Mathematics, School of
Our research groups cover a broad range of Mathematics, Statistics and Operational Research. Please visit the School of Mathematics home pages for more information.
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The Stability of Hyperbolic PDEs in String Theory, Particle Physics and Cosmology
(The University of Edinburgh, 20200901)In this thesis we study hyperbolic PDEs arising from general relativity and the standard model of particle physics. In particular we prove the asymptotic stability of special solutions of these PDEs against small initial ... 
Stochastic analysis and partial differential equations: theory and numerics
(The University of Edinburgh, 20201130)This thesis is concerned with problems at the interface of stochastic analysis and partial differential equations (PDEs). In particular, we focus on two different classes of problem: the transport of measures under the ... 
Granular media at multiple scales: mathematical analysis, modelling and computation
(The University of Edinburgh, 20200728)There are many challenges in modelling granular media, in particular due to hard particle interactions such as collisions. Modelling and simulating at a microscopic level produces very accurate results, but simulations ... 
Inertiagravitywaves in geostrophic turbulence
(The University of Edinburgh, 20200728)The dynamics of the atmosphere and ocean can be decomposed into a slow, largescale component, in near geostrophic and hydrostatic balance, and a fast component consisting mostly of inertia–gravity waves (IGWs). These waves ... 
Parameter clustering in Bayesian functional PCA of neuroscientific data
(The University of Edinburgh, 20200722)This thesis provides novel methodologies for functional Principal Component Analysis of dependent time series (curves) with particular emphasis on those arising from neuroscientific experiments. In this context, the ... 
Random geometric graphs and data structures: discrete estimations of the continuous
(The University of Edinburgh, 20200706)In recent years, there have been significant efforts to develop rigourous geometric and topological methodologies to better understand data sets, with the view of performing machine learning tasks such as spectral ... 
Evidence evaluation and functional data analysis
(The University of Edinburgh, 20200625)In forensic contexts, evidence gathered are valuable in suggesting possible crimes. The problems forensic scientists are often interested in are whether evidence found at the scene of the crime match those that are found ... 
McKeanVlasov equations: a probabilistic and pathwise approach
(The University of Edinburgh, 20200625)This thesis divides neatly into four collections of results. In the first (Part II), we provide existence and uniqueness results along with several properties for a class of McKeanVlasov Equations having random ... 
Analysis and applications of dynamic density functional theory
(The University of Edinburgh, 20200625)Classical ﬂuid mechanics and, in particular, the general compressible NavierStokesFourier equations, have long been of great use in the prediction and understanding of the ﬂow of ﬂuids in various scenarios. While the ... 
Multiplicative latent force models
(The University of Edinburgh, 20200625)Latent force models (LFM) are a class of ﬂexible models of dynamic systems, combining a simple mechanistic model with the ﬂexibility of an additive inhomogeneous Gaussian process (GP) forcing term. These hybrid models ... 
Accelerated sampling schemes for high dimensional systems
(The University of Edinburgh, 20200406)In this thesis we discuss accelerated sampling schemes for high dimensional systems, for example molecular dynamics (MD). The development of these methods is fundamental to the eﬀective study of a large class of problems, ... 
Some questions related to the Analyst’s TST and a conjecture of Carleson
(The University of Edinburgh, 20200317)This thesis is constituted by two independent parts. In the first part we discuss several problems which relate to the Analyst's Travelling Salesman Theorem of Peter Jones and the theories of uniformly rectifiable sets and ... 
Quantalic spectra of semirings
(The University of Edinburgh, 20200625)Spectrum constructions appear throughout mathematics as a way of constructing topological spaces from algebraic data. Given a localic semiring R (the pointfree analogue of a topological semiring), we define a spectrum of ... 
Derived contraction algebra
(The University of Edinburgh, 20191125)All minimal models of a given variety are linked by special birational maps called flops, which are a type of codimension two surgery. A version of the Bondal–Orlov conjecture, proved by Bridgeland, states that if X and ... 
MonteCarlo based numerical methods for a class of nonlocal deterministic PDEs and several random PDE systems
(The University of Edinburgh, 20191128)In this thesis, we will investigate McKeanVlasov SDEs (McKVSDEs) on Rd,: ; where coefficient functions b and σ satisfy sufficient regularity conditions and fWtgt2[0;T ] is a Wiener process. These SDEs correspond to a ... 
Landscape of Hamiltonian phase spaces: on the foundations and generalizations of one of the most powerful ideas of modern science
(The University of Edinburgh, 20191128)In this thesis we aim to revise the fundamental concept of phase space in modern physics and to devise a way to explicitly incorporate physical dimension into geometric mechanics. To this end we begin with a nearly sel ... 
Randomized iterative methods for linear systems: momentum, inexactness and gossip
(The University of Edinburgh, 20191128)In the era of big data, one of the key challenges is the development of novel optimization algorithms that can accommodate vast amounts of data while at the same time satisfying constraints and limitations of the problem ... 
Moduli space of supersymmetric black holes in five dimensions
(The University of Edinburgh, 20191128)This thesis presents a classification of all asymptotically at, supersymmetric and biaxisymmetric (i.e. possessing a U(1)2symmetry) soliton and black hole solutions to fivedimensional minimal supergravity. In particular, ... 
Mutation frequencies in a birthdeath branching process
(The University of Edinburgh, 20190820)A growing population of cells accumulates genetic mutations. We study stochastic models of this process. Cells divide and die as a branching process, and a cell's genetic information is a sequence of nucleotides which ... 
Cancer recurrence times and early detection from branching process models
(The University of Edinburgh, 20191128)Cancer is among the leading causes of death worldwide. While primary tumors are often treated effectively, they can spawn secondary cancers called metastases which dramatically decrease chances of survival. In order to ...