Current opportunities

Opportunities across science and engineering

We offer a diverse range of science and engineering projects where you can get involved in research that changes lives and meets the challenges of the future.

For opportunities across science and engineering, explore our available projects through our FindAPhD directory below. 

You can also explore more information about our opportunities in mathematics and statistics.

Opportunities in mathematics and statistics

In addition to the EPSRC DLA funded doctoral studentships in mathematics and statistics advertised above, home candidates can also apply for an open studentship, working with a supervisor to develop their own proposal in one of our groups. Applications for these student-led proposals should be made through the EPSRC DLA portal. The application code should be listed as 'S3.5-MPS-Open' and include the name of the proposed supervisor.

If you are interested in applying for an open studentship, you are encouraged to contact prospective supervisors to discuss your interest in and suitability for a PhD project prior to submitting an application. You can find contact information for supervisors on the relevant group pages linked below.

Mathematics research in the School of Mathematical and Physical Sciences is structured in three clusters that span the mathematical sciences. Each cluster comprises several groups:

Pure Mathematics

Algebraic Geometry and Mathematical Physics group

The Sheffield Algebraic Geometry and Mathematical Physics group is a thriving and expanding group of geometers and mathematical physicists. Its area of interest broadly encompasses complex algebraic geometry and the geometric structures that underpin quantum field theory and string theory.

The group fits in a cross-disciplinary way in the strategic priority for mathematical sciences of the Engineering and Physical Sciences Research Council (EPSRC), spanning the Algebra, Geometry and Topology, and Mathematical Physics research areas of the EPSRC portfolio.

The group has received over £2 million in funding from the EPSRC through multiple grants in recent years, including two EPSRC standard grants, an Early Career EPSRC fellowship, a Royal Society-EPSRC Dorothy Hodgkin fellowship and an EPSRC Programme Grant.

Student-led proposals are available in the following topic areas:

• Enumerative invariants, moduli spaces of curves, and allied enumerative theories
• Homological algebra, geometry and K-theory of singularities
• Derived categories and Grothendieck ring of varieties
• Relations between algebraic geometry and combinatorics
• Mirror Symmetry and Bridgeland stability conditions
• Hall algebras and Donaldson-Thomas Theory
• Topological field and string theory
• Relations of topological strings to integrable systems

Number theory group

A central focus of the Number theory group is the representation of Galois groups by homomorphisms to groups of matrices. Such Galois representations arise from the action of Galois groups on topological invariants of spaces defined by polynomial equations in several variables with rational coefficients (for example 'elliptic' curves y^2=x^3+ax+b), so algebraic geometry is important here.

Other key players are automorphic forms and associated automorphic representations, local components of which are representations of p-adic, or real, matrix groups as linear transformations on infinite dimensional spaces of functions.

Certain analytic functions called L-functions, generalising the Riemann zeta function, are also important. Still, what we do is perhaps better classified as algebraic, rather than analytic, number theory.

Influential conjectures of Langlands link Galois representations with automorphic forms (and different types of automorphic forms with each other), saying that their associated L-functions are the same. An instance is the modularity of elliptic curves, implicated in the proof of Fermat's Last Theorem. But the most basic example is the quadratic reciprocity law, when viewed the right way.

Recent PhD topics include:

• Representations of p-adic groups
• Applications of modularity to diophantine equations
• Deformation theory of Galois representations
• Explicit computation of automorphic forms
• Congruences between automorphic forms and their connections with values of L-functions at integer points.

The availability of supervisors in this group will vary from year to year.

Topology research group

Topology is the mathematical study of qualitative features of shapes, forms, and spaces. It is a fundamental research area that connects with nearly all subjects in the mathematical sciences. Topological methods are also essential for advancements in data science, quantum computing, robotics and autonomous systems.

The Topology research group's active research spans a wide range of topics in algebraic and geometric topology. It focuses on developing conceptual and computational tools using homotopy theory, category theory, homological algebra, and K-theory, addressing their algebraic, topological and analytic aspects. 

This field is also linked to the study of symmetries within group theory and its generalisations. The group's research in equivariant and chromatic homotopy theory is related to recent breakthroughs in resolving the Kervaire conjecture and the telescope conjecture.

Student-led proposals are available in the following topic areas:

• Algebraic topology and homotopy theory
• Homology of groups and generalisations
• Higher category theory and applications
• Homological and homotopical algebra
• K-theory: algebraic and analytic
• Manifolds and cell complexes

Algebra group

Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of statements within those systems.

Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several operations defined on that set. It is a generalisation of elementary and linear algebra, since it allows mathematical objects other than numbers and non-arithmetic operations.

The Algebra group studies both commutative and non-commutative structures.

Student led proposals are available in the following topic areas:

• Betti Cones of Stanley-Reisner Ideals
• A Study of the Local Cohomology of Binomial Edge Ideals
• Representations of quantum groups
• Algebras of differential operators on singular varieties and their modules
• Algebras of Poisson differential operators
• Poisson algebras and Poisson modules
• Modules of Noetherian algebras in prime characteristic
• Global dimension of Noetherian algebras

Applied Mathematics and Theoretical Physics

Gravitation and Cosmology research group

The Gravitation and Cosmology research group works on mathematical physics at the interface between quantum field theory, black hole physics, quantum gravity, quantum cosmology and non-linear dynamics. 

The group comprises six members of academic staff, three PDRAs and 12 PhD students. Below are some of the topics the group works on. We typically offer EPSRC funded studentships across the following areas:

• Mathematical foundations for quantum gravity: We develop models of quantum cosmology, quantum black holes and quantum gravity, using both continuum methods and discrete geometric approaches such as group field theory. We have recently shown that geometric singularities can be resolved within quantum gravity, allowing an exploration of non-linear dynamics in the early universe and the fundamental role of unitarity.
• Quantum field theories on curved spacetime: We explore the properties of quantum fields interacting with black holes on curved geometries, such as anti-de Sitter spacetime. This work requires the development of novel mathematics for renormalising quantum fields in non-linear, dynamical scenarios.
• Inflationary dynamics in cosmology: We develop and test mathematical models for the very early universe whose expansion is driven by quantum fields in a non-linear phase. We explore theories that go beyond Einstein gravity; for example, we have recently shown that in such theories a contracting universe can become an inflating universe after a bounce, thereby avoiding a spacetime singularity.
• Non-linear gravitational dynamics: We develop and apply new mathematical methods to model the non-linear dynamics of interacting black holes in strong-field gravity. New perturbative methods for general relativity will be applied to model binaries with unequal masses in the non-linear regime, to test Einstein’s theory of general relativity in the 2030s.

Plasma Dynamics Group

The Plasma Dynamics Group (PDG) is a cross-departmental research group comprising five academic members, two postdocs and 11 PhD students working on various aspects of the dynamics, evolution and stability of gravitationally stratified and/or magnetic fluids (plasmas).

Our research relies heavily on analytical methods, state-of-the-art numerical modelling (HPC and GPU computing), as well as on laboratory experiments. Our aim is to give answers to crucial questions such as:

• How is plasma heated to temperatures of million degrees by linear and nonlinear waves, magnetic reconnection/diffusion, and turbulence?
• How can we determine the formation, nature and evolution of coherent structures (swirls, vortices) in magnetic fluids and how these structures channel energy?
• How do instabilities affect the evolution of energy in plasmas?
• How do small scale processes (e.g. turbulences and/or waves) affect the dynamical and thermal state of the plasma?
• How can perturbations propagating in magnetic fluids help us to predict natural hazards: application to climate prediction and natural hazard risk assessment?

Student led proposals are available in the following topic areas:

• Lagrangian coherent structures in magnetic fluids.
• Formation of networks of vortices in plasmas.
• Linear and nonlinear waves (shocks, solitons) in partially ionised plasmas.
• Mathematical models to study waves and oscillations in stratified, rotating and magnetised plasmas/fluids.

Fluids group

The Sheffield Fluid Mechanics Group is a large cross-faculty research group that covers a broad range of perspectives in fluids research, from the study of fundamental mathematical properties of fluid flows, to experimental and industry-related research.  

Fluids appear in a huge variety of environmental and industrial applications, including the design of energy-efficient aircraft and wind turbines, optimisation of cooling systems, and forecasting of extreme events in environmental flows.

Fluid flows exhibit complex, often unpredictable behaviour. We use the theory of nonlinear dynamical systems to reveal the underlying structures that govern the evolution of turbulence and transitional flows, such as invariant manifolds and attractors. This approach helps explain how turbulence arises, persists, can be modelled or controlled. 

Our research also examines extreme fluctuations in turbulent flows, which cause high drag, noise, or structural stress. Using sensitivity analysis, statistical methods and optimisation, we seek to identify the mechanisms behind such events and explore strategies for their prediction and mitigation. We have applied optimisation and control techniques to reduce drag and enhance heat transfer in turbulent flows. 

We employ data assimilation, by combining simulations with experimental or observational data to reconstruct unmeasured flow fields and improve predictive accuracy. By integrating data assimilation with dynamical-systems analysis and optimisation, we explore better ways to forecast, synchronise, and control complex fluid systems.

Student-led proposals are available in the following topic areas:

• Dynamical-systems methods and numerical modelling
• Optimised models for suppression or enhancement of turbulence and heat transfer
• Enhancing simulations of fluid flows using data assimilation techniques
• Synchronisation of turbulent flows
• Prediction and characterisation of extreme events (dissipation, stress) in turbulence

Solar Physics and Space Plasma Research Centre

The Solar Physics and Space Plasma Research Centre (SP²RC) is at the forefront of addressing theoretical and experimental issues in high-temperature low density plasma physics, especially their continuum modelling and analysis. This includes magneto-seismology, dynamics of magnetised plasma and the continuum description of plasmas. 

SP²RC is one of the most dynamic research groups in the country and is well renowned internationally due to their scientific grant application and early career research training successes. The research centre is made up of two research groups, the Solar Wave Theory Group (SWAT) and the Space Systems Laboratory (SSL). 

SP²RC has a wide expertise in advanced analytical and numerical modelling, and analysing experimental (eg instrumental) data obtained by cutting-edge instrument development (eg MOF technology). We also have direct and guaranteed access to experimental facilities, where students may work or have training on instrument development and lab experiments. Students in this group may also have supplementary income from assistance in undergraduate tutorials.

Student-led proposals are available in the following topic areas:

• Mathematical theory of magnetised plasma seismology
• Dynamics of MHD jets
• Dynamics of plasma tornadoes: from nano- to giant MHD swirls
• MHD wave theory with applications
• MHD wave detection and Magneto-Seismology (SMS) in laboratory plasmas
• MHD and plasma instabilities with applications
• Heating processes of magnetised plasmas: waves, reconnection, etc.
• MHD and collisionless shocks in the magnetised plasmas
• The theory of plasmas beyond the continuum description
• Radiation modelling in MHD plasmas
• High-speed stereo Schlieren imaging and signal processing
• Applying Machine Learning (ML) and AI techniques to MHD and plasma diagnostics
• Instrument development of magneto-optical filters (MOFs)
• Pipeline development of image processing for magneto-optical filter (MOF) data
• Numerical modelling (HPC and GPU computing) of processes in MHD plasmas

Mathematical and Statistical Modelling

The Mathematical and Statistical Modelling research cluster uses a wide range of mathematical and statistical approaches to study questions across the physical, biological, and social sciences.

Our research covers the topics across mathematical biology, probability and statistics.

Topics within mathematical biology include:

• Animal movement and biological invasions
• Dynamics of genetic regulatory networks
• Dynamics of infectious diseases
• Pattern formation in plant and animal development
• Evolutionary modelling and statistical ecology

Topics within probability include:

• Evolution and genetics, and the study of genealogies, including spatial effects related to natural selection
• Random graphs and networks, including social networks, computer networks, and networks describing the interactions between genes and proteins
• Phase transitions in random graph models, such as the Erdős-Rényi model
• The preferential attachment model, describing a growing graph where new vertices are more likely to connect to existing vertices which already have a large number of neighbours

Topics within statistics include:

• Uncertainty in climate modelling and its visualisation
• Bayesian statistics in health economics
• Image analysis in microscopy
• Archeological statistics
• Modelling the properties of materials for engineering applications
• Machine learning and AI approaches to data analysis

Contact us

Got a question not answered here?

Email pgr-scholarships@sheffield.ac.uk