Research degrees

There are a number of funding opportunities available. Some cover both stipend (to cover living costs) and tuition fees, while others may cover fees only.

All our PhD students, including those who are self-funded, receive a Research and Training Support Grant per annum to support training, attending conferences and research-related costs. 

All scholarships, bursaries or other awards are offered on a competitive basis. Some funding opportunities may have earlier deadlines, so please check carefully.  

• Funded PhD projects

The Department offers a range of funded PhD projects through a variety of funders, and research grants – set out in the next tab and searchable in King’s funding database.

• EPSRC Doctoral Landscape Award Studentships

Candidates who are accepted onto a research degree programme will be considered for Engineering & Physical Sciences Research Council (EPSRC) funding via our Doctoral Landscape Awards.

All eligible PhD applicants are considered for these awards when they apply to the Department. Please cite the code EPSRC-Maths-2026-27 in the Funding section of the application form. Please select option 5 ‘I am applying for a funding award or scholarship administered by King’s College London’ and type the code into the ‘Award Scheme Code or Name’ box. Please copy and paste the code exactly.

• Science & Technology Facilities Council (STFC)

Applicants for the Applied Mathematics Research: Theoretical Physics MPhil/PhD are eligible to apply for STFC studentship funding. Please cite the code STFC-Maths-2026-27 in the Funding section of the application form. Please select option 5 ‘I am applying for a funding award or scholarship administered by King’s College London’ and type the code into the ‘Award Scheme Code or Name’ box. Please copy and paste the code exactly. Find out more here.

We offer PhD projects under a range of different research themes in pure and applied mathematics. Applicants can explore the funded studentships listed here and apply to ones that suit their interests.

In addition, applicants can also identify more topics and supervisors by looking at the individual research pages of potential supervisors who can be contacted directly. Please see our Funding tab for further information.

Please note the application deadlines on each funded project advert, as these may vary.

Projects with funding:

• Genome Data Science and AI for Therapeutic Target Discovery - King’s Prize Doctoral Programme

Other projects with potential funding

We welcome applications  from students who have secured  or are applying for other funding (within other studentships internal to the university or external schemes) and from self-funded students. Please see our funding tab (above) for information on PhD funding opportunities. When applying for these projects please specify the name of the supervisor and any relevant funding codes, found on the funding tab.'

Supervisor: Dionysios Anninos
Title: Group theory and the de Sitter universe

Abstract: The project aims to develop the interplay between group theoretic aspects of the de Sitter isometries and quantum fields on a fixed de Sitter background. A particular emphasis will be placed on gauge fields, both of integer and half-integer spin. The structure of entanglement of these fields will be considered.

Supervisor: Nadav Drukker
Title: Nonlocal observables, conformal anomalies and conformal manifolds

This project will straddle both sides of the AdS/CFT correspondence. On the field theory side we will study non-local observables, in particular surface operators in four dimensions. Their holographic duals are some high dimensional branes embedded in AdS_5 x S^5 space. To date, only the most symmetric configurations of this type have been studied and we will explore ways to find less symmetric ones. We will also compute using holography and conformal field theory techniques the conformal anomaly associated to them and correlation functions of local operators inserted on them. In the process you will learn a lot about  the AdS/CFT correspondence, conformal field theory in general and many different conceptual and computational techniques.

Supervisor: Chris Herzog
Title: Charting the Landscape of Defect and Boundary Quantum Field Theory

Abstract: Most of the major progress in theoretical physics over the last quarter century is associated with gravity and quantum field theory (QFT) in mixed dimensional systems -- whether that means black hole horizons, topological insulators, D-branes in string theory, twist defects in computations of entanglement entropy, or the interplay between the boundary and bulk in AdS/CFT correspondence.  This progress suggests major fundamental gaps in our formulation of gravity and QFT in mixed dimensional systems that cries out for reconsideration and development.  The aim of this PhD project will be to chart the renormalization group landscape of quantum field theories in the presence of boundaries and defects.  A variety of approaches will be used, from more conventional epsilon expansion and large N to newer AdS/CFT and numerical bootstrap techniques.  Results in this project may have direct experimental relevance for graphene and carbon nanotubes and also for flux tubes and Wilson lines in gauge theories.

Supervisor: Petr Kravchuk
Title: Numerical and analytical conformal field theory

Abstract: The goal of this research project is to explore the non-perturbative properties of general conformal field theories (mostly in 3 dimensions and higher) using both numerical and analytical approaches to self-consistency conditions (conformal bootstrap) or effective descriptions, with a view towards applications in critical phenomena, quantum field theory and AdS/CFT correspondence.

Supervisor: Neil Lambert
Title: Non-Lorentzian Field theories and Gauge/Gravity duality

Abstract: The mainstream examples of gauge/gravity duality involve Anti-de Sitter spacetimes and field theories with a conformal SO(2,D) symmetry. However, in recent years examples have arisen where the field theory is not Lorentzian, and the corresponding conformal group is not simply SO(2,d) for some d. This project will explore the construction of these theories and their symmetries at both the level of gauge field theory as well as their gravitational dual geometries. 

Supervisor: Sameer Murthy
Title: Black holes and the quantum structure of spacetime

Abstract: Black holes are known to have thermodynamic properties like temperature and entropy. This is an important clue towards a quantum theory of gravity, and explaining these thermodynamic features from a more fundamental quantum-statistical point of view is an active topic of research. The project aims to explore questions about quantum aspects of black holes, within the framework of string theory and AdS/CFT, of the following sort: 

(1) What is the nature of the microscopic states underlying a black hole? 
(2) How does one describe the collective behavior (phases) of these microstates?
(3) How does one describe the microscopic structure of spacetime from a gravitational path integral? 

A good knowledge of quantum field theory (including path integrals) and GR are prerequisites. Some knowledge of string theory is useful; this will be developed as one goes along. 

Supervisor: George Papadopoulos
Title: Geometry with applications to physics

Abstract: The projects I have on offer include an exploration of the geometry of the moduli space of connections with a view to apply the results in AdS/CFT. I am also interested in the application of the Perelman's ideas, used in the proof of the Poincare conjecture, to physics. 

Supervisor: Gerard Watts
Title: Defects and related structures in two-dimensional field theory

Abstract: Defects are ubiquitous in current studies of quantum field theory, providing both new results and fresh insights into old. This project could go in any number of directions in which I am currently working - the mathematical proof of the consistency of the topological defects in fermionic theories; the study of defects related to generalised Gibbs ensembles; the relation to non-invertible symmetries; numerical work on defect perturbations. It would start with the basis of two dimensional conformal and integrable field theory, and the subsequent direction would be decided by mutual agreement. 

Supervisor: Dr Marina Riabiz 
Title: Optimal postprocessing of sampling methods

Abstract: Markov Chain Monte Carlo (MCMC) methods are the main approach to sample from probability distributions with intractable normalizing constants, such as the parameter posterior in Bayesian computation. Whilst MCMC samples converge in the limit to the target distribution, in many real world applications it is possible to draw only a finite number of samples, due to restrictions in computing budget.

In this framework, Stein thinning is a recent method developed to optimally select a set of samples from an MCMC output of fixed length, by minimizing a measure of discrepancy between the empirical approximation produced and the target. This PhD project aims at exploring different research questions that are still open, in order to produce a robust version of the algorithm, both in traditional Bayesian setting and in modern post-Bayesian approaches, where not even the unnormalized target density is known. The project will be applied to challenging Bayesian inverse problems arising in cardiac electrophysiology.

Supervisor: Dr Vasiliki Koutra
Title: Designing Experiments in Large-Scale Networked Systems

Abstract: Modern experiments increasingly take place in highly interconnected systems: social networks, financial markets, biological pathways, and even the human brain. In these networks, what happens to one node rarely stays there: treatments can have direct effects on the target unit and spillover effects on its neighbours. Classical experimental designs, built on the assumption of independent units and simple responses, struggle to cope, and standard optimal design methods quickly become unworkable at realistic network scales.

A central challenge is that the network is often only partially known, subject to noise, or evolving over time. Real applications also involve treatments more complex than simple binary interventions, and responses can take diverse forms, including counts, time-to-event outcomes, or functional trajectories. Designing experiments that can account for these uncertainties and complexities is both practically important and scientifically challenging, offering rich opportunities for original research.

This PhD project will develop scalable, statistically principled methods for designing experiments in these complex settings. Drawing on optimal design, causal inference, and computational techniques, the research will explore ways to account for interference and complex dependencies without getting lost in the network’s size.
The methods are broadly applicable, from social media and digital platforms to epidemiology, biological networks, and brain connectivity studies. This project is well suited to candidates interested in statistics, network science, and large-scale experimental methodology, offering the chance to tackle genuinely challenging problems in a rapidly evolving field.

Supervisor: Professor Fabrizio Liesen 
Title: Autocompound Random Measures

Abstract: This research project is motivated by the need for complex models to tackle data heterogeneity. The Bayesian nonparametric approach offers a methodological framework to take into account the model uncertainty carried by heterogeneous datasets. Compound Random Measures (CoRMs) are a novel class of Bayesian nonparametric priors which have been introduced by Griffin and Leisen (2017). They can be employed for modelling data heterogeneity. Despite being recently introduced, CoRMs had an impact on different fields such as networks (Todeschini, Miscouridou and Caron, 2016), cybersecurity (Perusquia, Griffin and Villa, 2025) and survival analysis (Riva-Palacio and Leisen, 2018 and Riva-Palacio, Leisen and Griffin, 2022).

This project will introduce a new operation based on the CoRMs, called autocompounding, which will allow nonparametric modelling of complex structures in time and space with families of random measures. We will apply the new methodology to model trees, graph, networks and time series. CoRMs already proved that they have an impact in different disciplines beyond Statistics such as Machine Learning, Survival Analysis and Cybersecurity. Autocompounding will boost their use in different fields since they will allow the introduction of cutting-edge models to deal with data heterogeneity.

Supervisor: Professor Fabrizio Liesen
Title: Statistical Inference with Predictive Distribution

Abstract:
The standard Bayesian or Frequentist settings specify a model by assuming a distribution on the observations. However, this is not the only way to define a model for the observations. One way is to assign the predictive distributions. The Ionescu-Tulcea’s Theorem ensures that the model is well defined as the number of observations grow, ideally, to infinity. This is particularly suited if your focus is on prediction. However, this is not the only way to employ predictive distributions to perform inferential tasks. This project will explore different approaches to statistical inference which makes use of predictive distributions.

Classes of predictive distributions have been recently introduced.  One approach is to consider a type of dependence weaker than exchangeability, conditionally identity in distribution (CID), Berti, Pratelli and Rigo (2004). Models for CID sequences have been introduced in Bassetti, Crimaldi and Leisen (2010), Berti et al (2021, 2023). 

Recently, they have been employed in the RSS read paper of Fong, Holmes and Walker (2023) where they introduced a different view on statistical inference. In their setting, they assume that the uncertainty comes from what it is not observed. To model the missing data of the population they use a predictive approach. In  particular, they focus on CID sequences. Furthermore, CID sequences which have a “species sampling” form have been employed in Airoldi et al (2014) to introduce a novel (and flexible) Bayesian nonparametric prior.  

Supervisor: Professor Fabrizio Liesen
Title: Loss based approaches to Objective Bayesian Analysis

Abstract: This project focuses on two approaches that have been recently introduced for defining an Objective prior. The first one targets parameter on discrete space, see Villa and Walker (2015a, 2015b). It has been used to design objective priors in a variety of contexts. For instance, time series analysis (Leisen et al., 2020), change-point analysis (Hinoveanu et al., 2019), number of components in a mixture model (Grazian et al., 2020), variable selection in linear regression (Villa and Lee, 2020), and Gaussian graphical models (Hinoveanu et al., 2020). The second one target continuous parameters, see Leisen, Villa and Walker (2020). In particular, this approach is still in development. This project will explore research avenues within these two research lines.

Supervisor: Dr Davide Pigoli
Title: Statistical analysis of spatially and/or temporally dependent manifold-valued data

Abstract: There is a growing interest in the statistical analysis of data that take value on smooth manifolds (3D rotations, shapes, directions, positive definite matrices, points on a circle or a sphere, constrained curves to mention just a few examples).  The goal of the project is to develop a framework to analyse manifold-valued data that are observed over time and/or space, example including molecular dynamics simulations of protein folding in medicinal chemistry (temporally dependent),  permeability tensors in mining engineering (spatially dependent) or coordinates (latitude and longitude) of earthquakes aftershocks in earth science (spatio-temporally dependent).

Possible lines of research include (but are not limited to):- modelling of stochastic dependence in these non-linear spaces, where usual concepts of co-variability are not valid.- extension to dependent samples of existing techniques for dimensional reduction (Principal Geodesic Analysis, Principal Flows) and centerpoint estimation (Fréchet mean) for manifold-valued data. - multiresolution analysis and prediction for spatio-temporal manifold-valued processes.

Supervisor: Dr Davide Pigoli
Title: Time misalignment in additive models with random smooth

Abstract: In generalised additive mixed effects models (GAMMs), it is possible to introduce group-level (or individual-level) variations in the functional relationship between predictor and responses. However, these models assume the groups/individuals to operate on the same time scale. Research in functional data analysis has demonstrated that variations in the time dynamics are common between individuals and these need to be considered in the analysis. This project goal is to introduce time misalignment in GAMMs, either by iterative procedures or by introducing a group/individual level random time warping.

Supervisor: Dr Davide Pigoli
Title: Statistical and machine learning models for multimodal bioimages

Abstract: Detailed molecular level understanding of spatial tissue state also holds tremendous promise to identify novel therapeutic targets and interventions for hard-to-treat diseases. However, there is a lack of statistical and computational methodologies to effectively analyse multimodal images of tissue samples. This is because tissue samples are highly complex and exhibit high spatial heterogeneity.

There is an urgent need to identify new tools to mathematically define spatial characteristics in feature spaces that provide a shared ‘dictionary’ across samples and to which statistical and machine learning methods can be applied to explore the association with human biology. The project will focus on the development of both features that capture the spatial behaviour within each image modality and of features that capture the interactions between different image modalities.

Examples of features to be considered are statistical summaries (intensity histograms, texture measures, spatial correlations, etc.) and deep-learning generated features such as ones based on visual autoencoders or convolutional Neural Network. This project will capitalise on exemplar, recently generated, state-of-the-art datasets from human tissue with unprecedented molecular detail, with a focus on understanding why many patients with triple negative breast cancer (TNBC) do not respond to the standard of care neoadjuvant chemotherapy (NACT). The aims of the project are to develop statistical and machine learning predictive models to associate biomarkers derived from the images to the patient’s response to NACT and to assess features importance and prediction uncertainty.

Supervisors: Dr Kalliopi Mylona and Dr Davide Pigoli
Title: Designs with functional responses and factors with functional levels 

Abstract: There are plenty of applications with functional data, from the science and the industry, for e.g. environmental, medical, meteorological and financial applications (see, Ramsay and Silverman [2005] for a good introduction to functional data analysis and several datasets and applications). Although, the literature for functional data is very broad, the connection to design of experiments is still very limited. In Dette and Schorning [2016], the authors proposed an optimality criterion for the construction of designs that can be used for the comparison of two regression curves, the dose response relationships of two groups. In Aletti et al. [2016] presented a way to design an experiment from the car industry, in which the responses were functions, but the factors had discrete levels. In the same proceedings, specifically in Zhang and Großmann [2016], a statistical analysis method for data from a restricted randomised experiment, in which both the response and the treatments were functions, was proposed.

The main question is:  how someone should design and analyse an experiment when the levels of the factors and/or the measured responses are functions instead of numerical or categorical variables? The goal of this project is to extend current results in the literature (see May et al., [2024], Michaelidis et al [2021]) to more complex scenarios.

For full details on how to apply to our research degrees, please see the course pages:

• Applied Mathematics Research: Disordered Systems/Financial Mathematics/Probability
• Applied Mathematics Research: Theoretical Physics
• Pure Mathematics Research
• Statistics Research