D'Math University | Education & Specialist Mathematics
PhD Number Theory
The oldest, deepest, and most celebrated branch of pure mathematics — studying the properties of the integers and their generalisations. From the distribution of prime numbers and the Riemann Hypothesis to Wiles's proof of Fermat's Last Theorem and the Langlands Programme, number theory sits at the absolute summit of mathematical achievement. This doctoral programme places candidates within reach of its most profound open problems.
Programme Overview
The Queen of Mathematics
Gauss called mathematics the queen of the sciences and number theory the queen of mathematics. After two millennia, the title holds. Number theory asks simple questions — How are the primes distributed? Which equations have integer solutions? What is the arithmetic of elliptic curves? — and the answers require the most powerful tools mathematics has ever developed. Wiles's proof of Fermat's Last Theorem mobilised a century of algebraic geometry, analytic number theory, and representation theory, and the Langlands Programme remains one of the grandest unifying visions in all of science.
- Riemann Hypothesis: The most famous unsolved problem in mathematics — studied directly in our analytic number theory track
- Langlands Programme: The deepest unifying vision in number theory — our doctoral curriculum provides genuine access
- Elliptic curves: The objects at the heart of Fermat's Last Theorem, BSD conjecture, and modern cryptography
- Computational tools: SageMath, Magma, and PARI/GP for experimental number theory research
Research Tracks
Number theory at the doctoral level organises into three broad but deeply interconnected research tracks, each with its own methods, open problems, and specialist literature. Our programme supports candidates in all three, with supervisory expertise spanning the full range of the field.
- Analytic number theory: L-functions, exponential sums, sieve methods, and the deep study of primes in arithmetic progressions
- Algebraic number theory: Class field theory, algebraic K-theory, Galois representations, and the arithmetic of number fields and function fields
- Arithmetic geometry: Elliptic curves, abelian varieties, p-adic methods, and the Birch and Swinnerton-Dyer conjecture
Research Areas
Analytic Number Theory
The Riemann zeta function, prime number theorem, the Riemann Hypothesis, zero-free regions, and the distribution of primes in arithmetic progressions via Dirichlet L-functions.
Algebraic Number Theory
Number fields and their rings of integers, ideal class groups, unit groups, ramification theory, class field theory, and Artin reciprocity — the algebraic heart of the Langlands Programme.
Elliptic Curves & Modular Forms
The arithmetic of elliptic curves over Q and number fields, modular forms and their L-functions, the modularity theorem (Taniyama-Shimura-Weil), and the BSD conjecture.
L-functions & Automorphic Forms
The Langlands Programme — automorphic representations, functoriality conjectures, and the grand unification of number theory through the spectral theory of arithmetic groups.
Diophantine Equations
Rational and integral points on varieties, Faltings' theorem (Mordell conjecture), Baker's theory of linear forms in logarithms, and the geometry of numbers approach to Diophantine problems.
Arithmetic Geometry
p-adic numbers and p-adic analysis, etale cohomology, the Weil conjectures, Galois representations attached to geometric objects, and the Fontaine-Mazur conjectures.
Cryptographic Number Theory
Elliptic curve discrete logarithm problem, pairing-based cryptography, isogeny-based post-quantum cryptography, and the number-theoretic foundations of secure communications.
Sieve Theory
Selberg sieve, large sieve inequality, applications to twin primes (Zhang, Maynard, Polymath8), Goldbach-type problems, and almost-prime results in arithmetic progressions.
Iwasawa Theory
The Iwasawa main conjecture, p-adic L-functions, the structure of class groups in Zp-extensions, and the connection to the Birch and Swinnerton-Dyer conjecture via Kolyvagin's Euler systems.
Computational Number Theory
Algorithms for factoring, primality testing, elliptic curve point counting, L-function computation, and the use of databases (LMFDB) in modern number-theoretic research.
Click any course to view its objective and learning outcomes.
NTH 701 Research Methods +
Objective
To prepare doctoral candidates for number theory research.
Learning Outcomes
- Apply rigorous research design.
- Use specialised databases.
- Apply LaTeX for academic writing.
- Critique published research.
- Write proposals.
NTH 702 Algebraic Number Theory +
Objective
To master algebraic number theory.
Learning Outcomes
- Apply class field theory.
- Use Galois cohomology.
- Apply Iwasawa theory.
- Use elliptic curves.
- Discuss modular forms.
NTH 703 Analytic Number Theory +
Objective
To master analytic number theory.
Learning Outcomes
- Apply zeta and L-functions.
- Use sieve methods.
- Apply circle method.
- Use exponential sums.
- Discuss prime distribution.
NTH 704 Modular Forms +
Objective
To research modular forms and L-functions.
Learning Outcomes
- Apply Hecke theory.
- Use modular curves.
- Apply Eisenstein series.
- Use theta series.
- Discuss Langlands programme.
NTH 705 Arithmetic Geometry +
Objective
To research the arithmetic of algebraic varieties.
Learning Outcomes
- Apply elliptic curves over number fields.
- Use abelian varieties.
- Apply Galois representations.
- Use étale cohomology.
- Discuss BSD conjecture.
NTH 706 Doctoral Seminar +
Objective
To engage with current research.
Learning Outcomes
- Present and critique papers.
- Engage with international research.
- Participate in peer review.
- Build a network.
- Develop presentation skills.
NTH 707 Teaching Practicum +
Objective
To develop teaching skills.
Learning Outcomes
- Plan and deliver lectures.
- Design assessments.
- Apply pedagogical theory.
- Mentor undergraduates.
- Engage in curriculum design.
AND OR NOT XOR -> <->
NTH 708 PhD Thesis I +
Objective
To produce original research.
Learning Outcomes
- Identify an original problem.
- Conduct literature review.
- Develop methodology.
- Produce preliminary results.
- Present at conferences.
NTH 709 PhD Thesis II +
Objective
To advance the research.
Learning Outcomes
- Develop original methodology.
- Generate substantial findings.
- Publish in journals.
- Develop thesis structure.
- Defend methodology.
NTH 710 PhD Thesis III +
Objective
To consolidate research.
Learning Outcomes
- Write 80,000-100,000 word thesis.
- Synthesise contributions.
- Defend viva voce.
- Publish multiple articles.
- Contribute to the field.
Career Outcomes
University Professor (Pure Mathematics)
The premier academic destination for number theory PhDs. Professorships at research-intensive universities worldwide — particularly in number theory, algebra, and arithmetic geometry groups. A career of research, doctoral supervision, and teaching at the highest level.
Government Cryptographer
National security agencies in the UK, USA, Australia, Canada, and allied nations (the Five Eyes community) actively recruit number theorists for cryptographic research, protocol design, and the analysis of cryptographic threats including quantum attacks on current systems.
GCHQ / NSA Mathematical Advisor
Senior mathematical advisory roles at signals intelligence agencies — working on the deepest cryptographic and mathematical problems with national security implications. Number theory training is the highest-value credential for these positions.
Research Institute Fellow
Post-doctoral and permanent positions at elite mathematical research institutes — the Institute for Advanced Study (Princeton), IHES (Paris), the Fields Institute (Toronto), MSRI (Berkeley), and the Isaac Newton Institute (Cambridge).
Fields Medal Researcher
Number theory and arithmetic geometry have produced a disproportionate share of Fields Medallists. Tate, Deligne, Faltings, Wiles, Bhargava, Scholze — a PhD in number theory from a top institution places candidates in this lineage.
Mathematical Finance (Elliptic Curves)
Quantitative roles in financial mathematics and cryptographic infrastructure — blockchain mathematical design, post-quantum financial security systems, and risk modelling using number-theoretic random processes.
Top Universities for PhD Number Theory
Why D'Math University for Doctoral Number Theory?
Supervisors at the Frontier
Our 30+ number theory supervisors are active contributors to the most important problems in the field — publishing in Annals of Mathematics, Inventiones Mathematicae, and JAMS. Students engage directly with live research from day one.
The Full Breadth of the Field
Most programmes have depth in one or two sub-areas. Ours spans analytic, algebraic, and arithmetic geometry tracks simultaneously — enabling doctoral students to build the breadth that characterises the most successful number theorists.
International Collaborations
Joint seminars with partner institutions in Paris, Princeton, Cambridge, and Bonn. Doctoral students are expected to attend summer schools and present at international workshops from their second year.
Computational Infrastructure
Access to LMFDB, Magma, PARI/GP, and SageMath running on high-performance computing clusters. Computational experimentation is increasingly central to modern number theory research and our infrastructure supports it fully.
PhD Number Theory — the pursuit of the deepest truths about integers, primes, and structure. Applications reviewed year-round.