Merit Award Recipients
20222023 awards
Hiroyuki Miyoshi My PhD research uses complex analysis techniques to tackle transport problems in multiply connected domains. A key tool in my research is the SchottkyKlein prime function developed by Darren Crowdy. I am particularly interested in finding analytical solutions to transport problems that arise in engineering applications, such as flows in channels with superhydrophobic or liquidinfused surfaces. These challenges fall into the category of mixed boundary value problems, for which finding an analytical solution is known to be a formidable task. Together with my supervisor, we have developed a novel formula, called "Generalized Schwarz Integral Formula for multiply connected domains", to solve mixed boundary value problems. Numerical results show that this method has broad applicability, extending to the determination of the shape of hollow vortices and the capacitance of periodic electrodes. 

Daniel Goodair I have broad interests in the theory of stochastic partial differential equations, with a particular emphasis on the evolution equations of fluid dynamics under transport and advection noise on a bounded domain. My main contributions have been towards the wellposedness of stochastic partial differential equations, for an abstract nonlinear equation as well as the Euler and NavierStokes equations perturbed by differential noise with physical boundary. The introduction of a boundary gives these results novelty, a direction I have pursued further by studying the inviscid limit of stochastic NavierStokes with different boundary conditions and noise scaling. I am keen to explore the relationship between stochasticity and the boundary layer, including the regularising effect that noise could have. 

Henry Alston In my research, I aim to quantify the dynamics and thermodynamic properties of nonequilibrium systems, paying particular attention to systems with nonequilibrium (i.e. fluctuating or nonreciprocal) interactions. Biological examples of such systems include pilimediated interactions between bacteria and chemotactic interactions between cells. I employ techniques from statistical mechanics, condensed matter physics and stochastic thermodynamics theory to answer questions on the physics of living systems. 

Augusto Del Zotto My area of research is related to the hydrodynamic stability of steady solutions of the NavierStokes equations. In particular, during my Ph.D., I studied the transition threshold problem, which describes the transition between stable and turbulent regimes and links the size of perturbations to the consequent generated regime. The longtime behaviour of perturbed solutions is established by analysing stability and instability mechanisms that may occur at both a linear and nonlinear level. A key feature in the study of the transition threshold is the enhanced dissipation phenomenon, which is generated by the mixing properties of the steady states considered and provides stability. I have studied the transition problem relative to the 2D Poiseuille flow using hypocoercivity methods and the 3D stably stratified Couette flow in the Boussinesq equations, taking advantage of the dispersive nature of the equations. These analysis lead to an improvement of the known thresholds for both cases. 

Lancelot Da Costa My goal is to advance our understanding of intelligent systems by modelling cognitive systems and improving artificial systems. Accordingly, my research lies at the intersection of applied mathematics, cognitive science, and artificial intelligence. Current projects include: Probabilistic foundations of machine learning, Bayesian modelling of cognitive systems, Decisionmaking in autonomous agents, World model and causal representation learning. 
Previous awards
20202021
Esther Bou Dagher I am interested in the study of functional inequalities (Hardy, Sobolev, Poincaré, Logarithmic Sobolev, MoserTrudinger, etc.) which play a fundamental role in obtaining a priori estimates for solutions of PDEs and in analysing the long time behaviour of solutions of evolution problems. In my PhD thesis, I am concerned with proving and applying Poincaré and Logarithmic Sobolev inequalities in the setting of Carnot groups. In particular, in the setting of steptwo Carnot groups, I prove those inequalities for probability measures as a function of various homogeneous norms. To do that, the key idea is to obtain an intermediate inequality called the UBound inequality (based on joint work with my supervisor B. Zegarliński). Using this UBound inequality, we show that certain infinite dimensional Gibbs measures with unbounded interaction potentials as a function of homogeneous norms on an infinite product of Carnot groups satisfy the Poincaré inequality (based on joint work with Y. Qiu, B. Zegarliński, and M. Zhang). I also enlarge the class of measures as a function of the CarnotCarathéodory distance that gives us the q−Logarithmic Sobolev inequality in the setting of Carnot groups. As an application, I use the HamiltonJacobi equation in that setting to prove the p−Talagrand inequality and hypercontractivity. 

Rodolfo Brandao Macena Lira I use asymptotic and singularperturbation techniques to analyse contemporary problems in fluid mechanics and wave motion. I am particularly interested in modelling the dynamics of Leidenfrost droplets, namely evaporating liquid droplets levitated by their vapour above a superheated substrate. Recent experiments have revealed that these droplets, well known for their high mobility, can also undergo "symmetry breaking", leading to a spontaneous rolling motion in the absence of external forces. Using matched asymptotic expansions, my supervisor and I have derived and analysed a dynamic model of a Leidenfrost droplet that predicts symmetrybreaking spontaneous motion and rationalises several other experimental observations. I am also interested in problems involving wave diffraction in "acoustic metasurfaces", namely devices formed by arrays of resonant microstructures. During my PhD, we have used asymptotic methods to develop new analytical models for acoustic metasurfaces that systematically account for thermoviscous dissipative effects—which are key to understanding the extraordinary sound absorption capabilities of these surfaces. 
20182019
Rishabh Gvalani "My area of research lies at the intersection of probability, partial differential equations (PDEs), and the calculus of variations. Specifically, I study large systems of identical interacting particles which are driven by pairwise interactions between the individual particles and white noise. These systems are mathematical formulations of toy models commonly used in statistical physics to describe the behaviour of molecules in a rarefied gas. In the large particle limit (also known as the meanfield limit), their mean behaviour is governed by a nonlocal parabolic aggregationdiffusion type PDE. My research involves studying the qualitative properties of this equation (mainly existence and multiplicity of stationary solutions), fluctuations of the particle behaviour around it, and other interesting mathematical questions such as homogenisation and the existence of phase transitions. 

Riccardo Passeggeri During my PhD I explored various research topics in the interface between probability and mathematical statistics: signatures (rough path theory), limit theorems and quasiinfinite divisibility. The signature "summarises" the behaviour of a stochastic process and is a fundamental object in the theory of rough paths. I obtained various properties for the signature of the fractional Brownian motion (fBm) and I extended the cubature method, which is a deterministic method for solving SDEs, to the fBm case. 

Francesco Sanna Passino My research is on latent feature representations of dynamic networks, with applications to statistical cybersecurity. During the first two years of my PhD, I have worked on statistical methods for separation of human and automated activity in network flow data, models for network dynamics using PitmanYor processes, new link prediction in large networks using extensions of the Poisson matrix factorisation model, and selection of the latent dimension and number of communities in stochastic blockmodels, interpreted as generalised random dot product graphs. Statistical approaches to cybersecurity related problems have recently emerged as powerful tools for network intrusion detection systems. Statistical models have the ability to learn the normal behaviour of users and machines in an enterprise network, and identify deviations from the model as evidence potentially malicious behaviour, complementing signaturebased methods. The aim of my PhD work is to develop models for statistical analysis of computer networks at three different levels of resolution: entire network, nodes, and edges. 

Sarab Sethi For a long time scientists have relied on the information contained in animal vocalisations when assessing the biodiversity of an area. This is especially true when working in tropical forests where visual surveys are limited by high canopies and dense undergrowth. However, manual surveys are costly, provide poor sampling resolution and suffer from observer bias. We have been working on completely automating the process of biodiversity surveying, from data capture and transmission in real time from remote field sites, to the analysis required to derive information relating to the health of the ecosystem. Our work is based in the tropical forests of Malaysian Borneo where we hope to gain insight into the short and longterm effects of changing landuse on natural ecosystems. Beyond ecological studies we are developing methods to identify anomalous sounds such as chainsaws or gunshots to help detect and prevent illegal logging and poaching in these protected regions. 

Yang Li My research area lies in differential geometry and geometric analysis, with an emphasis on CalabiYau metrics, G2 metrics and their degeneration behaviours. Some of my previous results include the construction of an exotic CalabiYau metric on C3 with singular tangent cone at infinity, the description of collapsing CalabiYau metrics on K3 fibred CalabiYau 3folds, and the solution to the Dirichlet problem of the maximal graph system which arises from adiabatic limits of G2 manifolds. More recently I constructed the generalisation of the TaubNUT and OoguriVafa metrics in complex dimension 3, related to the StromingerYauZaslow conjecture. Besides, I am broadly interested in complex and algebraic geometry, gauge theory, special holonomy, and their connections to physics. 

Rosalba Garcia Millan In my research I study correlations in space and time to answer questions in physics and biology. I am involved in a number of projects and collaborations, ranging from reactiondiffusion processes to epigenetics. For example, I have studied a model known as branching process, which has applications in population dynamics, avalanches and neuronal activity. I looked at characterising its universal features and calculating a number of observables, such as the temporal profile (or avalanche shape). In a project with the NonEquilibrium Systems group, we let the branching particles diffuse on a general graph and calculated the explored volume. This has applications to the space travelled by a colony of infectious bacteria or the spread of tumorous cells in our body. On the experimental side of my research, I am collaborating with the Merkenschlager lab at the Hammersmith campus to investigate the spatial organisation of DNA in the cell nucleus and what the impact of this organisation in gene transcription is. The methods I use in my research are field theory, probabilistic methods, computer simulations and data analysis. Other problems I have researched are: the Oslo rice pile model, runandtumble motion and the concealed voter model. 
20172018
Francesca CarocciSUPERVISOR: PROFESSOR Richard ThomasThesis title: Moduli spaces and enumerative invariants arising from them I am interested in algebraic geometry and more specifically I have been studying moduli spaces and enumerative invariants arising from them. 

Maxime MorariuPatrichiSUPERVISOR: Dr Mikko pakkanenThesis title: Highfrequency financial data modelling with statedependent Hawkes processes Hawkes processes are a class of selfexciting point processes in which events of different types can precipitate each other, breaking the memorylessness property of Poisson processes. Given their ability to capture clustering phenomena, they have found numerous applications in finance in the last decade, in particular in the area of highfrequency data modelling. Indeed, the last twenty years have also seen the emergence of electronic orderdriven markets, where agents submit buy and sell orders to a virtual exchange via their computers. As billions of orders are submitted each day, this brought a profusion of new data to study, with a chance to understand the price formation mechanism at the smallest timescales. In this context, Hawkes processes have been used as a model of the order flow, the core idea being to specify a list of order types (i.e., orders with different effects on the market) and fit a Hawkes process to their timestamps to gain insights on the market dynamics. However, the main limitation of this approach is that Hawkes processes do not model the state of the market and its influence on the arrival rates of orders. That is why I have introduced the class of statedependent Hawkes processes, an extension of Hawkes processes where the events can now interact with an auxiliary state process. I have worked on both the theoretical foundations of this new class (existence and uniqueness) and its application (statistical inference from real market data). 

Markus SchmidtchenSUPERVISOR: Professor Jose CarrilloSystems of many interacting particles are ubiquitous in nature. In fact we encounter such systems every day in our lives, probably without even noticing. Be it in form of the traffic jam we get stuck in every morning, the crowded hallways of Imperial College on our way to the office at the beginning of term, or maybe, even when we let our minds wander watching bird flocks in the sky. All these systems have something very peculiar in common — the emergence of collective behaviour. In the absence of any leader they can form greater structures like flocks, mills, swarms, fish schools, to name a few. My research revolves around the formation of (morphological) patters arising from interspecific and intraspecific interactions in manyagent systems of two different species. A striking phenomenon is the phase segregation leading to disjoint regions where only one of the species is represented (such as the yellow xantophores and the black melanophores in zebrafish). A big mathematical challenge is the loss of regularity when this unmixing occurs. Initially smooth data may soon lose most of their regularly and exhibit discontinuities which is why classical estimates fail. I study these systems from different angles including modelling, numerical simulations, numerical analysis as well as more classical questions such as providing existence. 

Jakub WitaszekSUPERVISOR: PROFESSOR Paolo CasciniThe study of geometric shapes in mathematics is often divided into two steps. First, we identify the most basic geometric objects and, thereafter, we describe how to build more complex shapes from them (for instance, imagine a cylinder formed from a line swept along a circle). This basic idea is the premise of the Minimal Model Program, which provides a precise recipe for building algebraic varieties  geometric objects described by polynomials  from shapes with simple geometries: negativelycurved, flat, and positivelycurved. An important part of my research pertains to the development of the Minimal Model Program in positive characteristic, that is, for algebraic varieties defined as solutions of polynomial equations modulo a prime number p. Furthermore, I have been pursuing research on Frobenius liftings (special automorphisms of algebraic varieties) and their relation to various problems in classical algebraic geometry. 

Nurgissa YessirkegenovSUPERVISOR: Professor Michael RuzhanskyMy research is on subelliptic functional inequalities of different types and the corresponding function spaces on homogeneous groups. Inequalities of Hardy, Rellich, GagliardoNirenberg, CaffarelliKohnNirenberg, TrudingerMoser play a fundamental role in many subjects, most importantly allowing one to obtain apriori estimates for the wellposedness analysis of partial differential equations. As a byproduct linking Hardy inequalities in the integral form to the graded structure by considering Riesz kernels of Rockland operators gave a new proof of Sobolev inequalities but, most importantly, provided a new way of using Sobolev and other differential structures in various weighted estimates, opening up wide perspectives of further applications to hypoelliptic partial differential equations of different types, both linear and nonlinear. I am also interested in applications of these theory, most importantly, for higher order hypoelliptic evolution equations (nonlinear diffusion, Schrodinger, wave) where the traditional heat kernel methods largely fail due to the heat kernel being no longer realvalued." 

Pierrick BousseauSUPERVISOR: PROFESSOR Richard ThomasThesis title: Quantum mirrors of log CalabiYau surfaces and higher genus curve counting. I am interested in parts of algebraic geometry having close interactions with theoretical physics. This includes mirrorsymmetry, derived categories, curve counting theories, 

Tom McGrathSUPERVISORs: Dr Nick Jones and PROFESSOR Kevin Murphy
My research focuses on developing new methods to understand homeostatic behaviour, specifically the regulation of food intake (although the methods could be adapted to other problems). Feeding behaviour is of both theoretical and practical importance: it is a fundamental behaviour required by almost all organisms, and the brain areas responsible for controlling food intake are strongly evolutionarily conserved, with great similarities between disparate organisms. However, the mismatch between evolutionarily successful feeding strategies and the modern environment has left us with a growing obesity epidemic, and understanding how to reduce food intake would be of great health benefit. Working with collaborators in the Department of Medicine, we have developed models of food intake suitable for use with highresolution data from mice and rats, and have used these models to carry out an extensive study of the effects of different anorectic drugs on the regulation of food intake, as well as investigating the possible impact of behavioural and diet changes on feeding. 
20162017
Rauan AkylzhanovSUPERVISOR: PROFESSOR MICHAEL RUZHANSKYThesis title: $L^pL^q$ Fourier multipliers on locally compact groups Summary: Partial differential equations PDE describe physical processes. Pseudodifferential operators is the main tool in studying nonconstant 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 BochnerRiesz, 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 MikhlinHormander theorem and Lizorkin theorem. These have been generalized to various contexts by many authors mainly adapting the CalderonZygmund 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 AlgeriSUPERVISOR: PROFESSOR DAVID VAN DYKThesis 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 falsepositive 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 pvalue 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 longrange independence. This work is mainly motivated by the problem of the detection of particle dark matter. The solution proposed addresses both nested and nonnested frameworks and extends to one or more dimensions. 
Jake DunnSUPERVISOR: DR CLAUDE WARNICKThesis title: Black Hole stability problems in anti deSitter 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 deSitter 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 EngelSUPERVISOR: PROFESSOR JEROEN LAMBThesis Title: Noiseinduced 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 noiseinduced synchronisation to noiseinduced 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 GregorySUPERVISOR: DR COLIN COTTERThesis title: Multilevel ensemble based dataassimilation for weather forecasting Summary: Ensemble forecasts are used to quantify the uncertainty of weather systems and provide probabilistic predictions. Dataassimilation (e.g. filtering) is a framework in which observations are incorporated into these probabilistic forecasts. Despite being very flexible prediction tools, these ensemble based dataassimilation 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 wellresearched tool that can significantly bolster the efficiency of statistical estimation; its application to ensemble based dataassimilation 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 HoffmanSUPERVISOR: DR NICK JONESTitle: 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 highdimensional 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 wellstudied 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 PetracciSUPERVISOR: PROFESSOR ALESSIO CORTIThesis 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 WeidnerSUPERVISOR: DR TOM CASSThesis 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 wellstudied 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. 
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
Radu CimpeanuSupervisor: Prof Demetrios Papageorgiou


Fjodor GainullinSupervisor: Dr Dorothy Buck


Andrew McRaeSupervisor: Dr Colin Cotter and dr david ham


Thomas PrinceSupervisor: Prof Tom Coates
