PhD in 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 Mathematics?
A PhD in Mathematics is a research degree in which you aim to prove new theorems, develop new methods or build new theory under the guidance of a supervisor. Entry usually needs a master's, or an excellent bachelor's in countries that admit students directly. The outcome is a thesis containing original results, examined by specialists and often defended in an oral examination, with the main results usually submitted to journals. Mathematics PhDs tend to involve more solitary thinking than laboratory sciences, although collaboration is increasingly common.
How the PhD is organised depends on the country. UK and Australian PhDs normally last three to four years, with research from the outset. North American programmes run around five to six years, with graduate courses and qualifying examinations first, and students often choose a supervisor after a year or two. In Germany doctoral candidates are often employed on a research contract and teach exercise classes, while in Japan a three-year doctoral course follows a two-year master's. Funding is common in mathematics but competitive everywhere.
Research areas span number theory, algebra, geometry and topology, analysis and partial differential equations, probability, combinatorics, logic and mathematical physics, as well as computational and applied directions. Daily life involves reading papers, working at hard problems through long stretches of slow progress, attending seminars and presenting at conferences. The degree leads most directly to postdoctoral research and academic posts, but trading firms, technology research groups, security agencies and research institutes also recruit mathematics PhDs for their ability to make progress on problems nobody has yet solved.
Curriculum
What you may study
Doctoral research in Mathematics often falls within areas such as these.
- Number theory and arithmetic geometry
- Algebra and representation theory
- Algebraic and differential geometry
- Topology and geometric group theory
- Harmonic and functional analysis
- Partial differential equations
- Probability theory and stochastic analysis
- Combinatorics and graph theory
- Mathematical logic and set theory
- Mathematical physics and dynamical systems
Published programmes for Mathematics at PhD
No programmes for 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 Mathematics
Panels look for excellent results in advanced mathematics, a master's dissertation or other research experience, and strong references from mathematicians who can judge your potential for research. UK and Australian applicants often apply for an advertised project or to a named supervisor, sometimes with a short proposal, and early contact with that supervisor is expected. US and Canadian departments admit students to the programme as a whole, so the statement of purpose matters more than a detailed proposal, and some still request scores from the GRE Mathematics Subject Test.
Deadlines for funded places often fall between November and February for an autumn start, so begin preparing well over a year ahead. You will need transcripts, a CV, a research statement, references and usually an English test result. In Germany many positions are advertised as research assistant posts and you apply much as you would for a job. In Japan, applicants usually agree a supervisor before applying and may sit an entrance examination. Check what any funding covers, including tuition, stipend, health insurance and teaching duties, and whether it lasts the full expected length.
Careers
Career pathways
Most Mathematics PhD graduates stay in research, at universities or in industry, in positions like these.
- Postdoctoral Researcher
- Assistant Professor
- Lecturer in Mathematics
- Research Fellow
- Research Mathematician
- Quantitative Researcher
- Cryptographer
- Machine Learning Research Scientist
- Computational Scientist
- Operations Research Scientist
- Mathematical Modeller
- Data Scientist
Sectors
Industries
Universities and Research Institutes
Most PhD graduates begin with postdoctoral posts on the way to lectureships, and some join research institutes that host visiting and permanent mathematicians.
Quantitative Finance
Trading and investment firms recruit mathematics PhDs as quantitative researchers who study market data and design trading strategies.
Technology Research
Research groups in technology companies hire doctoral mathematicians to work on machine learning theory, optimisation, algorithms and quantum computing.
Security and Cryptography
National security agencies and technology firms employ PhDs to design and analyse cryptographic systems, including schemes intended to resist quantum computers.
Science and Engineering Modelling
Weather services, energy companies and engineering firms use applied mathematics PhDs to develop numerical models and simulation methods.
Who hires
Employers
- Max Planck Society
- RIKEN
- Jane Street
- Two Sigma
- Microsoft Research
- Google DeepMind
- National Security Agency (NSA)
- GCHQ
- Defence Research and Development Organisation (DRDO)
- Indian Space Research Organisation (ISRO)
Named for illustration only; not a ranking, endorsement or partnership.
Outlook
Looking ahead
Mathematical research is changing in how it is done as well as in what it studies. Formal proof assistants such as Lean are being used to verify major results and to build shared libraries of checked mathematics, and AI systems are beginning to suggest conjectures and help with parts of proofs, which raises real questions about how mathematicians will work. Large collaborative projects, once rare in the subject, are becoming more common.
Permanent academic jobs are scarce everywhere and usually come after one or more postdoctoral positions, so it helps to build skills that also matter outside universities, such as programming and clear teaching. Problems arising from machine learning, quantum computing and post-quantum cryptography are opening new research directions and new employers for doctoral mathematicians. Being able to explain to a non-specialist what your result means, and what it leaves unsettled, will serve you in any setting.
Specializations
Other Mathematics & Statistics specializations at PhD
Subject
Mathematics & Statistics at PhD
Questions
Questions about PhD in Mathematics
How long does a PhD in Mathematics take in different countries?
Typically three to four years in the UK, Australia and Germany, three years after a two-year master's in Japan, and around five to six years in North America, where the programme includes graduate coursework and qualifying exams. Actual completion times vary with the project and the funding period.
How do I find a supervisor for a Mathematics PhD?
Read recent papers from departments that work in your area, and look for academics whose problems interest you. Where the system expects it, as in the UK, Australia, Germany and Japan, email them with a short, specific message about their work and your background, attaching a CV and transcript. In the United States and Canada you usually apply to the department first and choose a supervisor later.
Do I need the GRE Mathematics Subject Test for a PhD?
Only for some programmes, mainly in the United States, and several have made it optional or dropped it. Universities in the UK, Europe, Australia and Japan rarely ask for it. Check each department's current requirements, as policies have changed frequently in recent years.
Can I move into industry after a PhD in pure mathematics?
Yes. Pure mathematicians regularly move into quantitative finance, cryptography, software, data science and technology research, where employers value deep problem-solving ability over knowledge of a particular field. Learning to program well during the PhD makes this move much easier.
Plan
Plan your PhD
Keep exploring
More on Mathematics
Next step
Planning PhD in Mathematics?
Tell us about your academic profile, goals and budget to explore your options.
Free assessmentPersonalised guidanceNo obligation