Research groups & mentors

A methodology layer, recommended by experienced advisors: don't pick a university by faceless rankings — read the professors' profiles and publications, find where the top specialists in your area of interest are, and effectively choose a scientific advisor in advance. Different departments lead in different sub-fields.

How to use this

1. Decide which sub-field(s) excite you most (analysis, number theory, geometry/topology, algebra, logic, combinatorics, probability).
2. Find departments with a genuine school in that area — a cluster of faculty, seminars, and PhD students — not just one lone professor.
3. Go to the official people pages (linked below), read 3–5 professors' recent papers, and identify a realistic future advisor before you apply. Strength-of-fit beats prestige.

Affiliations checked against official department pages (as of May 2026). Professors move; common confusions are flagged. Items marked [unverified] should be confirmed on the linked staff page.

Worked example — this method on maths.ox.ac.uk

Running the method below live on Oxford's directory (2026-05-25):

1. Open maths.ox.ac.uk → People List and filter Status = Academic Faculty.
2. Open each senior figure's profile — it lists their research group and recent publications ("read their science").
3. Cross-check chairs and honours against the live profiles, because secondary sources go stale (this pass caught two outdated chair attributions).

The six current Oxford star professors found this way are in the Oxford section below. The same three steps work on any department's people page.

Cambridge — DPMMS

Deep strength in number theory & arithmetic geometry, algebraic geometry, differential geometry & topology, combinatorics, analysis/PDE & mathematical relativity, and a strong probability & statistics wing. (Applied maths is separate, in DAMTP.)

Standout: Sir Timothy Gowers (combinatorics/analysis; Fields 1998); Jack Thorne (number theory); Wendelin Werner (probability; Fields 2006); Mihalis Dafermos (analysis of general relativity); Richard Samworth (statistics, FRS). Specialism: UK's deepest bench in number theory and geometry, plus elite probability. Pure-leaning. People: https://www.dpmms.cam.ac.uk/people/academic

Oxford — Mathematical Institute

One of the largest UK departments. Flagship pure groups: Number Theory (global powerhouse), Geometry, Topology, Algebra, Mathematical Logic / model theory (historic strength), Combinatorics, Functional Analysis.

Standout — verified on live maths.ox.ac.uk profiles, 2026-05-25:

• Sir Andrew Wiles FRS — Number Theory; proof of Fermat's Last Theorem, Abel Prize. profile
• James Maynard FRS — Number Theory; Fields Medal 2022 (gaps between primes). profile
• Ben Green FRS — Waynflete Professor of Pure Mathematics; additive combinatorics / number theory (Green–Tao). profile
• Dame Frances Kirwan FRS — Savilian Professor of Geometry (2017–; first woman in the chair); algebraic & symplectic geometry. profile
• Jon Keating FRS — Sedleian Professor of Natural Philosophy (2019–); quantum chaos, random matrix theory, Riemann zeta. profile
• Sir Martin Bridson FRS — Whitehead Professor of Pure Mathematics, President of the Clay Mathematics Institute; geometric group theory, low-dimensional topology. profile

Specialism: number theory at the very top globally; strong geometry, topology / geometric group theory, logic, and mathematical physics. Balanced, strongly pure. Browse: People List (filter Status = Academic Faculty); each profile lists research group + recent papers. Correction: earlier secondary sources listed Hitchin (Savilian Geometry) and John Ball (Sedleian) — both retired from those chairs; current holders are Kirwan and Keating.

Warwick — Mathematics Institute

Internationally famous for Ergodic Theory & Dynamical Systems (edits the leading journal) and Probability / Stochastic Analysis. Also strong Algebraic Geometry & Number Theory, Geometric Group Theory, Geometric Analysis/PDE, Combinatorics.

Standout: Miles Reid (algebraic geometry, FRS); Karen Vogtmann (geometric group theory, FRS); Keith Ball (convex geometry, FRS); Mark Pollicott (ergodic theory); Samir Siksek (number theory). Specialism: dynamical systems + probability, alongside top-tier algebraic geometry. Balanced. People: https://warwick.ac.uk/fac/sci/maths/people/staff/ Note: Martin Hairer (Fields 2014) is now at EPFL with a joint Imperial post — not current Warwick.

Imperial — Department of Mathematics

Strong Geometry (gauge theory, symplectic, algebraic), Number Theory & Arithmetic Geometry, Algebra, Analysis; especially renowned for Stochastic Analysis / Probability and applied/computational maths.

Standout: Sir Simon Donaldson (geometry/gauge theory; Fields 1986); Richard Thomas (algebraic geometry, FRS); Toby Gee (number theory, FRS); Martin Hairer (stochastic PDE; Fields 2014; joint with EPFL). Specialism: geometry (Donaldson school) and arithmetic geometry/number theory, plus elite stochastic analysis. Slightly applied-leaning overall, with a top-tier pure core. People: https://www.imperial.ac.uk/natural-sciences/departments/mathematics/research/pure-mathematics/people/

Durham — Mathematical Sciences

Broad: Pure Mathematics (Algebra & Number Theory, Geometry, Topology), Probability, Statistics, and a strong Mathematical & Theoretical Particle Physics group.

Standout: Sunil Chhita (probability — random tilings, KPZ, dimers); geometry/topology faculty across Riemannian geometry and low-dimensional topology [specific honours not individually verified]. Specialism: geometry/topology + a distinctive pure maths ↔ mathematical physics overlap. Balanced. People: https://www.durham.ac.uk/departments/academic/mathematical-sciences/about-us/staff/ Note: Tom Bridgeland is at Sheffield, not Durham.

Edinburgh — School of Mathematics

Pure maths concentrated in the Hodge Institute (algebra, geometry, number theory, topology — representation theory, noncommutative & derived methods, arithmetic geometry). Large Analysis & Probability theme.

Standout (verified 2026-05-25): Iain Gordon (representation theory; LMS Berwick) · Arend Bayer (algebraic geometry — stability conditions; FRSE, Adams & Whitehead Prizes) · Agata Smoktunowicz (noncommutative ring theory; FRSE, EMS/Whitehead) · Tom Leinster (category theory) · Clark Barwick (homotopy theory & arithmetic; LMS Berwick). Specialism: algebra, geometry & representation theory (Hodge Institute); also hosts the ICMS (Director: arithmetic geometer Minhyong Kim). Strongly pure, with a large analysis & probability wing. People: https://www.maths.ed.ac.uk/school-of-mathematics/people/academic-staff

St Andrews — Mathematics & Statistics

Pure signatures: Algebra & Combinatorics (finite & infinite groups, semigroups, computational algebra) and Analysis (fractal geometry & dimension theory, ergodic theory). Home of CIRCA, the GAP software project, and the MacTutor history archive.

Standout: Kenneth Falconer (fractal geometry; textbook author; Regius Professor); Colva Roney-Dougal (group theory / computational algebra; OBE; Director of CIRCA); Nik Ruškuc (semigroups) [confirm title]. Specialism: group/semigroup theory & computational algebra (GAP/CIRCA) and fractal geometry. Balanced. People: https://www.st-andrews.ac.uk/mathematics-statistics/people/

Bristol — School of Mathematics

Globally distinctive for quantum chaos, random matrix theory, and links to analytic number theory. Strong pure (analytic & probabilistic number theory, ergodic theory, combinatorics), probability, and mathematical physics. Co-hosts the Heilbronn Institute.

Standout (verified 2026-05-25): Jens Marklof FRS (ergodic theory / number theory; Henry Overton Wills Professor) · Nina Snaith (random matrix theory & the Riemann zeta — CFKS heuristics) · Tim Dokchitser (arithmetic geometry, BSD; Heilbronn Chair) · Francesco Mezzadri (random matrix theory). Specialism: analytic number theory ↔ random matrix theory / quantum chaos. Balanced. People: https://www.bristol.ac.uk/maths/people/ Note: Jon Keating → Oxford (2019); Tim Browning → ISTA Austria (2018) — neither current Bristol.

Bath — Mathematical Sciences

Six groups; best known in pure/probability for its Probability Laboratory (Prob-L@B), plus solid Analysis and Algebra & Geometry.

Standout (verified 2026-05-25): Manuel del Pino (nonlinear PDE; Royal Society Research Professor) · Mathew Penrose (probability — random geometric graphs) · Monica Musso (nonlinear elliptic PDE) · Juan Dávila (nonlinear analysis/PDE) · Alexander Cox (probability, optimal transport). Specialism: a strong probability group (Prob-L@B) paired with a high-calibre nonlinear-PDE / analysis group. Mixed tilt; the pure strengths are analysis/PDE and probability, not algebra/number theory. People: https://www.bath.ac.uk/teams/department-of-mathematical-sciences-academic-staff/

TU Delft — Delft Institute of Applied Mathematics (DIAM)

Explicitly applied. Strengths: stochastic analysis / SPDEs and functional analysis, applied probability & queueing, discrete/combinatorial optimization, scientific computing / numerical PDEs.

Standout (verified 2026-05-25): Mark Veraar (stochastic analysis / SPDEs — DIAM's most pure-facing strength) · Kees Vuik (numerical analysis / scientific computing; SIAM Fellow 2026) · Frank Redig (statistical physics) [unverified]. Specialism: applied analysis, optimization, numerical maths — not a classical pure-maths department. The maths BSc is Dutch-taught; the English undergraduate route is CS & Engineering (see TU Delft). People: https://www.tudelft.nl/en/eemcs/the-faculty/departments/applied-mathematics/about-diam/research-groups

École Polytechnique (France) — CMLS & CMAP

Pure maths is at CMLS (Centre de Mathématiques Laurent Schwartz, École Polytechnique–CNRS): Algebra & Arithmetic, Analysis & PDE, Geometry & Dynamics. Applied maths is at CMAP.

Standout (verified at CMLS, 2026-05-25): Yvan Martel (nonlinear PDE, solitons; ICM 2018 speaker) · Charles Favre (CNRS; arithmetic & holomorphic dynamics) · Frank Pacard (geometric analysis; former director) · Nicolas Perrin (algebraic geometry; current director). Specialism: a compact, high-level pure centre (PDE/analysis, arithmetic dynamics) in the Laurent Schwartz tradition. The 3-year English-taught BSc (double majors incl. Maths & Physics / Maths & CS) is a route into this environment — one of the shortlisted continental options. People: https://cmls.ip-paris.fr/ Note: legacy lists still show Claude Viterbo (symplectic geometry) at CMLS — he has moved to the ENS (DMA), Paris.

ETH Zürich — Department of Mathematics (D-MATH)

The key exchange target from the TU Delft CS route, and a world-top reference department.

Strong across geometric analysis & PDE, number theory & arithmetic geometry, algebraic/symplectic geometry & topology, probability, and financial maths.

Standout (verified 2026-05-25): Alessio Figalli (Fields Medal 2018; optimal transport, PDE) · Rahul Pandharipande (algebraic geometry — GW/DT theory; Clay Research Award) · Sarah Zerbes (number theory — Euler systems, BSD; Whitehead Prize) · Emmanuel Kowalski (analytic number theory) · Tristan Rivière (geometric analysis). Specialism: a world-top, broad pure department — a current Fields Medallist plus elite geometry / number-theory groups. People: https://math.ethz.ch/the-department/people/faculty-and-lecturers/professors.html Note: Maryna Viazovska (Fields 2022) is at EPFL, not ETH Zürich.

Quick fit summary by sub-field

• Number theory: Oxford, Cambridge (top), then Imperial; Bristol for the number-theory/random-matrix interface.
• Geometry & topology: Imperial (Donaldson school), Cambridge; Edinburgh; Oxford; Warwick; Durham.
• Algebra / representation theory: Edinburgh (Hodge), St Andrews (groups/semigroups).
• Logic / model theory: Oxford.
• Combinatorics: Cambridge, Oxford, Warwick.
• Analysis / PDE: Cambridge, Edinburgh, Imperial, Bath.
• Dynamical systems / ergodic theory: Warwick (flagship), Bristol, St Andrews.
• Probability / statistics: Cambridge, Warwick, Bath (Prob-L@B), Imperial, Edinburgh; TU Delft for applied probability.
• Continental: ETH Zürich world-top across geometric analysis, number theory and algebraic geometry (Figalli, Pandharipande, Zerbes); École Polytechnique (CMLS) a compact pure centre in PDE & arithmetic dynamics.

Verification flags

• Martin Hairer → EPFL (joint Imperial), not Warwick.
• Jon Keating → Oxford (2019), not Bristol.
• Tim Browning → ISTA Austria (2018), not Bristol.
• Tom Bridgeland → Sheffield, not Durham.
• Claude Viterbo → ENS (DMA), Paris — no longer at École Polytechnique (CMLS).
• Maryna Viazovska (Fields 2022) → EPFL, not ETH Zürich.
• Jens Marklof now holds Bristol's Henry Overton Wills Professorship (the chair Keating vacated).
• Individual honours marked [unverified] (e.g. TU Delft's Frank Redig; some Durham / St Andrews faculty) should be checked on the linked staff pages.