Esther Bou Dagher
SUPERVISOR: Professor Boguslaw Zegarlinski

I am interested in the study of functional inequalities (Hardy, Sobolev, Poincaré, Logarithmic Sobolev, Moser-Trudinger, 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 step-two 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 U-Bound inequality (based on joint work with my supervisor B. Zegarliński). Using this U-Bound 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 Carnot-Carathéodory distance that gives us the q−Logarithmic Sobolev inequality in the setting of Carnot groups. As an application, I use the Hamilton-Jacobi equation in that setting to prove the p−Talagrand inequality and hypercontractivity.

Rodolfo Brandao Macena Lira
SUPERVISOR: Dr. Ory Schnitzer

I use asymptotic and singular-perturbation 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 symmetry-breaking 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.

Rishabh Gvalani
SUPERVISOR: Professor Greg Pavliotis, Prof. José A. Carrillo

"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 mean-field limit), their mean behaviour is governed by a nonlocal parabolic aggregation-diffusion 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.
Other topics I am interested in include equations with degenerate diffusion, kinetic equations such as the Boltzmann and Landau equations, the theory of gradient flows in metric spaces, and optimal transport and its applications to nonlinear PDE.

Riccardo Passeggeri
SUPERVISOR: Professor Almut Veraart
Thesis title: On quasi-infinite divisibility, limit theorems and signatures

During my PhD I explored various research topics in the interface between probability and mathematical statistics: signatures (rough path theory), limit theorems and quasi-infinite 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.
Concerning limit theorems, my supervisor and I explored the ergodic properties of infinitely divisible (ID) random fields, where a random field is a generalisation of a stochastic process and a probability distribution is ID if it can be expressed as the probability distribution of the sum of an arbitrary number of i.i.d. random variables. For example, we showed that moving averages are mixing (a property stronger than ergodicity). Then, we studied limit theorems for ambit processes, which are processes developed for the study of turbulences. Our results allows to make inference about the intermittency component (i.e. the volatility) of turbulences.
Finally, I investigated the class of quasi-infinite divisible distributions. They are dense in the space of all probability distributions with respect to convergence in distribution.

Francesco Sanna Passino
SUPERVISOR: Dr Nicholas A Heard

My research is on latent feature representations of dynamic networks, with applications to statistical cyber-security. 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 Pitman-Yor 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 cyber-security 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 signature-based 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
SUPERVISOR: Dr Nick Jones
Thesis title: Fully automated acoustic monitoring in the tropical forests of Borneo

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 long-term effects of changing land-use 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
SUPERVISOR: Professor Simon K Donaldson

My research area lies in differential geometry and geometric analysis, with an emphasis on Calabi-Yau metrics, G2 metrics and their degeneration behaviours. Some of my previous results include the construction of an exotic Calabi-Yau metric on C3 with singular tangent cone at infinity, the description of collapsing Calabi-Yau metrics on K3 fibred Calabi-Yau 3-folds, 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 Taub-NUT and Ooguri-Vafa metrics in complex dimension 3, related to the Strominger-Yau-Zaslow conjecture. Besides, I am broadly interested in complex and algebraic geometry, gauge theory, special holonomy, and their connections to physics.

Rosalba Garcia Millan
SUPERVISOR: Dr Gunnar Pruessner

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 reaction-diffusion 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 Non-Equilibrium 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, run-and-tumble motion and the concealed voter model.

Francesca Carocci

SUPERVISOR:  PROFESSOR Richard Thomas

Thesis 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.
Over the past 30 years, motivated by physicists predictions obtained via string theory and mirror symmetry, algebraic geometers constructed a number of different moduli spaces
attempting to answer in a satisfactory way, i.e. matching the physicist results, the following question: given a smooth projective variety, how many algebraic curves lie on it?
My current research is mostly devoted to study:
local contributions of singular curves to a version of Donaldson-Thomas invariants on CY 3-folds which conjecturally give a mathematical definition of genus 0 Gopakumar -Vafa invariants, and alternate compactification of Konsietvich moduli space of stable maps whose associated invariants should avoid the contribution of the boundary components.

Maxime Morariu-Patrichi

SUPERVISOR: Dr Mikko pakkanen

Thesis title: High-frequency financial data modelling with state-dependent Hawkes processes

Hawkes processes are a class of self-exciting 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 high-frequency data modelling. Indeed, the last twenty years have also seen the emergence of electronic order-driven 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 state-dependent 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 Schmidtchen 

SUPERVISOR: Professor Jose Carrillo

Systems 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 many-agent 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 un-mixing 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 Witaszek

SUPERVISOR: PROFESSOR Paolo Cascini

The  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: negatively-curved,  flat,  and  positively-curved.

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 Yessirkegenov 

SUPERVISOR: Professor Michael Ruzhansky 

My research is on subelliptic functional inequalities of different types and the corresponding function spaces on homogeneous groups. Inequalities of Hardy, Rellich, Gagliardo-Nirenberg, Caffarelli-Kohn-Nirenberg, Trudinger-Moser play a fundamental role in many subjects, most importantly allowing one to obtain a-priori estimates for the well-posedness 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 real-valued."

Pierrick Bousseau 

SUPERVISOR: PROFESSOR Richard Thomas

Thesis title: Quantum mirrors of log Calabi-Yau 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, 
Donaldson-Thomas invariants, stability conditions, quiver representations and cluster varieties. In particular, I worked on a correspondence theorem between q-refined tropical curve counting and higher genus Gromov-Witten invariants, with an application to the construction of deformation quantizations of families of log Calabi-Yau surfaces coming from mirror symmetry.

Tom McGrath 

SUPERVISORs: 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 high-resolution 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.

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

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

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

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

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

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

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 

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.

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.

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

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

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.