Mathematics, School of
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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 ... 
Stochastic dispersive PDEs with additive spacetime white noise
(The University of Edinburgh, 20191128)In this thesis, we will discuss the Cauchy problem for some nonlinear dispersive PDEs with additive spacetime white noise forcing. We will focus on two different models: the stochastic nonlinear beam equation (SNLB) ... 
Quantitative propagation of chaos of McKeanVlasov equations via the master equation
(The University of Edinburgh, 20191128)McKeanVlasov stochastic differential equations (MVSDEs) are ubiquitous in kinetic theory and in controlled games with a large number of players. They have been intensively studied since McKean, as they pave a way to ... 
Stochastic PDEs beyond standard monotonicity: well posedness and regularity of solutions
(The University of Edinburgh, 20191128)Nonlinear stochastic partial differential equations (SPDEs) are used to model wide variety of phenomena in physics, engineering, finance and economics. In many such models the equations exhibit superlinear growth. In ... 
Tilting theory of contraction algebras
(The University of Edinburgh, 20190701)This thesis focuses on a class of finite dimensional symmetric algebras arising in geometry, known as contraction algebras. The main results presented here combine to give a complete description of the derived equivalence ... 
Applications of the blowup technique in singularly perturbed chemical kinetics
(The University of Edinburgh, 20190701)This thesis addresses the geometric analysis of traveling front propagation in singularly perturbed dynamical systems. The study of front propagation in reactiondiffusion systems has received a significant amount of ... 
Basis preconditioning in interior point methods
(The University of Edinburgh, 20190701)Solving normal equations AAᵀx = b, where A is an m x n matrix, is a common task in numerical optimization. For the efficient use of iterative methods, this thesis studies the class of preconditioners of the form BBᵀ , where ...