Merit Award Recipients
20152016
Alexis ArnaudonSUPERVISOR: PROF DARRYL HOLM Sometimes, when deriving idealized mathematical models for the description of physical processes one encounters the socalled 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 NataleSUPERVISOR: DR COLIN COTTER 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 structurepreservation 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 NguyenSUPERVISOR: DR ALMUT VERAART 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 spacetime, for example, in air pollution data. In these cases, modelling the spatiotemporal stochastic volatility will not only be useful for better representation, but also for prediction. This project focuses on models in the socalled 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 spatiotemporal OrnsteinUhlenbeck (STOU) process where carefully chosen integration sets determine spatiotemporal 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 ButlerSUPERVISOR: PROF XUESONG WU 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 higherorder correction, but here plays a leadingorder role. I have then moved on to investigating how further downstream these vortices can interact with surface roughness in order to amplify oneanother, and themselves. 

Alexander RushSUPERVISOR: DR EVAMARIA GRAEFE My research is on semiclassical methods for nonHermitian 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 PTsymmetric 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 nonHermitian systems. I have also been developing a new numerical propagator for nonHermitian 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 BadikovSUPERVISOR: PROF MARK DAVIS 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 20072008. Instead modelindependent 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 modelindependent superhedging 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 infinitedimensional 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 finitedimensional approximations to the original problem. 

Giacomo PlazzottaSUPERVISOR: DR CAROLINE COLIJN 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 subshape without reconstruction of the whole tree, bypassing several issues of the current treereconstruction methods. In doing so, we have developed the first methods for treefree phylodynamics. The approach is also unique in being suitable for very large data sets. 
20142015






