Mathematics PhD/iPhD/MPhil/MSc (Research)
The School of Mathematics & Statistics combines worldleading research expertise in pure and applied mathematics and statistics in areas such as mathematical biology, fluid dynamics, geometry and topology and environmental statistics.
 PhD: 34 years fulltime; 68 years parttime; Thesis of Max 80,000 words
 MSc (Research): 12 years fulltime; 23 years parttime;
 MPhil: 23 years fulltime; 34 years parttime;
 IPhD: 4 years fulltime;
Research projects
PhD postgratuate opportunities by research group
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Algebra
Quantum spinchains and exactly solvable lattice models (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
Quantum spinchains and 2dimensional statistical lattice models, such as the Heisenberg spinchain and the six and eightvertex models remain an active area of research with many surprising connections to other areas of mathematics.
Some of the algebra underlying these models deals with quantum and Hecke algebras, the TemperleyLieb algebra, the Virasoro algebra and KacMoody algebras. There are many unanswered questions ranging from very applied to more pure topics in representation theory and algebraic combinatorics. For example, recently these models have been applied in combinatorial representation theory to compute GromovWitten invariants (enumerative geometry) and fusion coefficients in conformal field theory (mathematical physics).
Integrable quantum field theory and Ysystems (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
The mathematically rigorous and exact construction of a quantum field theory remains a tantalising challenge. In 1+1 dimensions exact results can be found by computing the scattering matrices of such theories using a set of functional relations. These theories exhibit beautiful mathematical structures related to Weyl groups and Coxeter geometry.
In the thermodynamic limit (volume and particle number tend to infinity while the density is kept fixed) the set of functional relations satisfied by the scattering matrices leads to socalled Ysystems which appear in cluster algebras introduced by Fomin and Zelevinsky and the proof of dilogarithm identities in number theory.
Operator algebras associated to selfsimilar actions. (PhD)
Supervisors: Mike Whittaker
Relevant research groups: Algebra, Analysis, Geometry and Topology
This project will focus on selfsimilar groups and their operator algebras. The primary aim will be to examine a new class of groups that act selfsimilarly on the path space of a graph and to study the noncommutative geometry of a natural class of operator algebras associated to these selfsimilar groups.
Selfsimilar groups are an important and active new area of group theory. The most famous example is the Grigorchuk group, which was the first known example of a group with intermediate growth. This makes investigating C*algebras associated to them particularly interesting. In particular, these groups are often defined by their action on a graph, and the associated C*algebra encodes both the group and path space of the graph in a single algebraic object, as well as the interaction between them.

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Analysis
Operator algebras associated to selfsimilar actions. (PhD)
Supervisors: Mike Whittaker
Relevant research groups: Algebra, Analysis, Geometry and Topology
This project will focus on selfsimilar groups and their operator algebras. The primary aim will be to examine a new class of groups that act selfsimilarly on the path space of a graph and to study the noncommutative geometry of a natural class of operator algebras associated to these selfsimilar groups.
Selfsimilar groups are an important and active new area of group theory. The most famous example is the Grigorchuk group, which was the first known example of a group with intermediate growth. This makes investigating C*algebras associated to them particularly interesting. In particular, these groups are often defined by their action on a graph, and the associated C*algebra encodes both the group and path space of the graph in a single algebraic object, as well as the interaction between them.
Aperiodic substitution tilings and their C*algebras. (PhD)
Supervisors: Mike Whittaker
Relevant research groups: Analysis, Geometry and Topology
A tiling is a collection of subsets of the plane, called tiles, for which any intersection of the interiors of two distinct tiles is empty and whose union is all of the plane. A tiling said to be aperiodic if it lacks translational periodicity. The most common method of producing aperiodic tilings is to use a substitution rule; a method for breaking each tile into smaller pieces, each of which is a scaled down copy of one of the original tiles, and then expanding so that each tile is congruent to one of the original tiles.
This project will focus on a natural class of operator algebras associated with an aperiodic substitution tiling. These algebras were first considered by Kellendonk and reflect the symmetries of a tiling in an algebraic object that allows up to consider invariants in a noncommutative framework. A key area of study are spectral triples associated with aperiodic tilings, which allow us to think of tilings as noncommutative geometric objects.

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Continuum Mechanics
Continuous production of solid metal foams (PhD)
Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics  Modelling and Analysis of Material Systems
Porous metallic solids, or solid metal foams, are exceedingly useful in many engineering applications, as they can be manufactured to be strong yet exceedingly lightweight. However, industrial processing methods for producing such foams are problematic and unreliable, and it is not currently possible to control the porosity distribution of the final product a priori.
This project will consider a new method of solid foam production, where bubbles of gas are introduced continuously into a molten metal flowing through a heat exchanger; foaming and solidification then occur almost simulatanously, allowing the foam structure to be controlled pointwise. The aim of this project is to construct a simple mathematical model for a gas bubble moving in a liquid filled channel ahead of a solidification front, to predict optimal conditions whereby the gas bubble is drawn toward the phase boundary, hence forming a porous solid.
This project will require some background in fluid mechanics and a combination of analytical and numerical techniques for solving partial differential equations.
Radial foam fracture (PhD)
Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics  Modelling and Analysis of Material Systems
Gasliquid foams are a useful analgoue of crystalline atomic solids. 2D foam fracture has been used to study the mechanisms of fracture in metals. A twodimenisonal network model (formed from a large system of differential equations) has recently been produced to study foam fracture in a rectangular channel which is pressurised along one edge. This model has helped to explain the origin of the velocity gap (a range of inadmissable steady fracture velocities), observed both in foam fracture experiments and in atomistic simulations of brittle fracture. This project will apply this network modelling approach to study radial foam fracture in a HeleShaw cell, to mimick recent experiments. This system has strong similarity to radial SaffmannTaylor fingering, where fingering has been observed when a less viscous fluid displaces a more viscous fluid in a confined geometry. This project will involve studying systems of ordinary and partial differential equations using both numerical and analytical methods.
Numerical simulations of planetary and stellar dynamos (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics
Using Fluid Dynamics and Magnetohydrodynamics to model the magnetic fields of the Earth, planets, the Sun and stars. Involves highperformance computing.
Mathematical models of vasculogenesis (PhD)
Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics  Modelling and Analysis of Material Systems, Mathematical Biology
Vasculogenesis is the process of forming new blood vessels from endothelial cells, which occurs during embryonic development. Viable blood vessels facilitate tissue perfusion, allowing the tissue to grow beyond the diffusionlimited size. However, in the absence of vasculogenesis, efforts to engineer functional tissues (eg for implantation) are restricted to this diffusionlimited size. This project will investigate mathematical models for vasculogenesis and explore mechanisms to stimulate blood vessel formation for in vitro tissues. The project will involve collaboration with Department of Biological Engineering at MIT, as part of the SofTMech^{MP} project.
A coupled cardiovascularrespiration model for mechanical ventilation (PhD)
Supervisors: Peter Stewart, Nicholas A Hill
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics, Mathematical Biology
Mechanical ventilation is a clinical treatment used to draw air into the lungs to facilitate breathing, used in treatment of premature babies with respiratory distress syndrome and in the treatment of severe Covid pneumonia. The aim is to oxygenate the blood while simultaneously removing unwanted byproducts. However, overinflation of the lungs can reduce the blood supply to the gas exchange surfaces, leading to a ventilationperfusion mismatch. This PhD project will give you the opportunity to develop a mathematical model to describe the coupling between blood flow in the pulmonary circulation and air flow in the lungs (during both inspiration and expiration). You will devise a coupled computational framework, capable of testing patientspecific ventilation protocols. This is an ideal project for a postgraduate student with an interest in applying mathematical modelling and image analysis to predictive healthcare. The project will give you the opportunity to join a crossdisciplinary Research Hub that aims to push the boundaries of quantitative medicine and improve clinical decision making using innovative mathematical and statistical modelling.
Observationallyconstrained 3D convective spherical models of the solar dynamo (Solar MHD) (PhD)
Supervisors: Radostin Simitev, David MacTaggart, Robert Teed
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics
Solar magnetic fields are produced by a dynamo process in the Solar convection zone by turbulent motions acting against Ohmic dissipation. Solar magnetic activity affects nearEarth space environment and can harm modern technology and endanger human health. Further, Solar magnetism poses fundamental physical and mathematical problems, e.g. about the nature of plasma turbulence and the topology of magnetic field generation. Current models of the global Solar dynamo fall in two classes (a) meanfield dynamos (b) convectiondriven dynamos. The meanfield models are only phenomenological as they replace turbulent interactions by adhoc source and quenching terms. On the other hand, spherical convectiondriven dynamo models are derived from basic principles with minimal assumptions and potentially offer true predictive power; these can also be extended to other stars and giant planets. However, at present, convection driven dynamo models operate in a wrong dynamical regime and have limited success in reproducing a number of important 1 observations including (a) the sunspot cycle period, polarity reversals and the sunspot butterfly diagram, (b) the poleward migration of diffuse surface magnetic fields, (c) the polar field strength and phase relationships between poloidal/toroidal components. The aims of this project are to (a) develop a threedimensional convectiondriven Solar dynamo model constrained by assimilation of helioseismic data, and (b) start to use the model to estimate turbulent properties that determine the internal dynamics and activity cycles of the Sun.
Force balances in planetary cores and atmospheres (PhD)
Supervisors: Robert Teed, Radostin Simitev
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics
Current dynamo simulations are run, not under the conditions of planetary cores and atmospheres, but in a regime idealised for computations. To forecast changes in planetary magnetic fields such as reversals and dynamo collapse, it is vital to understand the actual fluid dynamics of these regions.
The aim of this project is to produce simulations of planetary cores and atmospheres with realistic force balances and, in doing so, understand how such force balances arise and affect the dynamics of the flow. Force balances control many aspects of the fluid dynamics, and hence the dynamo process itself, including the size of flow structure, the buoyancy flux and zonal flows, so an understanding of the force balance available in various planetary cores and atmospheres is vital for understanding their dynamo processes. To achieve this the project will use a different technique to that typically used in dynamo simulations. The approach is to perform global simulations in a spherical shell with a magnetic field imposed by explicitly setting a component (or components) of the field at one of the boundaries. Within the interior the field is evaluated as normal using the induction equation. This setup amounts to a model of magnetoconvection where the dynamics of the flow and magnetic field can be studied independently of the dynamo mechanism.
Magnetic helicity as the key to dynamo bistability (PhD)
Supervisors: David MacTaggart, Radostin Simitev
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics
The planets in the solar system exhibit very different types of largescale magnetic field.The Earth has a strongly dipolar field, whereas the fields of other solar system planets, such as Uranus and Neptune, are far more irregular. Although the different physical compositions of the planets of the solar system will influence the behaviour of the largescale magnetic fields that they generate, the morphology of planetary magnetic fields can depend on properties of dynamos common to all planets. Here, we refer to an important and recent discovery from dynamo simulations. Remarkably, two very different types of chaotic dipolar dynamo solutions have been found to exist over identical values of the basic parameters of a generic model of convectiondriven dynamos in rotating spherical shells. The two solutions mentioned above can be characterised as ‘mean dynamos’, MD, where a strong poloidal field dominates and ‘fluctuating dynamos’, FD, where the poloidal component is weaker and the largescale field can be described as multipolar. Although these two states have been shown to be bistable (coexist) for a wide range of identical parameters, it is not clear how a particular state, MD or FD, is chosen and how/when one state can change to the other. Some of the bifurcations of such states has been investigated, but a deep understanding of the dynamics that cause the bifurcations remains to be developed. Since the magnetic topology of MD and FD states are fundamentally different, an important part of this project will be to probe the nature of MD and FD states by studying magnetic helicity, a magnetohydrodynamic invariant that combines information on the topology of the magnetic field with the magnetic flux. The role of magnetic helicity and other helicities (e.g. cross helicity) is currently not well understood in relation to MD and FD states, but these quantities are conjectured to be very important in the development of MD and FD states.
Bistability is also related to a very important phenomenon in dynamos  global field reversal. A strongly dipolar (MD) field can change to a transitional multipolar (FD) state before a reversal and then settle into another dipolar equilibrium (of opposite polarity) again after the reversal.This project aims to develop a coherent picture of how bistability operates in spherical dynamos. Since bistability is a fundamental property of dynamos, a characterisation of how bistable solutions form and develop is key for any deep understanding of planetary dynamos and, in particular, could be crucial for understanding magnetic field reversals.
Stellar atmospheres and their magnetic helicity fluxes (PhD)
Supervisors: Simon Candelaresi, Radostin Simitev, David MacTaggart, Robert Teed
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics
Our Sun and many other stars have a strong largescale magnetic field with a characteristic time variation. We know that this field is being generated via a dynamo mechanism driven by the turbulent convective motions inside the stars. The magnetic helicity, a quantifier of the field’s topology, is and essential ingredient in this process. In turbulent environments it is responsible for the inverse cascade that leads to the largescale field, while the build up of its smallscale component can quench the dynamo.
In this project, the student will study the effects of magnetic helicity fluxes that happen below the stellar surface (photosphere), within the stellar atmosphere (chromosphere and corona) and between these two layers. This will be done using twodimensional mean field simulations that allow parameter studies for different physical parameters. A fully threedimensional model of a convective stellar wedge will then be used to provide a more detailed picture of the helicity fluxes and their effect on the dynamo. Using recent advancements that allow us to extract surface helicity fluxes from solar observations, the student will make use of observations to verify the simulation results. Other recent observational results on the stellar magnetic helicity will be used to benchmark the findings.

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Geometry and Topology
Operator algebras associated to selfsimilar actions. (PhD)
Supervisors: Mike Whittaker
Relevant research groups: Algebra, Analysis, Geometry and Topology
This project will focus on selfsimilar groups and their operator algebras. The primary aim will be to examine a new class of groups that act selfsimilarly on the path space of a graph and to study the noncommutative geometry of a natural class of operator algebras associated to these selfsimilar groups.
Selfsimilar groups are an important and active new area of group theory. The most famous example is the Grigorchuk group, which was the first known example of a group with intermediate growth. This makes investigating C*algebras associated to them particularly interesting. In particular, these groups are often defined by their action on a graph, and the associated C*algebra encodes both the group and path space of the graph in a single algebraic object, as well as the interaction between them.
Aperiodic substitution tilings and their C*algebras. (PhD)
Supervisors: Mike Whittaker
Relevant research groups: Analysis, Geometry and Topology
A tiling is a collection of subsets of the plane, called tiles, for which any intersection of the interiors of two distinct tiles is empty and whose union is all of the plane. A tiling said to be aperiodic if it lacks translational periodicity. The most common method of producing aperiodic tilings is to use a substitution rule; a method for breaking each tile into smaller pieces, each of which is a scaled down copy of one of the original tiles, and then expanding so that each tile is congruent to one of the original tiles.
This project will focus on a natural class of operator algebras associated with an aperiodic substitution tiling. These algebras were first considered by Kellendonk and reflect the symmetries of a tiling in an algebraic object that allows up to consider invariants in a noncommutative framework. A key area of study are spectral triples associated with aperiodic tilings, which allow us to think of tilings as noncommutative geometric objects.

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Geophysical & Astrophysical Fluid Dynamics
Postdoctoral and PhD projects with specific funding will appear on this page. However, there is also the possibility to apply for funding via various schemes, so please get in touch with one of the group members if you are interested.

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Integrable Systems and Mathematical Physics
Quantum spinchains and exactly solvable lattice models (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
Quantum spinchains and 2dimensional statistical lattice models, such as the Heisenberg spinchain and the six and eightvertex models remain an active area of research with many surprising connections to other areas of mathematics.
Some of the algebra underlying these models deals with quantum groups and Hecke algebras, the TemperleyLieb algebra, the Virasoro algebra and KacMoody algebras. There are many unanswered questions ranging from very applied to more pure topics in representation theory and algebraic combinatorics. For example, recently these models have been applied in combinatorial representation theory to compute GromovWitten invariants (enumerative geometry) and fusion coefficients in conformal field theory (mathematical physics).
Integrable quantum field theory and Ysystems (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
The mathematically rigorous and exact construction of a quantum field theory remains a tantalising challenge. In 1+1 dimensions exact results can be found by computing the scattering matrices of such theories using a set of functional relations. These theories exhibit beautiful mathematical structures related to Weyl groups and Coxeter geometry.
In the thermodynamic limit (volume and particle number tend to infinity while the density is kept fixed) the set of functional relations satisfied by the scattering matrices leads to socalled Ysystems which appear in cluster algebras introduced by Fomin and Zelevinsky and the proof of dilogarithm identities in number theory.
Cherednik Algebras and related topics (PhD)
Supervisors: Misha Feigin
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
The project is aimed at clarifying certain questions related to Cherednik algebras. These questions may include study of homomorphisms between rational Cherednik algebras for particular Coxeter groups and special multiplicity parameters, defining and studying of new partial spherical Cherednik algebras and their representations related to quasiinvariant polynomials, study of differential operators on quasiinvariants related to nonCoxeter arrangements. Relations with quantum integrable systems of CalogeroMoser type may be explored as well. Some other possible topics may include study of quasiinvariants for nonCoxeter arrangements in relation to theory of free arrangements of hyperplanes.
qDT invariants and deformations of hyperKahler geometry (PhD)
Supervisor: Ian Strachan
Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics
The project seeks to understand and exploit the integrable structure behind quantum DonaldsonThomas invariants in terms of deformation of hyperKahler geometry and quantumRiemannHilbert problems.
Almostduality for arbitrary genus Hurwitz spaces (PhD)
Supervisor: Ian Strachan
Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics
The space of rational functions (interpreted as the space of holomorphic maps from the Riemann sphere to itself) may be endowed with the structure of a Frobenius manifolds, and hence there also exists an almostdual Frobenius manifold structure. The class of examples include Coxeter and ExtendedAffineWeyl orbit group spaces. This extends to spaces of holomorphic maps between the torus and the sphere, where one can proved stronger results than just existence results. The project will seek to extend this to the explicit study of the space of holomorphic maps from an arbitrary genus Riemann surface to the Riemann sphere.

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Mathematical Biology
Efficient asymptoticnumerical methods for cardiac electrophysiology (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology
The mechanical activity of the heart is controlled by electrical impulses propagating regularly within the cardiac tissue during one's entire lifespan. A large number of very detailed ionic current models of cardiac electrical excitability are available.These realistic models are rather difficult for numerical simulations. This is due not only to their functional complexity but primarily to the significant stiffness of the equations.The goal of the proposed project is to develop fast and efficient numerical methods for solution of the equations of cardiac electrical excitation with the help and in the light of newlydeveloped methods for asymptotic analysis of the structure of cardiac equations (Simitev & Biktashev (2006) Biophys J; Biktashev et al. (2008), Bull Math Biol; Simitev & Biktashev (2011) Bull Math Biol)
The student will gain considerable experience with the theory of ordinary and partial differential equations, dynamical systems and bifurcation theory, asymptotic and perturbation methods,numerical methods. The applicant will also gain experience in computerprogramming, scientific computing and some statistical methods for comparison with experimental data.
Fastslow asymptotic analysis of cardiac excitation models (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology
Mathematical models of cardiac electrical excitation describe processess ocurring on a wide range of time and length scales.
Mathematical models of vasculogenesis (PhD)
Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics  Modelling and Analysis of Material Systems, Mathematical Biology
Vasculogenesis is the process of forming new blood vessels from endothelial cells, which occurs during embryonic development. Viable blood vessels facilitate tissue perfusion, allowing the tissue to grow beyond the diffusionlimited size. However, in the absence of vasculogenesis, efforts to engineer functional tissues (eg for implantation) are restricted to this diffusionlimited size. This project will investigate mathematical models for vasculogenesis and explore mechanisms to stimulate blood vessel formation for in vitro tissues. The project will involve collaboration with Department of Biological Engineering at MIT, as part of the SofTMech^{MP} project.
A coupled cardiovascularrespiration model for mechanical ventilation (PhD)
Supervisors: Peter Stewart, Nicholas A Hill
Relevant research groups: Continuum Mechanics  Fluid Dynamics and Magnetohydrodynamics, Mathematical Biology
Mechanical ventilation is a clinical treatment used to draw air into the lungs to facilitate breathing, used in treatment of premature babies with respiratory distress syndrome and in the treatment of severe Covid pneumonia. The aim is to oxygenate the blood while simultaneously removing unwanted byproducts. However, overinflation of the lungs can reduce the blood supply to the gas exchange surfaces, leading to a ventilationperfusion mismatch. This PhD project will give you the opportunity to develop a mathematical model to describe the coupling between blood flow in the pulmonary circulation and air flow in the lungs (during both inspiration and expiration). You will devise a coupled computational framework, capable of testing patientspecific ventilation protocols. This is an ideal project for a postgraduate student with an interest in applying mathematical modelling and image analysis to predictive healthcare. The project will give you the opportunity to join a crossdisciplinary Research Hub that aims to push the boundaries of quantitative medicine and improve clinical decision making using innovative mathematical and statistical modelling.

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Number Theory
Our group has an active PhD student community, and every year we admit new PhD students. We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.

Overview
With a sizeable complement of academic staff and postgraduate students, the School of Mathematics and Statistics is one of the largest in the UK.
Our research interests cover several areas of Pure and Applied Mathematics. These areas are not mutually exclusive and there are considerable benefits from interactions between the different areas that enhance the research environment.
All our research areas are highly rated internationally and most members of the School have ongoing collaborations with mathematicians overseas and elsewhere in the UK. Many overseas mathematicians spend periods in Glasgow working with members of the School.
There are several series of research seminars with invited speakers from the UK and overseas. There is also a regular postgraduate seminar where our PhD students can develop their presentational skills. All this helps to generate a lively and mutually supportive research environment, which has led to the award of coveted prizes and competitive fellowships for several of our younger staff.
Research areas
Study options
fulltime (years) 
parttime (years) 

Phd  34  68 
Integrated PhD  4  n/a 
MSc (Res)  12  23 
MPhil  23  34 
Integrated PhD programme (4 years)
Our PhD with Integrated Study in Mathematical Sciences is a fouryear PhD programme in the School of Mathematics and Statistics.
Completion of taught Masters level courses in the first nine months will provide you with a valuable introduction to academic topics and research methods, whilst providing key training in the critical evaluation of research data.
Upon successful completion of the taught component, you will progress to your research degree. You will submit a thesis to be examined by the end of your fourth year.
Entry requirements
PhD programmes
2.1 Honours degree or equivalent
Required documentation
Applicants should submit:
 Transcripts/degree certificate
 Two references
 CV
 Name of potential Supervisor
Integrated PhD programmes
2.1 Honours degree or international equivalent in a relevant subject area
English language requirements
For applicants whose first language is not English, the University sets a minimum English Language proficiency level.
International English Language Testing System (IELTS) Academic module (not General Training)
 6.5 with no subtest under 6.0.
 Tests must have been taken within 2 years 5 months of start date. Applicants must meet the overall and subtest requirements using a single test.
Common equivalent English language qualifications
All stated English tests are acceptable for admission to this programme:
TOEFL (ib, my best or athome)
 90 with minimum R 20, L 19, S 19, W 23.
 Tests must have been taken within 2 years 5 months of start date. Applicants must meet the overall and subtest requirements using a single test, this includes TOEFL mybest.
PTE (Academic)
 60 with minimum 59 in all subtests.
 Tests must have been taken within 2 years 5 months of start date. Applicants must meet the overall and subtest requirements using a single test.
Glasgow International College English Language (and other foundation providers)
 65%.
 Tests are accepted for academic year following sitting.
University of Glasgow Presessional courses
 Tests are accepted for 2 years following date of successful completion.
Alternatives to English Language qualification
 Degree from majorityEnglish speaking country (as defined by the UKVI including Canada if taught in English).
 Students must have studied for a minimum of 2 years at Undergraduate level, or 9 months at Master’s level, and must have completed their degree in that majorityEnglish speaking country and within the last 6 years.
 Undergraduate 2+2 degree from majorityEnglish speaking country (as defined by the UKVI including Canada if taught in English).
 Students must have completed their final two years study in that majorityEnglish speaking country and within the last 6 years.
For international students, the Home Office has confirmed that the University can choose to use these tests to make its own assessment of English language ability for visa applications to degree level programmes. The University is also able to accept an IELTS test (Academic module) from any of the 1000 IELTS test centres from around the world and we do not require a specific UKVI IELTS test for degree level programmes. We therefore still accept any of the English tests listed for admission to this programme.
Presessional courses
The University of Glasgow accepts evidence of the required language level from the English for Academic Study Unit Presessional courses. We also consider other BALEAP accredited presessional courses:
Fees and funding
Fees
2024/25
 UK: To be confirmed by UKRI [23/24 fee was £4,712]
 International & EU: £30,240
Prices are based on the annual fee for fulltime study. Fees for parttime study are half the fulltime fee.
Irish nationals who are living in the Common Travel Area of the UK, EU nationals with settled or presettled status, and Internationals with Indefinite Leave to remain status can also qualify for home fee status.
Alumni discount
We offer a 20% discount to our alumni on all Postgraduate Research and full Postgraduate Taught Masters programmes. This includes University of Glasgow graduates and those who have completed Junior Year Abroad, Exchange programme or International Summer School with us. The discount is applied at registration for students who are not in receipt of another discount or scholarship funded by the University. No additional application is required.
Possible additional fees
 Resubmission by a research student £540
 Submission for a higher degree by published work £1,355
 Submission of thesis after deadline lapsed £350
 Submission by staff in receipt of staff scholarship £790
Depending on the nature of the research project, some students will be expected to pay a bench fee (also known as research support costs) to cover additional costs. The exact amount will be provided in the offer letter.
Funding
 View a full list of our current scholarships
Support
Our postgraduate students join a community of academic experts across a wide range of pure and applied mathematics and statistics and develop a mature understanding of fundamental theories and analytical skills applicable to many solutions.
There is a lively seminar program with members of the Schools involved in a number of networks including the North British Functional Analysis Seminar, The North British Differential Equations Seminar and the LMS Network on Classical and Quantum Integrability.
The School also welcomes many guest speakers from the UK and overseas. We hold regular postgraduate seminars, where our PhD students can develop their presentation skills, and international conferences and workshops.
Students can also attend our hugely popular weeklong training course each year, which provides practically motivated training in key statistical analysis and modelling skills.
You will be part of a Graduate School which provides the highest level of support to its students.
The overall aim of our Graduate School is to provide a worldleading environment for students which is intellectually stimulating, encourages them to contribute to culture, society and the economy and enables them to become leaders in a global environment.
We have a diverse community of over 750 students from more than 50 countries who work in innovative and transformative disciplinary and interdisciplinary fields. An important part of our work is to bring our students together and to ensure they consider themselves an important part of the University’s academic community.
Being part of our Graduate School community will be of huge advantage to you in your studies and beyond and we offer students a number of benefits in addition to exceptional teaching and supervision, including:
 A wideranging and responsive research student training programme which enables you to enhance your skills and successfully complete your studies.
 Mobility scholarships of up to £4000 to enable you to undertake work in collaboration with an international partner.
 A diverse programme of activities which will ensure you feel part of the widerresearch community (including our biannual science slam event).
 A residential trip for all new research students.
 The opportunity to engage with industrypartners through training, placements and events.
 Professionally accredited programmes.
 Unique Masters programmes run in collaboration with other organisations.
 Stateoftheart facilities including the James Watt Nanofabrication Centre and the Kelvin Nanocharacterisation Centre.
 Highlyrated support for international students.
Over the last five years, we have helped over 600 students to complete their research studies and our students have gone on to take up prestigious posts in industries across the world.
How to apply
Identify potential supervisors
All Postgraduate Research Students are allocated a supervisor who will act as the main source of academic support and research mentoring. You may want to identify a potential supervisor and contact them to discuss your research proposal before you apply. Please note, even if you have spoken to an academic staff member about your proposal you still need to submit an online application form.
You can find relevant academic staff members with our staff research interests search.
Gather your documents
Before applying please make sure you gather the following supporting documentation:
 Final or current degree transcripts including grades (and an official translation, if needed) – scanned copy in colour of the original document.
 Degree certificates (and an official translation, if needed): scanned copy in colour of the original document.
 Two references on headed paper and signed by the referee. One must be academic, the other can be academic or professional. References may be uploaded as part of the application form or you may enter your referees contact details on the application form. We will then email your referee and notify you when we receive the reference. We can also accept confidential references direct to rioresearchadmissions@glasgow.ac.uk, from the referee’s university or business email account.
 Research proposal, CV, samples of written work as per requirements for each subject area.
Contact us
 If you have any questions about your application before you apply: email scienggradschool@glasgow.ac.uk
 If you have any questions after you have submitted your application: contact our Admissions team
 Any references may be submitted by email to: rioresearchadmissions@glasgow.ac.uk
International Students
 Advice on visa, immigrations and the Academic Technology Approval Scheme (ATAS) can be found at 'Apply for a visa outside the UK or come to the UK as an EU / EEA national'