Analysis PhD Level: Postgraduate research Subject: Mathematics and Statistics Year of entry: 2026 Study options Part-time PhD | 8 years | Start date: multiple dates Full-time PhD | 4 years | Start date: September 2026 Key facts School School of Mathematics College College of Science and Engineering Programme website Analysis PhD by Distance Not available School location King's Buildings campus Apply now 19 to 21 November 2025: Postgraduate Virtual Open Days Join us live to explore postgraduate taught and research study at Edinburgh and gain key insights before you apply. Register now Overview About the degree programme Your passion for mathematical analysis will be rewarded by contact with, and supervision by, world-leading academic staff, and a rich seminar and working group programme, resulting in a qualification that boasts an internationally respected pedigree.Our analysis research group is one of the UK’s top centres for research in the field, especially in linear and nonlinear PDEs and harmonic analysis. Find out what it's like studying for a PhD in the School of Mathematics. Hear from some of our current PhD students as they share their experience of researching mathematics. Applying How to apply You apply online for this programme. After you read the application guidance, select your preferred programme, then choose 'Start your application' to begin. When to apply All applications received by 19 January 2026 will receive full consideration for funding. Later applications will be considered until all positions are filled. Application fee There is no fee to apply to this programme. What you need to apply As part of your online application, you will need to provide: a personal statementrelevant knowledge/training skillsand, if applicable:your research topicdetails of any published workYou will also need to provide some or all of the following supporting information and documents:a research proposalname of supervisor (if known)research project (if applying to an advertised project)copies of your degree certificates and academic transcripts that confirm your current or final marksreferencesevidence you meet the English language requirementscertified translations if your original documents are not in EnglishWhen you start your application, you will be able to see the full list of documents you need to provide. ReferencesYou must submit two references with your application. Apply Select the award, duration and delivery mode you want to study. Then select the start date you want to apply for. Select programme - Select a programme -PhD Analysis with Internship (8 years)PhD Analysis (8 years)PhD Analysis (4 years) After you apply Once you have applied for this programme, you will be able to track the progress of your application and accept or decline any offers.Checking the status of your applicationWe will notify you by email once we have made a decision. Due to the large number of applications we receive, it might take a while until you hear from us.Receiving our decisionWhat to do if you receive an offer:Receiving an offerAccepting an offerSubmitting supporting documents Academic Technology Approval Scheme If you need to apply for a student visa, you may need an Academic Technology Approval Scheme clearance certificate in order to study this programme. Academic Technology Approval Scheme Funding, fees and costs Funding Most PhD students in Mathematics receive full scholarships to cover their tuition fees and living costs.Others may use a combination of the following funding to pay their tuition fees and living costs:scholarshipsborrowing moneytaking out a loanfamily supportpersonal savingsincome from workemployer sponsorshipExplore sources of funding for postgraduate study Featured fundingSchool of Mathematics funding opportunities Search for other funding opportunities You can find scholarships, bursaries and other funding you might be eligible for on our Scholarships and Student Funding site.Postgraduate funding opportunities Tuition fees Tuition fees by award and duration Tuition fees for full-time and part-time options are listed for one academic year. Part-time PhD Analysis with Internship (8 years) tuition fees PhD Analysis (8 years) tuition fees Full-time PhD Analysis (4 years) tuition fees Graduate discount If you are a University of Edinburgh graduate, you will be eligible for a 10% discount on your tuition fees for this programme. You may also be eligible if you were a visiting undergraduate student.Find out how to receive your graduate discount Costs Accommodation and living costs You need to cover your accommodation and living costs for the duration of your programme.We estimate that you might spend £1,167 to £2,330 per month if you are a single student.Living costs include:foodutility billstravel, clothes, books and stationeryrecreational costs (for example, TV subscriptions and social events)Living costsAccommodation costs depend on where you live while studying and the type of accommodation you choose.University postgraduate accommodation options and costs Entry requirements These entry requirements are for the 2026-27 academic year and requirements for future academic years may differ. Entry requirements for the 2027-28 academic year will be published on 1 Oct 2026. Qualifications 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 To find international equivalent qualifications, select where you studied from the country or region list. Where you studied - Select a country or region -AfghanistanAlbaniaAngolaArgentinaArmeniaAustraliaAustriaAzerbaijanBahamasBahrainBangladeshBelarusBelgiumBelizeBeninBoliviaBosnia and HerzegovinaBotswanaBrazilBulgariaBurkina FasoBurundiCambodiaCameroonCanadaChileChinaColombiaCosta RicaCôte d'IvoireCroatiaCubaCyprusCzech RepublicDemocratic Republic of CongoDenmarkDominican RepublicEcuadorEgyptEl SalvadorEritreaEstoniaEswatiniEthiopiaFinlandFranceGabonThe GambiaGeorgiaGermanyGhanaGreeceGuatemalaGuineaHaitiHondurasHong Kong-SAR ChinaHungaryIcelandIndiaIndonesiaIranIraqIrelandIsraelItalyJamaicaJapanJordanKazakhstanKenyaKorea, Republic ofKuwaitLatviaLebanonLesothoLiberiaLibyaLithuaniaLuxembourgMacau-SAR ChinaMadagascarMalawiMalaysiaMaldivesMaliMaltaMauritiusMexicoMoldovaMongoliaMontenegroMoroccoMozambiqueMyanmarNamibiaNepalNetherlands, TheNew ZealandNicaraguaNigeriaNorth MacedoniaNorwayOmanPacific IslandsPakistanPalestinian TerritoriesPanamaParaguayPeruPhilippinesPolandPortugalQatarRomaniaRussiaRwandaSaudi ArabiaSenegalSerbiaSierra LeoneSingaporeSlovakiaSloveniaSomaliaSouth AfricaSouth SudanSpainSri LankaSudanSwedenSwitzerlandSyriaTajikistanTaiwanTanzaniaThailandTogoTrinidad and TobagoTurkeyTurkmenistanUgandaUkraineUnited Arab EmiratesUnited States of AmericaUruguayUzbekistanVenezuelaVietnamWest Indies and CaribbeanZambiaZimbabweMy country or region is not listed English language requirements You must prove that your English language abilities are at a high enough level to study this degree programme.This is the case for all applicants, including UK nationals.You can meet our English language requirements with one of the following:an English language testa degree that was taught and assessed in Englishcertain professional qualifications English language tests we accept We accept any of the following English language tests, at the specified grade or higher: 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. Oxford ELLT: total 7 with at least 6 in each component. Oxford Test of English Advanced: total 145 with at least 135 in each component. How old your English language tests can be Tests no more than two years oldThe following English language tests must be no more than two years old on the 1st of the month in which your programme starts, regardless of your nationality:IELTS AcademicTOEFL-iBT (including Home Edition)Trinity ISEOxford ELLTOxford Test of English AdvancedTests no more than three and a half years oldAll other English language tests must be no more than three and a half years old on the 1st of the month in which your programme starts, regardless of your nationality. Degrees taught and assessed in English We 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 countriesWe 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-MESCHow old your degree can beIf you are not a national of a majority English-speaking country, then your degree must be no more than five years old on the 1st of the month in which your programme starts.This time limit does not apply to your degree if you are a national of a majority English-speaking country. Find out more about our English language requirements Find out about other English language qualifications we accept, including professional qualifications.English language requirements Programme details Research profile The School of Mathematics is a vibrant community of more than 100 academic and related staff supervising over 200 PhD students. You would be joining a welcoming and diverse group of people offering a rich learning experience, thanks to their depth of understanding and passion for research.The Graduate School has a dynamic and empathetic team of administrative staff, who are committed to supporting both students and staff through the challenges of a PhD.Learn more about the Maxwell Institute Graduate SchoolIn addition, the School of Mathematics has several postgraduate research student and staff networks, who meet regularly off and on campus to support and empower underrepresented groups. Research areasWe have a unique focus on the interplay of classical Euclidean harmonic analysis with the modern theory of PDEs. We study harmonic analytic ideas in number theory, geometric measure theory, combinatorics, and discrete geometry and geometrically invariant inequalities; and we investigate applications of harmonic analysis to elliptic and parabolic PDEs with rough coefficients and/or on rough domains.We also study:nonlinear hyperbolic, dispersive and kinetic equations and systems arising in the classical field theories of mathematical physics, mathematical biology and, in connection with black holes, mathematical general relativityfree-boundary problems, optimal mass transportation and Monge-Ampère equations in nonlinear elasticity and other continuum theorieswell-posedness for supercritical initial value problems with noisy initial data Our research community Mathematics is a discipline of high intellect with connections stretching across all the scientific disciplines and beyond, and in Edinburgh you can be certain of thriving in a rich academic setting. Our School is one of the country’s largest mathematics research communities in its own right, but you will also benefit from Edinburgh’s high-level collaborations, both regional and international.Research students will have a primary and secondary supervisor and the opportunity to network with a large and varied peer group. You will be carrying out your research in the company of eminent figures and be exposed to a steady stream of distinguished researchers from all over the world.Our status as one of the most prestigious schools in the UK for mathematics attracts highly respected staff. Many of our 100+ current academics are leaders in their fields and have been recognised with international awards.Researchers are encouraged to travel and participate in conferences and seminars. You’ll also be in the right place in Edinburgh to meet distinguished researchers from all over the world who are attracted to conferences held at the School and the various collaborative centres based here. You’ll find opportunities for networking that could have far-reaching effects on your career in mathematics.Piscopia InitiativeWe particularly encourage women and those who identify as women plus other underrepresented groups in mathematics to apply, and to work with groups such as the Piscopia Initiative to improve the representation and inclusion of women and minorities in mathematics.Piscopia Initiative Find out more about pursuing a PhD in the School of Mathematics in this Q&A with the Director of the Graduate School. Programme structure To graduate from this PhD programme, you will need to complete taught courses worth 120 credits in total. 60 credits should be achieved during Year 1 of the programme. Supervisors and research theme staff will guide you through this process. Facilities You will enjoy excellent facilities, ranging from one of the world’s major supercomputing hubs to libraries for research at the leading level.As a research student, you will have access to dedicated desk space with monitors and a laptop computer, in prime locations such as the Bayes Centre. This state-of-the-art building is home to over 400 internationally recognised scientists and leading industry experts.About the Bayes CentreWe provide all our mathematics postgraduates with access to software packages such as:MapleMatlabMathematica Research students are allocated parallel computing time on ‘Eddie’, the Edinburgh Compute and Data Facility. You can also request use of the BlueGene/Q supercomputer facility for your research. Careers Careers Service Our Careers Service can help you to fully develop your potential and achieve your future goals. The Careers Service supports you not only while you are studying at the University, but also for up to two years after you finish your studies. With the Careers Service, you can: access digital resources to help you understand your skills and strengthstry different types of experiences and reflect on how and what you developget help finding work, including part-time jobs, vacation work, internships and graduate jobsattend careers events and practice interviewsget information and advice to help you make informed decisions Visit the Careers Service website Life at Edinburgh Accommodation We guarantee an offer of University accommodation for all new, single postgraduate taught students from outside the UK and new, single postgraduate research (typically PhD) students who:apply for accommodation by 31 July in the year when you start your programmeaccept an unconditional firm offer to study at the University by 31 Julystudy at the University for the whole of the academic year starting in SeptemberUniversity accommodation websiteAccommodation guarantee criteriaWe also offer accommodation options for couples and families.Accommodation for couples and familiesIf you prefer to live elsewhere, we can offer you advice on finding accommodation in Edinburgh.Accommodation information from the Edinburgh University Students' Association Advice Place Societies and clubs Our societies and sports clubs will help you develop your interests, meet like-minded people, find a new hobby or simply socialise.SocietiesSport Clubs The city of Edinburgh Scotland's inspiring capital will form the background to your studies — a city with an irresistible blend of history, natural beauty and modern city life. Find out more about living in Edinburgh Exercise, leisure and support facilities Outside of your studies, we offer a range of facilities that you might find useful day-to-day, including:sport and exercise facilitiesUniversity cafes and cateringEdinburgh University Students' Association (EUSA) venues and shopsa multi-faith chaplaincy for all faiths and nonea University nursery (based at King’s Buildings campus) Health and wellbeing support You will have access to free health and wellbeing services throughout your time at university if you need them.The support services we offer include: a student counselling service a health centre (doctor's surgery) support if you're living in University accommodation dedicated help and support if you have a disability or need adjustmentsHealth and wellbeing support services Disability and Learning Support Contact General enquiries Graduate School Administrator pgresearch@maths.ed.ac.uk
