Mathematics thesis and dissertation collection
This collection contains a selection of the latest doctoral theses completed at the School of Mathematics. Please note this is not a comprehensive record.
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Recent Submissions
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Solution methods for some variants of the vehicle routing problem
(The University of Edinburgh, 2023-05-19)The Vehicle Routing Problem (VRP) is among the most important and widely researched problems in the field of combinatorial optimization. It fits a large variety of applications across many industries faced with the ... -
Estimates of space derivatives for functions of non-autonomous and McKean-Vlasov processes. Application to uniform weak error bounds induced by the approximating subsampled particle system
(The University of Edinburgh, 2023-05-16)This thesis is split in three parts, all of which obtain derivative estimates for the solution to the backward Kolmogorov equation associated to diverse stochastic process and study their application to uniform weak error ... -
Numerical framework for solving PDE-constrained optimization problems from multiscale particle dynamics
(The University of Edinburgh, 2023-04-21)In this thesis, we develop accurate and efficient numerical methods for solving partial differential equation (PDE) constrained optimization problems arising from multiscale particle dynamics, with the aim of producing a ... -
Regularised variational schemes for non-gradient systems, and large deviations for a class of reflected McKean-Vlasov SDE
(The University of Edinburgh, 2023-03-27)This thesis consists of two parts. The first part constructs entropy regularised variational schemes for a range of evolutionary partial differential equations (PDEs), not necessarily in gradient flow form, with a focus ... -
Non-Lorentzian geometry of fluids and strings
(The University of Edinburgh, 2023-03-16)Non-Lorentzian geometry is a branch of geometry where, roughly speaking, the notion of a metric is replaced by something else. We begin by providing an overview of non-Lorentzian geometries, and we describe how they as ... -
Moment polyptychs and the equivariant quantisation of hypertoric varieties
(The University of Edinburgh, 2023-03-01)In this thesis, we develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde formula for Higgs bundles. We do this by first ... -
Fukaya category and (open) Gromov-Witten invariants
(The University of Edinburgh, 2023-02-22)This thesis contributes to the problem of obtaining both open and closed Gromov-Witten invariants from the Fukaya category. For closed invariants, the main ingredient is the cyclic open-closed map which maps the cyclic ... -
Problems related to uniform rectifiability and bi-Lipschitz images
(The University of Edinburgh, 2023-02-15)In the first part of this thesis we give a new sufficient condition for a set to be uniformly d-rectifiable. The condition is that through each point of the set, there are d-many uniformly spread out line segments contained ... -
Computational modelling and optimal control of interacting particle systems: connecting dynamic density functional theory and PDE-constrained optimization
(The University of Edinburgh, 2023-02-07)Processes that can be described by systems of interacting particles are ubiquitous in nature, society, and industry, ranging from animal flocking, the spread of diseases, and formation of opinions to nano-filtration, ... -
Deep-water and shallow-water limits of the intermediate long wave equation: from deterministic and statistical viewpoints
(The University of Edinburgh, 2023-02-06)In this thesis, we study the convergence problem for the intermediate long wave equation (ILW) from deterministic and statistical viewpoints. ILW models the internal wave propagation of the interface in a two-layer fluid ... -
Power system adequacy: on two-area models and the capacity procurement decision process
(The University of Edinburgh, 2023-01-20)In this work, we explore methodological extensions to modelling practices in power system adequacy for single-area and two-area systems. Specifically, we build on top of some of the practices currently in use in Great ... -
Conditional densities of partially observed jump diffusions
(The University of Edinburgh, 2023-01-19)In this thesis, we study the fi ltering problem for a partially observed jump diffusion (Zₜ)ₜɛ[ₒ,T] = (Xₜ, Yₜ)tɛ[ₒ,T] driven by Wiener processes and Poisson martingale measures, such that the signal and observation noises ... -
Donaldson-Thomas theory and cohomological Hall algebras of character stacks
(The University of Edinburgh, 2023-01-18)Given a smooth finitely generated algebra with a potential one can study the refined Donaldson-Thomas theory of its moduli stack of representations via motivic or cohomological methods. In this thesis we focus on fundamental ... -
Presolve, crash and software engineering for HiGHS
(The University of Edinburgh, 2023-01-18)The efficient computational solution of linear optimization problems is generally enhanced significantly by using a "presolve" procedure to process the problem logically in order to reduce the dimension of the problem to ... -
Optimising heating and cooling of smart buildings
(The University of Edinburgh, 2023-01-10)This thesis is concerned with optimization techniques to improve the efficiency of heating and cooling of both existing and new buildings. We focus on the thermal demand-side and we make novel contributions to the ... -
Spatial and temporal hierarchical decomposition methods for the optimal power flow problem
(The University of Edinburgh, 2023-01-09)The subject of this thesis is the development of spatial and temporal decomposition methods for the optimal power flow problem, such as in the transmissiondistribution network topologies. In this context, we propose novel ... -
Accelerating Bayesian computation in imaging
(The University of Edinburgh, 2022-12-19)The dimensionality and ill-posedness often encountered in imaging inverse problems are a challenge for Bayesian computational methods, particularly for state-of-the-art sampling alternatives based on the Euler-Maruyama ... -
Bayesian analysis of jointly heavy-tailed data
(The University of Edinburgh, 2022-11-29)This thesis develops novel Bayesian methodologies for statistical modelling of heavy-tailed data. Heavy tails are often found in practice, and yet they are an Achilles heel of a variety of main-stream random probability ... -
Size and shape of things: magnitude, diversity, homology
(The University of Edinburgh, 2022-11-29)This thesis has to do with magnitude: a numerical invariant of enriched categories which, when specialized to examples such as posets, groupoids or metric spaces, turns out to encompass a variety of size-related quantities ... -
Pre-trained solution methods for unit commitment
(The University of Edinburgh, 2022-11-16)This thesis aims to improve the solution methods for the unit commitment problem, a short-term planning problem in the energy industry. In particular, we focus on Dantzig-Wolfe decomposition with a column generation ...
