2016-2017 awards

Rauan Akylzhanov

Rauan Akylzhanov

SUPERVISOR:  PROFESSOR MICHAEL RUZHANSKY

Thesis title: $L^p-L^q$ Fourier multipliers on locally compact groups

Summary: Partial differential equations PDE describe physical processes. Pseudo-differential operators is the main tool in studying non-constant coefficients partial differential equations. An important subclass is Fourier multipliers corresponding to the constant coefficient PDEs. Examples include Laplace operator, the heat operator, the Schrodinger evolution operator, the wave equation operator, the Bochner-Riesz, the resolvent operators. A fundamental problem in the study of Fourier multipliers is to relate regularity of the symbol and the boundedness of the operator. The classical results are Mikhlin-Hormander theorem and Lizorkin theorem. These have been generalized to various contexts by many authors mainly adapting the Calderon-Zygmund theory to various settings. As a consequence, these theorems require certain regularity of the sybmol. My PhD research focuses on Fourier multipliers on locally compact groups. The main insight and the tool is the theory of von Neumann algebras.

Sara Algeri

Sara Algeri

SUPERVISOR: PROFESSOR DAVID VAN DYK

Thesis title: Statistical signal identification by Testing One Hypothesis Multiple times

Summary: The identification of an individual signal with unspecified parameters can be posited statistically as a multiple hypothesis testing problem, one test for each possible value of the parameters. Unfortunately, the most common inferential procedures to correct for multiple testing may not be appropriate and/or feasible in this setting. Stringent significance requirements, for example, may be employed when the cost of a false-positive is enormous, limiting the use of simulation and resampling methods. Statistically, this setting is as an example of a hypothesis test where a nuisance parameter is present only under the alternative. The goal of this project is to propose a general method to address this problem by combining the outcomes of several dependent tests via a global test statistic. This allows us to derive an upper bound for the resulting global p-value which is shown to be less conservative than classical correction methods such as Bonferroni's correction,  while being equally generalizable, easy to compute, and sharp under long-range independence. This work is mainly motivated by the problem of the detection of particle dark matter. The solution proposed addresses both nested and non-nested frameworks and extends to one or more dimensions.

Jake Dunn

Jake Dunn

SUPERVISOR: DR CLAUDE WARNICK

Thesis title: Black Hole stability problems in anti de-Sitter spacetimes

Summary: Perhaps the most striking result from Einstein’s theory of relativity is the prediction of black holes. These are objects so massive that not even light can escape their gravitational effects. When Einstein’s field equations are phrased as an initial value problem one can evolve to a black hole solution for specific initial data. One of the main questions in mathematical relativity is: ‘What happens when one perturbs this initial data?’, or `Are the black hole solutions stable?’

My research is concerned with studying a black hole in an anti de-Sitter setting. This is where the Einstein equations have had a negative cosmological constant added to them. I have been studying a variety of problems within this setting including: the decay of solutions to the Klein.

Maximilan Engel

Maximilan Engel

SUPERVISOR: PROFESSOR JEROEN LAMB

Thesis Title: Noise-induced Phenomena in Hopf Bifurcations

Summary: The Hopf bifurcation of a dynamical system implies a transition from an attracting equilibrium to an attracting limit cycle. In a huge range of examples of such behaviour comprising laser, climate or fluid models noise is present.  I study mathematically the effect of the interaction between the noise excitation and other components of the models such as a phase amplitude coupling, also called shear.  For a certain class of such models I can quantify the bifurcation from noise-induced synchronisation to noise-induced chaos depending on the level of shear, replacing  the deterministic bifurcation by a stochastic Hopf bifurcation. Furthermore, I embed such bifurcation phenomena into the context of killed processes, studying only trajectories that survive on a bounded domain corresponding with actual physical observability of the dynamics.

Alastair Gregory

Alastair Gregory

SUPERVISOR: DR COLIN COTTER

Thesis title: Multilevel ensemble based data-assimilation for weather forecasting

Summary: Ensemble forecasts are used to quantify the uncertainty of weather systems and provide probabilistic predictions. Data-assimilation (e.g. filtering) is a framework in which observations are incorporated into these probabilistic forecasts. Despite being very flexible prediction tools, these ensemble based data-assimilation techniques are very computationally expensive to implement as they require many simulations of a random trajectory within the weather system. Thankfully, multilevel Monte Carlo is now a well-researched tool that can significantly bolster the efficiency of statistical estimation; its application to ensemble based data-assimilation is therefore of great importance. My research has proposed several methods to do exactly this, and be amongst the rapidly expanding area of literature applying multilevel Monte Carlo to this type of ensemble forecasting. The work includes solutions to problems ranging from coupling multiple filtering trajectories together, to greatly increasing the variance reduction (that controls efficiency in the statistical estimates) in filtering.

Till Hoffman

Till Hoffman

SUPERVISOR: DR NICK JONES

Title: Blau space models for social networks

Summary: Your friends are probably quite similar to you with respect to a range of attributes including age, gender, ethnicity and political orientation. Peter Blau postulated that people inhabit a high-dimensional space of social traits named in his honour, and that they are more likely to connect with one another if they are close in the space. Whilst this phenomenon is well-studied in the social sciences, explicit spatial network models have rarely been used to improve our understanding of Blau space. My research is concerned with adapting network models for Blau space, developing the statistical techniques to fit such models to data, and using the fitted models to understand which dimensions of the social space have the largest impact on how social networks form.

Andrea Petracci

Andrea Petracci

SUPERVISOR: PROFESSOR ALESSIO CORTI

Thesis title: On Mirror Symmetry for Fano varieties and for singularities

Summary: Algebraic geometry studies geometric shapes that can be defined by polynomial equations. Among these shapes, Fano varieties have a prominent role as they are 'positively curved' and are the basic building blocks.

It has been proved that in any dimension the number of Fano varieties, up to deformation, is finite, but a complete classification is known only up to dimension 3. Recent ideas, which are generally called Mirror Symmetry, coming from theoretical physics have allowed Tom Coates, Alessio Corti and their collaborators to establish a classification programme for Fano varieties in terms of the discrete geometry of polyhedra.

My research work lies in this programme. I verified a conjecture about the number of rational curves in a certain class of Fano varieties of dimension 2 and I have been studying the deformation theory of toric singularities.

Martin Weidner

Martin Weidner 

SUPERVISOR: DR TOM CASS

Thesis title: Rough differential equations on manifolds

Rough differential equations can be seen as differential equations which are subject so some external forcing or noise. Usually the noise is highly random which makes it very irregular and hard to study from an analytic point of view. For example, most stochastic differential equations (be it Ito or Stratonovich equations) can be interpreted as rough differential equations. One part of my project is concerned with the question of how rough differential equations can be understood when the solutions live on a manifold, possibly in an infinite dimensional setting. This involves some interesting methods from the study of Hopf algebras. Furthermore I try to find criteria on the vector fields of the equation which guarantee the existence of a global solution. It turns out that this leads to results that are interesting and helpful even in the well-studied case where the manifold happens to be a vector space. Finally my aim is to combine those findings with methods from Malliavin calculus in order to improve and generalise existing results on the existence of smooth densities for the solution of equations that are driven by certain types of Gaussian noise.

2016-2017

Previous awards

2015-2016

Alexis Arnaudon 

Alexis Arnaudon

SUPERVISOR: PROF DARRYL HOLM
GEOMETRIC DEFORMATIONS OF INTEGRABLE SYSTEMS

Sometimes, when deriving idealized mathematical models for the description of physical processes one encounters the so-called integrable systems. These systems have the remarkable property that powerful mathematical methods can be developed in order to explicitly compute their solutions, thus providing us with a deeper understanding of the mathematics and the physics of these models. Of course these integrable systems are almost always too simplistic and need further refinements for a use in modern applications.   My research focuses on a particular type of extensions of integrable systems that uses geometrical methods to deform them and retain or not their property of being integrable. These methods are based on hidden symmetries of the equations and provide powerful and systematic tools to perform such deformations. They led me to the discovery of interesting new nonlocal and nonlinear  partial differential equations as well as a better understanding of their geometrical structure.

Andrea Natale 

Andrea Natale 

SUPERVISOR: DR COLIN COTTER
STRUCTURE-PRESERVING DISCRETISATIONS FOR FLUIDS

Many fluid models share a common mathematical structure which is usually ignored by the standard algorithms used in computer simulations. As a consequence, numerical solutions may often yield unphysical behaviours, and fail to reproduce certain features of the governing equations, such as conservation laws or symmetries. My research is on devising finite element discretisations for fluid systems that preserve as much as possible of their mathematical structure. By addressing structure-preservation in numerical simulations, we are able to better capture the qualitative character of the solutions, even when high resolution numerics is not feasible (this is the case, for instance, in atmosphere simulations, where the range of scales involved is too wide to be fully resolved even on modern supercomputers). Moreover, preserving the structure of the equation at the discrete level allows us to derive discretisations for different models in a unified and systematic way.

Michele Nguyen

Michele Nguyen 

SUPERVISOR: DR ALMUT VERAART
MODELLING SPATIO-TEMPORAL STOCHASTIC VOLATILITY

In financial modelling, stochastic volatility is often used to account for the volatility clusters in stock prices. This refers to the alternating periods of large and small fluctuations about the mean trend. Such behaviour has also been observed in space-time, for example, in air pollution data. In these cases, modelling the spatio-temporal stochastic volatility will not only be useful for better representation, but also for prediction.

This project focuses on models in the so-called ambit framework. We model solutions of stochastic partial differential equations directly using random fields written as stochastic integrals. In the first part of the project, we study a spatio-temporal Ornstein-Uhlenbeck (STOU) process where carefully chosen integration sets determine spatio-temporal correlation structure. Secondly, we look at a volatility modulated moving average (VMMA). This is a Gaussian process convolution with an additional volatility term in the integrand. While the STOU process acts as a model of volatility, the VMMA is a model with volatility. For each model, we develop theoretical properties as well as methods for simulation and statistical inference. Since the two models can be seen as building blocks of the general ambit field, we hope to motivate similar work for the latter.

Adam Butler 

Adam Butler 

SUPERVISOR: PROF XUESONG WU
THE EARLY DEVELOPMENT OF STATIONARY CROSSFLOW VORTICES: NON-PARALLELISM, RECEPTIVITY AND MUTUAL INTERACTION

As air passes over an aircraft wing, the viscous boundary layer near the surface of the wing transitions from laminar to turbulent, producing more drag on the aircraft. The aim of Laminar Flow Control is to investigate this process and develop methods to delay this transition to turbulence, and so reduce the aircraft's fuel usage. One of the most common paths to transition is through the growth of stationary crossflow vortices. My research is focused on the initial development of these vortices close to the leading edge of the wing, through the use of asymptotic analysis and triple deck theory. The first part of my work has been to study the generation of these vortices by roughness on the surface of the wing, as well as the effect the variation of the background flow has on their subsequent growth - an effect that elsewhere can be treated as a higher-order correction, but here plays a leading-order role. I have then moved on to investigating how further downstream these vortices can interact with surface roughness in order to amplify one-another, and themselves.

Alexander Rush 

Alexander Rush

SUPERVISOR: DR EVA-MARIA GRAEFE
SEMICLASSICAL PHASE-SPACE METHODS FOR HERMITIAN AND NON-HERMITIAN QUANTUM DYNAMICS

My research is on semiclassical methods for non-Hermitian quantum mechanics. This means that I'm studying systems that don't necessarily satisfy the conservation of energy because they may be coupled to external influences such as particle gains and losses. The applications of this theory are extensive, not only in quantum mechanics but in other classical wave mechanics such as optics and electronics, where experimental applications have recently surged with the development of PT-symmetric theory. Quantum systems are very difficult in general to solve, so I'm focusing on semiclassical methods which provide simpler approaches to quantum dynamics and also tell us something about the classical limit for these non-Hermitian systems.  I have also been developing a new numerical propagator for non-Hermitian systems, informed by known methods for Hermitian systems, which is based on the semiclassical propagation of coherent states and exactly captures the quantum dynamics.

Sergey Badikov 

Sergey Badikov

SUPERVISOR: PROF MARK DAVIS
INFINITE-DIMENSIONAL LINEAR PROGRAMMES AND ROBUST HEDGING OF CONTINGENT CLAIMS

In the classical setting of mathematical finance prices of derivatives are calculated using a specific model for the underlying assets, calibrated to market prices of traded derivatives. Misspecification of a model might lead to large losses as was exhibited during the financial crisis of 2007-2008. Instead model-independent approach is designed to find bounds on the price of an untraded derivative given market prices of traded derivatives without assuming any model for the underlying. Such bounds can be computed by studying model-independent super-hedging strategies for the untraded derivative based only on traded securities. The dual of this problem becomes finding a martingale measure such that the price of the untraded derivative is maximized in the presence of market constraints given by the prices of traded securities.  In this work we formulate the problem as an infinite-dimensional linear programme as this framework is advantageous both theoretically and computationally. On the theoretical side we study strong duality and existence of optimal solutions. In particular we focus on attainment of optimal solutions in the primal problem as the topic so far has been studied in only few special cases. On the computational side we perform sensitivity analysis with respect to input parameters and study convergence rates of finite-dimensional approximations to the original problem.

Giacomo Plazzotta

Giacomo Plazzotta 

SUPERVISOR: DR CAROLINE COLIJN
LINKING TREE SHAPES TO THE SPREAD OF INFECTION USING GENERALISED BRANCHING PROCESSES

Where there is evolution, there are phylogenetic trees, i.e. branching processes. When a pathogen spreads through hosts, it evolves and adapts, and the differences in the pathogens' genomes can be captured with next generation DNA sequencing. A number of pathogen samples collected from different hosts can therefore represent the tips of a branching tree, and the structure of the branching tree can give insights into the infection dynamics. The field of inferring pathogen dynamics from genomic data is broadly termed "phylodynamics". We found analytical convergence of the frequency of shapes inside a growing branching tree; for simple shapes and homogeneous time processes this limit can be expressed with a simple function of the basic reproduction number of the pathogen. This results in a new method of inference of the reproduction number, based solely on the frequency of tree shapes, and with a precision increasing with the number of taxa. We developed an algorithm that provides an estimate of the frequency of the tree sub-shape without reconstruction of the whole tree, by-passing several issues of the current tree-reconstruction methods. In doing so, we have developed the first methods for tree-free phylodynamics. The approach is also unique in being suitable for very large data sets.

2015-16

2014-2015

Radu Cimpeanu

Radu Cimpeanu

Supervisor: Prof Demetrios Papageorgiou
Modelling, Analysis and Simulation of Incompressible Multi-Fluid Flows

Multi-fluid flows are omnipresent in our lives, from the fabrication of integrated circuit components in most electronics to the miniature laboratories inside medical tools, and even as a drop of rain splashes onto the wing of an aeroplane. In this thesis we use theoretical and numerical tools to investigate topics from the fascinating world of interfacial flows. The first part of this dissertation is dedicated to the study of multi-fluid systems in small scale channel geometries in the presence of electric fields. We use theoretical and numerical tools to design electric field protocols that target applications such as microfluidic mixing, flux generation and polymer self-assembly in simple geometrical configurations where this has not been achieved before. In the second part of this thesis we turn our attention to interfacial flows in aerodynamics, in particular to the process of formation of liquid layers on aircraft surfaces under adverse weather conditions. We then extend the powerful asymptotic framework of triple-deck theory to analyse changes in the process of flow separation induced by the resulting liquid films.

Fyodor Gainullin

Fjodor Gainullin

Supervisor: Dr Dorothy Buck
Applications of Heegaard Floer homology to Dehn surgery

My research is about surgery on 3-manifolds (i.e. 3-dimensional spaces). When doing surgery, we cut out a piece from a manifold and attach another one into its place. In 20th century, 3-manifold surgeons used relatively simple tools like scalpels and forceps to do surgery on manifolds. More than 10 years ago some people came up with a new exciting theory, called Heegaard Floer homology. As an application they built a fancy robot (called ‘Mapping cone formula’) that does surgery on 3-manifolds. Even though this robot is not precise enough to do everything, it still has enabled many new things in 3-manifold surgery. In my research I am using this new tool for doing surgery on 3-manifolds. (Disclaimer for those who got too excited reading this: all the physical tools mentioned above are just figures of speech. No 3-manifold was hurt during my PhD.)

Andrew McRae

Andrew McRae

Supervisor: Dr Colin Cotter and dr david ham
Compatible finite element methods for atmospheric dynamical cores

My research is on numerical methods for geophysical fluid simulations, particularly applied to weather forecasting.  As single-core computing performance has stagnated in the last decade, current supercomputers are being built with ever-larger numbers of processing cores.  Many current weather models use finite different methods with a latitude-longitude grid structure, which leads to a parallel load imbalance due to the convergence of gridlines at the poles.  In my research, I am exploring the use of a particular class of finite element methods, which we label 'compatible' or 'mimetic'.  These allow us to obtain many of the favourable numerical properties of the existing methods, but on arbitrary grids of the atmosphere -- in particular, quasi-uniform meshes, which avoids problems with parallel scaling.

Thomas Prince

Thomas Prince

Supervisor: Prof Tom Coates
Applications of mirror symmetry to the classification of Fano varieties

My research is primarily motivated by the classification of Fano manifolds, important objects in algebraic geometry, utilising the phenomenon of mirror symmetry. This phenomenon conjectures a correspondence between a Fano manifold and a 'mirror-dual' manifold with a function (superpotential) which encodes the geometry of the Fano manifold in a significantly different way. Recently a great deal of progress has been made toward understanding the phenomenon of mirror symmetry for Fano manifolds. In particular in joint work with a group based at Imperial, we have a large number of candidate mirror-dual models to Fano manifolds. My work has been centered on developing techniques to reconstruct a Fano manifold out of these candidate mirror models, as well as giving geometric significance to our candidates. For example in joint work with Professors Coates and Kaspryzk we have found 527 new Fano 4-folds and, in the case of surfaces, have used an algorithm of Gross-Siebert to construct Fano (orbifolds) from their mirror-dual via toric degeneration.