91̽»¨

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. Explore information about our opportunities in mathematics and statistics.

Opportunities in mathematics and statistics

We are offering a limited number of EPSRC DLA funded doctoral studentships in mathematics and statistics and welcome applications from both Home and Overseas students.

Mathematics research in the School of Mathematical and Physical Sciences is structured in three clusters that span the mathematical sciences:

Pure Mathematics

Algebraic Geometry and Mathematical Physics group

The 91̽»¨  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.

Possible PhD projects are available in the following topics:

• 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

The has six permanent academic staff, one postdoc and six PhD students. Basic tools are number fields (finite extensions of the rationals, generated by roots of polynomials with rational coefficients) and their local completions such as p-adic fields (which package congruences modulo powers of a prime p).

A central focus of our interests 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 projects include:

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

The availability of supervisors 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 key fundamental research area, interconnected with nearly all subject areas in the mathematical sciences. Topological methods also play an underpinning role in the advances of data science, robotics and autonomous systems.

The active research in the  spans a wide range of topics in algebraic and geometric topology. Category theory and homological algebra are areas where we develop crucial conceptual and computational tools, as is K-theory in its algebraic, topological, and analytic flavours.

The subject is fundamentally linked to the study of symmetries in the theory of groups and their generalisations, and our research in equivariant and chromatic homotopy theory is related to recent breakthroughs towards the resolution of the Kervaire conjecture and the telescope conjecture. 

91̽»¨ is a great place to do a PhD in topology, and the members of the Topology group offer PhD studentships in all of the areas mentioned above.

Applied Mathematics and Theoretical Physics

Gravitation and Cosmology research group

The  works on several aspects of mathematical physics, in particular classical relativity, quantum field theory, black hole physics, quantum gravity and quantum cosmology.

The group comprises five members of academic staff, one PDRA and 17 PhD students. We typically offer EPSRC funded studentships across all these areas.

We explore several directions at the interface between quantum physics and relativity. In quantum field theory in curved spacetime, we have studied the properties of quantum fields near black holes and in curved backgrounds such as anti-de Sitter spacetime.

These studies involve complicated mode-sum calculations and often use recently improved numerical techniques.

We also explore models of quantum cosmology and quantum gravity, for instance discrete geometric approaches such as group field theory.

We have shown various ways in which classical geometric singularities can be resolved within quantum gravity, using common techniques from quantum physics such as the study of coherent or squeezed states.

We study theoretical models for the very early universe, such as those motivated from theories beyond Einstein gravity. For example, we have recently shown that in such theories a contracting universe can lead to an inflationary universe after a bounce, thereby avoiding a spacetime singularity. We are also interested in field theory models of inflation.

In classical relativity we study topics such as the propagation of waves on curved spaces, the stability of rotating black holes under perturbation, and the scattering, absorption and amplification of fields by black holes.

Of particular interest is modelling the dynamics of binary systems with one light and one heavy black hole. Many of these directions require developing advanced perturbative methods for general relativity.

Plasma Dynamics Group (PDG)

The is a cross-departmental research group comprising five academic members, one postdoc 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 the plasma heated to temperatures of million degrees (by linear and nonlinear waves, magnetic reconnection/diffusion, turbulences)?
• 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?

Possible PhD projects are available in the following 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

Fluids appear in a huge range of environmental and industrial applications. Our research interests cover a broad range of systems, such as turbulence, atmospheric dispersion, microfluidics, bubble dynamics and related topics.

Fluid flows exhibit a wide range of interesting phenomena, and the study of fundamental fluid dynamics is a traditional testbed for the development of methods in applied mathematics.

Examples of our research include:

• optimised models for suppression or enhancement of turbulence
• reconstruction of flows from limited observational data using data assimilation techniques
• enhancement of transfer rates via the use of novel technologies incorporating microbubbles
• modelling of contaminant concentrations.

Mathematical and Statistical Modelling

Probability group

The works in stochastic processes and applied probability. 

This has included evolution and genetics, and the study of genealogies - work in our group has focused on spatial effects related to natural selection, in particular on the relationship between spatial structure and the rate at which natural selection progresses.

A common theme across much of our work is that of random graphs and networks. These can be used to model relationships between various types of entity, such as social networks, computer networks and networks describing the interactions between genes and proteins.

The group has also studied scaling limits of random graphs, looking at how the structure appears as the number of vertices tends to infinity.

Other models of random graphs, which have been studied in the group include the classical model of Erdős and Rényi, which has an interesting phase transition between a phase with a poorly connected network with a lot of components separated from each other and a phase with a better connected "giant component" containing a significant proportion of the vertices.

Another is the preferential attachment model, which describes a growing graph where new vertices are more likely to connect to existing vertices which already have a large number of neighbours.

Variations and extensions of preferential attachment, such as where vertices have an intrinsic attractiveness which also influences the connection probabilities, have been the subject of a number of publications.

Mathematical Biology group

The research cluster uses a wide range of mathematical and statistical approaches to study questions in biology, medicine and ecology.

Our research sits within the Mathematical Biology theme of EPSRC’s portfolio, with connections to Artificial Intelligence Technologies, Biophysics and Soft Matter Physics, Continuum Mechanics, Non-Linear Systems, and Statistics and Applied Probability.

The group comprises four academic staff and 10-15 PDRAs and PhD students. Current research interests 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.

Statistics group

The main research in the Statistics group is within the statistics and applied probability area of the EPSRC remit, developing new statistical methodology and probabilistic models with applications across UKRI.

Related work includes quantifying the uncertainty in modelling, whether based on a single model or on an ensemble of different models of the same underlying phenomenon.

Particular applications include:

• uncertainty in climate modelling and its visualisation
• Bayesian statistics in health economics
• image analysis in microscopy
• modelling the properties of materials for engineering applications.

Candidates can apply to work with a supervisor to develop their own project in one of our groups.

If you are interested in applying, you are strongly 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 in the relevant group pages linked above.

Contact us

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Email pgr-scholarships@sheffield.ac.uk