Mathematics, School of
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Parabolic boundary value problems with rough coefficients
(The University of Edinburgh, 20181129)This thesis is motivated by some of the recent results of the solvability of elliptic PDE in Lipschitz domains and the relationships between the solvability of different boundary value problems. The parabolic setting has ... 
Crepant resolution conjecture for DonaldsonThomas invariants via wallcrossing
(The University of Edinburgh, 20181129)Let Y be a smooth complex projective Calabi{Yau threefold. DonaldsonThomas invariants [Tho00] are integer invariants that virtually enumerate curves on Y. They are organised in a generating series DT(Y) that is interesting ... 
Wellposedness of the onedimensional derivative nonlinear Schrödinger equation
(The University of Edinburgh, 20181129)This thesis is concerned with the wellposedness of the onedimensional derivative nonlinear Schrodinger equation (DNLS). In particular, we study the initialvalue problem associated to DNLS with lowregularity initial ... 
Many server queueing models with heterogeneous servers and parameter uncertainty with customer contact centre applications
(The University of Edinburgh, 20181129)In this thesis, we study the queueing systems with heterogeneous servers and service rate uncertainty under the HalfinWhitt heavy traffic regime. First, we analyse many server queues with abandonments when service rates ... 
Singularities of noncommutative surfaces
(The University of Edinburgh, 20181129)The primary objects of study in this thesis are noncommutative surfaces; that is, noncommutative noetherian domains of GK dimension 2. Frequently these rings will also be singular, in the sense that they have infinite ... 
On knormality and regularity of normal projective toric varieties
(The University of Edinburgh, 20180702)We study the relationship between geometric properties of toric varieties and combinatorial properties of the corresponding lattice polytopes. In particular, we give a bound for a very ample lattice polytope to be knormal. ... 
Bridgeland stability conditions, stability of the restricted bundle, BrillNoether theory and Mukai's program
(The University of Edinburgh, 20181129)In [Bri07], Bridgeland introduced the notion of stability conditions on the bounded derived category D(X) of coherent sheaves on an algebraic variety X. This topic is originally inspired by concepts in string theory and ... 
Stochastic, distributed and federated optimization for machine learning
(The University of Edinburgh, 20171130)We study optimization algorithms for the finite sum problems frequently arising in machine learning applications. First, we propose novel variants of stochastic gradient descent with a variance reduction property that ... 
Design of reliable aerospace system architecture
(The University of Edinburgh, 20181129)Reliability and redundancy of safetycritical network systems is a paramount issue in system engineering. Be it in evaluating existing network systems or solving optimization problems for designing network systems, it ... 
Statistical methods for the testing and estimation of linear dependence structures on paired highdimensional data: application to genomic data
(The University of Edinburgh, 20180702)This thesis provides novel methodology for statistical analysis of paired highdimensional genomic data, with the aimto identify gene interactions specific to each group of samples as well as the gene connections that ... 
Heuristic methods for solving two discrete optimization problems
(The University of Edinburgh, 20180702)In this thesis we study two discrete optimization problems: Traffic Light Synchronization and Location with Customers Orderings. A widely used approach to solve the synchronization of traffic lights on transport networks ... 
Data sampling strategies in stochastic algorithms for empirical risk minimization
(The University of Edinburgh, 20180702)Gradient descent methods and especially their stochastic variants have become highly popular in the last decade due to their efficiency on big data optimization problems. In this thesis we present the development of data ... 
Controlling optical beams in nematic liquid crystals
(The University of Edinburgh, 20180702)A major area of research recently has been the study of nonlinear waves in liquid crystals. The availability of commercial liquid crystals and the formation of solitons at mWpower levels has meant that experimental ... 
Generalised Langevin equation: asymptotic properties and numerical analysis
(The University of Edinburgh, 20180702)In this thesis we concentrate on instances of the GLE which can be represented in a Markovian form in an extended phase space. We extend previous results on the geometric ergodicity of this class of GLEs using Lyapunov ... 
Structure and semantics
(The University of Edinburgh, 20171130)Algebraic theories describe mathematical structures that are defined in terms of operations and equations, and are extremely important throughout mathematics. Many generalisations of the classical notion of an algebraic ... 
On numerical approximations for stochastic differential equations
(The University of Edinburgh, 20171130)This thesis consists of several problems concerning numerical approximations for stochastic differential equations, and is divided into three parts. The first one is on the integrability and asymptotic stability with ... 
Improved computational approaches to classical electric energy problems
(The University of Edinburgh, 20171130)This thesis considers three separate but connected problems regarding energy networks: the load flow problem, the optimal power flow problem, and the islanding problem. All three problems are nonconvex non linear problems, ... 
Entanglement entropy of locally perturbed thermal systems
(The University of Edinburgh, 20171130)In this thesis we study the time evolution of Rényi and entanglement entropies of thermal states in Conformal Field Theory (CFT). These quantities are usually hard to compute but RyuTakayanagi (RT) and HubenyRangaman ... 
Birational geometry of Fano fibrations
(The University of Edinburgh, 20170710)An algebraic variety is called rationally connected if two generic points can be connected by a curve isomorphic to the projective line. The output of the minimal model program applied to rationally connected variety is ... 
Modelling sustainable intensification in Brazilian agriculture
(The University of Edinburgh, 20170710)At the United Nations Framework Conference on Climate Change COP15 (2009) Brazil presented ambitious commitments or Nationally Appropriate Mitigation Actions (NAMAs), to reduce greenhouse gases emissions (GHGs) mitigation ...