the heart of harmonic analysis. In three dimensions, a joint is a point of intersection of three lines that do not lie within a common plane. Given a collection of ... 
Quantum symmetric pairs in topological field theory
(The University of Edinburgh, 20191128)[No Deposit Agreement] 
Factorisation homology and skein categories of surfaces
(The University of Edinburgh, 20191128)[No Deposit Agreement] 
Fano varieties: positivity, Kstability and more
(The University of Edinburgh, 20220411)This thesis is about Fano varieties and their properties. We will determine the Kstability of certain singular del Pezzo surfaces and smooth Fano 3folds, the existence of cylinders in singular del Pezzo surfaces, and ... 
Iterative methods for solving stochastic optimal control problems
(The University of Edinburgh, 20220404)Optimal control problems are inherently hard to solve as the optimization must be performed simultaneously with updating the underlying system. Therefore, most of the time, they have to be solved numerically. In this ... 
Regularized interior point methods for convex programming
(The University of Edinburgh, 20220323)Interior point methods (IPMs) constitute one of the most important classes of optimization methods, due to their unparalleled robustness, as well as their generality. It is well known that a very large class of convex ... 
Bayesian methods for biomarker evaluation and disease diagnosis
(The University of Edinburgh, 20220322)Accurate diagnosis of disease is of fundamental importance in medical research and clinical practice. For such reason, the role that diagnostic testing and screening play is undeniable. The major goal of a diagnostic ... 
Methods for the analysis of oscillatory integrals and BochnerRiesz operators
(The University of Edinburgh, 20220321)For a smooth surface Γ of arbitrary codimension, one can consider the Lp mapping properties of the BochnerRiesz multiplier m(ζ) = dist(ζ,Γ)^α φ(ζ), where α > 0 and φ is an appropriate smooth cutoff function. Even for ... 
Constructing and classifying fivedimensional black holes using integrability
(The University of Edinburgh, 20220317)In this thesis we look at the problem of nding and classifying stationary and biaxisymmetric solutions in vedimensional theories of gravity, using particular hidden symmetries. We consider three theories: the electrostatic ... 
Statistical modelling of environmental extremes
(The University of Edinburgh, 20220316)This thesis is concerned with the development of theory and statistical methodologies that may be used to analyse environmental extremes. As extreme environmental events are often associated with large economic costs and ... 
Mixedmode oscillations in singularly perturbed threetimescale systems
(The University of Edinburgh, 20220222)In this work, we are concerned with the dynamics of threedimensional, threetimescale systems of ordinary differential equations. Systems with two timescales have been extensively studied, while the theory of systems ... 
Randomly assembled cyclic multiblock ADMM: a fast method for largescale linearly constrained quadratic optimization
(The University of Edinburgh, 20220222)This work is motivated by a simple question: how to find a relatively good solution to a very large optimization problem in a limited amount of time. We consider the linearly constrained convex minimization model with ... 
Bayesian inverse problems
(The University of Edinburgh, 20211127)We consider linear, mildly illposed inverse problems in separable Hilbert spaces under Gaussian noise, whose covariance operator is not identity (i.e. it is not a white noise problem), and use the Bayesian approach to ... 
Thermal effects on nonlinear optical beam propagation in nematic liquid crystals
(The University of Edinburgh, 20211127)This thesis examines in detail the study of nonlinear wave phenomena arising in different contexts in applied nonlinear optics. In particular, it explores the intricate nonlinear interplay between light and matter using ...