Mathematical Sciences MPhil/PhD

Major code: MPMR

About us

Mathematical Sciences

Mathematical Sciences at Liverpool offers wide opportunities for postgraduate study in an active research environment.

Research is organized in 9 research groups (or clusters) within Applied Mathematics, Pure Mathematics, Theoretical Physics, and Financial and Actuarial Mathematics and Applied Probability, in addition to 3 research centres (RCMM, CMIT and ERRC) and one Institute (IFAM).

There are 12 active research groups in many areas of Mathematics and Theoretical Physics, holding research grants from EPSRC, STFC and other sources. State of- the-art computing facilities are available. The Department’s teaching was evaluated in the most recent survey by the Quality Assurance Agency and awarded a total of 23 points out of 24 (excellent).

Teaching context

Since its formation in 1995 as a (combined) Department of Mathematical Sciences, the Department has sought to nurture its traditional research strengths on the one hand while, on the other, expand in new directions when the opportunity arises. Care has been taken to ensure critical mass in key areas, while simultaneously allowing top class research to inform a wide range of our teaching.

Research staff

The Faculty of Science and Engineering houses a number of Research Centres and Institutes which encourage interdisciplinary research, both between Science Departments and with departments in other faculties or even universities. Each year there are opportunities for a number of postgraduate research students in each centre. Further information can be obtained by contacting the centre directly.

Research Centre in Mathematics and Modelling

This multidisciplinary research centre acts as a forum and as a management tool for modelling projects in the University and for coordinating research proposals to attract external funding. RCMM runs active visits, seminars and workshop programmes for international exchanges in mathematical modelling.

Professor A B Movchan

T:+44 (0)151 794 4740


Centre for Mathematical Imaging Techniques

Imaging technology is increasingly used in diverse and interdisciplinary fields of mathematical, physical, biological, biomedical, dental, medical and engineering sciences. CMIT is established to tackle real-life problems by developing, using and promoting state-of- the-art mathematical techniques.

Professor K Chen

T:+44 (0)151 794 4741


Environmental Radioactivity Research Centre

This Centre provides a coherent base from which to deploy the wide range of interdisciplinary skills essential to studies of environmental radioactivity and its applications and includes staff from the Departments of Mathematical Sciences, Geography, Physics and Archaeology. Research opportunities exist for students with backgrounds in mathematical, physical or environmental sciences to work either on modelling or empirically based topics in environmental radioactivity.

Professor Peter G Appleby
T:+44 (0)151 794 4020

Research groups

Cluster of Algebraic Geometry

The Cluster carries out active research work in foliation singularities, Classification of higher dimensional algebraic varieties, Nagata's conjecture, division theorems for the cohomology of discriminant, automorphism groups, conical resolutions of discriminants, Moduli spaces, Algebraic cycles on K3-surfaces and Fano threefolds, Finite-dimensional motives and algebraic cycles, Zeta-functions in motivic stable categories, hyperbolic root systems, self-correspondences via moduli of sheaves of some K3 surfaces, birational geometry of higher-dimensional algebraic varieties, Singularities of discontinuous Hamiltonian systems, singular Fano varieties.

Cluster of Applied Probability

The Cluster carries out active research work in analysis of communication networks, optimal control, controlled stochastic jump processes, multi-criteria optimisation, infectious disease modelling, stochastic models for endemic infection, control theory applied to infectious disease models, statistical analysis for complex data, bootstrap, kernel, wavelet and local polynomial estimation, Bayesian statistical inference.

Cluster of Dynamical Systems

The Cluster carries out active research work in dynamics of surface homeomorphisms, Pruning theory, Train track algorithms, Ergodic theorems, Non-linear Poincaré and multiple recurrence, complex dynamics involving the Mandelbrot set, dynamics in parameter spaces, quadratic rational maps, transcendental dynamics, Fractal dimensions associated with entire and meromorphic functions, measurable dynamics of transcendental functions.

Cluster of Financial and Actuarial Mathematics

The Cluster carries out active research work in risky investments in insurance, extinction probability of interacting branching collision processes, risk theory, actuarial mathematics, ruin probabilities in models with dependence, ruin probabilities for Lévy processes, markets with uncertain parameters, optimal investment and consumption, pricing and hedging in incomplete markets, stability of infinite dimensional stochastic systems, stochastic systems with memory or delay.

Cluster of Mathematical Biology

The Cluster carries out active research work in theory and applications of discrete and continuous excitable media, including analytical and numerical study of spiral and scroll waves and excitation propagation in the heart; dynamics on and of networks, including epidemiology and population dynamics; biological fluid dynamics, including motile phytoplankton in aquatic environment and influence of turbulence and prey distribution on plankton predation; pattern formation in embryogenesis and by swimming algae; evolution and systems biology.

Cluster of Quantum Field Theory and Applications

The Cluster carries out active research on high-precision calculations in quantum gauge theories such as quantum chromodynamics, with particular applications to collider physics and the search for effects beyond the Standard Model of particle physics, for example supersymmetry. We are also engaged in the effort to understand the mechanism of quark confinement, and work on numerical simulations in the context of lattice gauge theory.

Cluster of Singularity Theory and its Applications

The Cluster carries out active research work in various aspects of singularity theory,  local first order invariants of smooth mappings, monodromy groups of functions with symmetry, Looijenga's theorem, Lorentzian geometry, links of singularities, complex hyperbolic geometry, Bridgeland stability conditions,  triangulated category and t-structures, structure of Witt groups of perverse sheaves, Hyperbolic reflection groups, Fano varieties and fibre spaces with singularities, unimodular function singularities, Lagrangian maps.

Cluster of String Phenomenology

The Cluster carries out active research work in Collider phenomenology, black hole entropy and partition functions, string phenomenology, vacuum selection in and geometry of string theory, Euclidean supersymmetry, particles in supersymmetric theories, phenomenology from String Theory brane intersections (observed at Large Hadron Collider), phenomenological and cosmological implications, quantum mechanics from an equivalence principle.

Cluster of Waves and Solid Mechanics

The Cluster carries out active research work in in Mathematical models of solids with imperfect interfaces, continuum mechanics; composite media, Asymptotic analysis of crack-defect interaction, Asymptotic analysis of fracture in composite materials, fracture mechanics,  asymptotics for eigenvalue problems posed in multi-structures, wave propagation in phononic crystals, Spectral problems related to sizing and location of defects in elastic structures, Electro-elastic waves in piezoelectric cubic crystals, surface waves in ferromagnetic media, numerical methods for partial differential equations, Image analysis.

Megan Selbach-Allen

Once you have a degree no one can take it away from you. Furthermore there always seems to be a need for more people who have studied technical disciplines such as mathematics. I know that where ever I end up in the future this experience and the degree I am earning will help me to be successful.


What are you studying?

I am doing an Msc in Mathematical Science

What does that entail?

I am doing a taught degree and thus take a few courses each term. I am also completing a dissertation in mathematical modelling of infectious diseases. This work involves using mathematical equations and objects such as networks to understand how an infectious disease such as flu might spread through a population.

What were your main reasons for choosing to undertake postgraduate study/research at University of Liverpool?

The University of Liverpool was the only place that would allow me to research infectious disease modelling, still pursue a degree in mathematics. The other programs I considered would have forced me into epidemiology or a specialized area of mathematics such as computational biology. I wanted to continue in a general maths program while studying infectious diseases modelling.

What do you enjoy most about the whole postgraduate experience?

I enjoy getting to study subjects that I am truly passionate about and getting to apply the knowledge I learn in the classroom to real problems I am trying to address in my research. It's not just about memorizing answers for an exam, but understanding concepts and ideas and seeing if I can apply them in new ways.

How do you believe undertaking postgraduate study/research will help your career prospects?

Once you have a degree no one can take it away from you. Furthermore there always seems to be a need for more people who have studied technical disciplines such as mathematics. I know that where ever I end up in the future this experience and the degree I am earning will help me to be successful. 

What advice would you give to anybody considering undertaking postgraduate study/research?

I would say to make sure you enjoy your subject. If you enjoy what you are learning and studying then your experience will be that much easier and more fun. Postgraduate study is challenging, but if you enjoy what you are doing and work hard than it is definitely doable. 

What do you think of Liverpool as a place to live and study?

Liverpool has gone through a lot of redevelopment recently and the City Centre and Docks are great places to hang out and walk around. I love living in the city and walking to University every day. The quality of life in Liverpool is great. It is the perfect sized city, big enough to have anything you could need, but not to big to be overwhelming.