PhD in Applied Mathematics
A doctoral degree built on original research under a supervisor, usually three to six years long, ending with a thesis that you defend.
Overview
Why study a PhD in Applied Mathematics?
A PhD in Applied Mathematics is a research degree. Working with a supervisor, you take an open problem, often one that comes from physics, engineering, biology or industry, and develop new models, new analysis or new numerical methods to address it. Most candidates enter with a master's, although some countries accept applicants with an outstanding bachelor's. At the end, your thesis is assessed by experts in the area, usually through an oral examination, and much of the work normally appears as journal papers.
Structure varies by country. A UK or Australian PhD usually lasts three to four years and research begins almost at once, sometimes within a doctoral training centre that adds taught courses in the first year. North American programmes are longer, commonly five years or more, because the first two years mix graduate courses with qualifying examinations. In Germany and Japan many candidates join a research group as employees or funded students, and in Germany the doctorate can also be carried out at a non-university research institute in partnership with a university.
Research topics are wide. They include fluid dynamics and turbulence, waves and acoustics, solid mechanics, mathematical biology and medicine, nonlinear dynamics, numerical analysis and scientific computing, inverse problems and imaging, and the mathematics of data and machine learning. Most doctoral students in this field receive funding of some kind, but funded places attract many applicants, so start contacting potential supervisors and checking deadlines a year ahead. The doctorate leads naturally to research careers in universities, national laboratories and forecasting centres, and also to specialist modelling and quantitative roles in industry.
Curriculum
What you may study
Doctoral research in applied mathematics often falls into areas such as these.
- Fluid dynamics and turbulence
- Waves, acoustics and electromagnetism
- Solid and continuum mechanics
- Mathematical biology and physiology
- Nonlinear dynamics and pattern formation
- Numerical methods for partial differential equations
- Inverse problems and imaging
- Data assimilation and uncertainty quantification
- Optimisation and control
- Mathematical foundations of machine learning
Published programmes for Applied Mathematics at PhD
No programmes for Applied Mathematics at PhD are published yet.
Not in our catalogue yet: that reflects what Universta lists, not what universities teach.
Programmes show each university’s published offering. Course guides describe the course across universities.
Admission
Getting into PhD in Applied Mathematics
Applicants usually need a master's in applied mathematics, mathematics, physics or engineering, or, where direct entry exists, as in many US programmes and some UK ones, a first degree with outstanding results. Committees look for evidence of research potential: a dissertation or project involving modelling or numerical work, strong marks in the relevant modules, and references from lecturers or supervisors familiar with your research. Programming experience is close to essential, since almost every applied mathematics thesis involves serious computation.
The route in differs by country. In the UK, Australia and Germany applicants commonly apply to a named, advertised project or approach a potential supervisor first, and a research proposal may be requested. In the United States admission is usually to the department as a whole, with an adviser chosen during the programme, whereas many Canadian departments expect a supervisor to agree to take you on. In Japan, contacting a professor before applying is normally expected. Funding is usually decided alongside admission, so check scholarship and studentship deadlines carefully.
Careers
Career pathways
A doctorate in applied mathematics leads most often to research posts, inside or outside universities, such as these.
- Postdoctoral Researcher
- Numerical Weather Prediction Scientist
- Lecturer or Assistant Professor
- Climate Modeller
- Research Mathematician
- Computational Scientist
- Computational Fluid Dynamics Specialist
- Quantitative Researcher
- Machine Learning Research Scientist
- Mathematical Biologist
- Scientific Computing Specialist
- Research Software Engineer
Sectors
Industries
Universities and Research Institutes
Postdoctoral posts are the usual first step towards a lectureship, while national laboratories and institutes offer longer-term research positions.
Weather, Climate and Energy
Forecasting centres, climate institutes and energy firms employ PhDs to develop numerical models of the atmosphere, oceans, reservoirs and power grids.
Aerospace and Engineering
Research and development teams hire doctoral graduates to create new simulation methods for aircraft, engines, vehicles and materials.
Quantitative Finance
Trading firms and asset managers recruit applied mathematics PhDs as quantitative researchers working on pricing, execution and risk.
Technology and AI Research
Research groups in technology companies employ PhDs on optimisation, scientific machine learning and large-scale numerical computation.
Space, Defence and Government
Space agencies, defence laboratories and public research bodies use doctoral graduates for trajectory design, signal processing and simulation.
Who hires
Employers
- Met Office
- RIKEN
- European Centre for Medium-Range Weather Forecasts
- NASA
- Max Planck Society
- Fraunhofer-Gesellschaft
- Google DeepMind
- Jane Street
- Indian Institute of Tropical Meteorology
- Defence Research and Development Organisation (DRDO)
Named for illustration only; not a ranking, endorsement or partnership.
Outlook
Looking ahead
Most PhD graduates who stay in academia spend several years in postdoctoral posts, sometimes in more than one country, before applying for permanent positions. Industry offers a parallel path, and many applied mathematicians move between the two, joining research teams in engineering, energy, finance or technology. Publishing during the doctorate, presenting at conferences and building well-documented research code all help, whichever route you take.
Several trends are shaping research. Exascale and other high-performance computers allow simulations at a level of detail that was out of reach a decade ago, while machine learning is being combined with physical models in weather forecasting, materials design and medicine. Questions about how far a model's prediction can be trusted are attracting growing attention, as are quantum algorithms for simulation. These areas reward researchers who are comfortable with both rigorous analysis and serious computation.
Specializations
Other Mathematics & Statistics specializations at PhD
Subject
Mathematics & Statistics at PhD
Questions
Questions about PhD in Applied Mathematics
Is a PhD in Applied Mathematics usually funded?
Often, but not automatically. Support takes several forms: a studentship that pays fees and a living stipend, an assistantship in return for teaching or research work, a salaried post, or a scholarship from a government or university. Applied mathematics projects are sometimes co-funded by industry or national laboratories. Certain awards are restricted to domestic applicants, so check who qualifies for each scheme and apply early.
Should I contact a supervisor before applying for an Applied Mathematics PhD?
In the UK, Australia, Germany, Canada and Japan, agreeing a supervisor or applying for a specific project is usually part of admission, so read recent papers by people in your area and email them briefly about the problems that interest you. In the United States you normally apply to the whole department and settle on an adviser after starting. Either way, a good fit with your supervisor's research matters a great deal.
How much computing will an Applied Mathematics PhD involve?
Usually a great deal. Even theoretical projects tend to test results numerically, and many theses are built around simulations that run on clusters or national supercomputers. You will be expected to write reliable, well-documented code, often in Python, C++, Fortran or Julia, and to understand the numerical methods behind it. Many doctoral programmes offer training in high-performance computing in the first year.
What do Applied Mathematics PhD graduates do outside universities?
Many join research and development teams in aerospace, energy, automotive and engineering software firms, or work in weather and climate centres, national laboratories and space agencies. Others become quantitative researchers in trading firms or research scientists in technology companies. These roles value the ability to develop new methods rather than only apply existing ones, which is exactly what a doctorate trains.
Plan
Plan your PhD
Keep exploring
More on Applied Mathematics
Next step
Planning PhD in Applied Mathematics?
Tell us about your academic profile, goals and budget to explore your options.
Free assessmentPersonalised guidanceNo obligation