Algebra, Geometry and Topology PhD

The School of Mathematics is a vibrant community of more than 100 academic and related staff supervising over 100 PhD students.

Working within one of the largest mathematics groups in the UK, you’ll be completing your degree in an environment that hums with a busy graduate school life, and you’ll have the chance to make your mark in seminars, workshops, clubs and outings.

The Hodge Institute is the home of algebra, geometry, number theory and topology research groups in the School of Mathematics.

• The Hodge Institute

Our interests include:

• non-commutative ring theory
• non-commutative algebraic geometry
• representation theory
• quantum algebra
• category theory
• algebraic geometry
• higher and derived categories
• symplectic geometry
• algebraic and geometric topology
• arithmetic geometry

While we offer a large community of researchers under one roof, we believe in encouraging you to gain as broad a perspective as possible. The best way to do this is to involve yourself in the international dialogue on your research area, through attending conferences and symposia, and visiting your peers in centres of research worldwide.

Throughout your studies, you’ll be given opportunities to travel to events and institutions that will allow you to gain this perspective and open up new areas of investigation.

You can find out more on the Maxwell Institute Graduate School website:

• Maxwell Institute Graduate School

These entry requirements are for the 2025/26 academic year and requirements for future academic years may differ. Entry requirements for the 2026/27 academic year will be published on 1 Oct 2025.

A UK first class honours degree, or its international equivalent, in an appropriate subject; or a UK 2:1 honours degree plus a UK masters degree, or their international equivalents; or relevant qualifications and experience.

International qualifications

Check whether your international qualifications meet our general entry requirements:

• Entry requirements by country

English language requirements

Regardless of your nationality or country of residence, you must demonstrate a level of English language competency which will enable you to succeed in your studies.

English language tests

We accept the following English language qualifications at the grades specified:

• IELTS Academic: total 6.5 with at least 6.0 in each component. We do not accept IELTS One Skill Retake to meet our English language requirements.
• TOEFL-iBT (including Home Edition): total 92 with at least 20 in each component. We do not accept TOEFL MyBest Score to meet our English language requirements.
• C1 Advanced (CAE) / C2 Proficiency (CPE): total 176 with at least 169 in each component.
• Trinity ISE: ISE II with distinctions in all four components.
• PTE Academic: total 65 with at least 59 in each component. We do not accept PTE Academic Online.
• Oxford ELLT: 7 overall with at least 6 in each component.

Unless you are a national of a majority English speaking country, your English language qualification must be no more than three and a half years old from the start of the month in which the programme you are applying to study begins. If you are using an IELTS, PTE Academic, TOEFL, Trinity ISE, or Oxford ELLT test, it must be no more than two years old on the first of the month in which the programme begins, regardless of your nationality. (Revised 14 January 2025 to include Oxford ELLT.)

Degrees taught and assessed in English

We also accept an undergraduate or postgraduate degree that has been taught and assessed in English in a majority English speaking country, as defined by UK Visas and Immigration:

• UKVI list of majority English speaking countries

We also accept a degree that has been taught and assessed in English from a university on our list of approved universities in non-majority English speaking countries (non-MESC).

• Approved universities in non-MESC

If you are not a national of a majority English speaking country, then your degree must be no more than five years old at the beginning of your programme of study.

Find out more about our language requirements:

• English language requirements