how to do phd in maths

To apply for admissions and financial aid, or for additional information on admissions requirements for the Ph.D. program in pure mathematics, please go to the appropriate Harvard Kenneth C. Griffin Graduate School of Arts and Sciences website listed below. All other inquiries may be directed to the Graduate Program Administrator of the Mathematics Department.

• Harvard Kenneth C. Griffin Graduate School of Arts and Sciences (Harvard Griffin GSAS)
• Mathematics Graduate Studies
• Financial Support

Graduate Program Administrator

The Department of Mathematics does not discriminate against applicants or students on the basis of race, color, national origin, ancestry or any other protected classification.

Preparing the Application The statement of purpose for graduate applications is carefully weighted by the admissions committee. The applicant’s statement should convince the committee that they are able to communicate effectively and with a deep understanding of mathematics. It is not intended to be a biographical sketch or a reflection on one’s decision to enter the field.

Three letters of recommendation are required. Letter writers should be faculty or others qualified to evaluate the applicant’s potential for graduate study in mathematics. The letters must be submitted online and by the application deadline.

Applicants should include any research papers, publications, and other original works they would like to have evaluated by the admissions committee.

The department requests that applicants submit GRE Mathematics Subject Test scores if practical. Applicants should check the Educational Testing Service (ETS) website for test dates in their area to ensure the scores will be submitted before the application deadline. An official score report should be sent to the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences using code 3451.

While the admissions committee reviews all applications submitted before the deadline, missing math subject test scores provide one less data point available to evaluate the application. Depending on the strength of the application, the missing subject test scores may put the application at a disadvantage.

Applicants who are non-native English speakers and who do not hold an undergraduate degree from an institution at which English is the primary language of instruction must submit scores from the internet based test (iBT) of the Test of English as a Foreign Language (TOEFL) or the International English Language Testing System (IELTS) Academic test.

Harvard Griffin GSAS requires applicants to upload an electronic copy of undergraduate transcripts. Hard copies of official transcripts are not required at the time of application.

Ph.D. Program in Pure Mathematics The department does not grant a terminal Master’s degree, but the Master’s can be obtained “on the way” to the Ph.D. by fulfilling certain course and language exam requirements.

In general, there is no transfer status application to the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences or to the Department of Mathematics. No formal credit is given for an MSc or MA earned elsewhere. All applicants are considered to be applying as first-year graduate students. The only difference Master’s study may make is to better prepare students for the Qualifying Exam.

All graduate students are admitted to begin their studies in the fall term. The department plans on an entering class of about twelve students. Since the admissions committee receives a few hundred applications, the competition is keen.

Funding Graduate Study Applicants are urged to apply for all funding available to them. If no outside funding is available to the applicant, financial aid in the form of scholarships, research assistantships, and teaching fellowships is available. In general, students who do not have outside support will get scholarship support in their first year, but students are required to act as a teaching fellow for one course in their second through fifth years.

The department strongly recommends applicants to seek out and apply for all sources of financing available to them for graduate study. Recommended sources for funding US graduate students are NSF Graduate Fellowships and NDSEG Fellowships . Applicants from the UK are urged to also apply for the Kennedy fellowships and applicants from UK, New Zealand, Canada and Australia for Knox fellowships . International students may apply for the Fulbright IIE or any home country fellowships available for study abroad.

Harvard John A. Paulson School of Engineering and Applied Sciences The Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) offers programs for both the Master’s degree and the Ph.D. degree in Applied Mathematics. Please visit the SEAS website for more information on degrees in applied mathematics.

Overview of the PhD Program

For specific information on the Applied Mathematics PhD program, see the navigation links to the right. 

What follows on this page is an overview of all Ph.D. programs at the School; additional information and guidance can be found on the  Graduate Policies  pages. 

General Ph.D. Requirements

• 10 semester-long graduate courses, including at least 8 disciplinary.   At least 5 of the 10 should be graduate-level SEAS "technical" courses (or FAS graduate-level technical courses taught by SEAS faculty), not including seminar/reading/project courses.  Undergraduate-level courses cannot be used.  For details on course requirements, see the school's overall PhD course requirements  and the individual program pages linked therein.
• Program Plan (i.e., the set of courses to be used towards the degree) approval by the  Committee on Higher Degrees  (CHD).
• Minimum full-time academic residency of two years .
• Serve as a Teaching Fellow (TF) in one semester of the second year.
• Oral Qualifying Examination Preparation in the major field is evaluated in an oral examination by a qualifying committee. The examination has the dual purpose of verifying the adequacy of the student's preparation for undertaking research in a chosen field and of assessing the student's ability to synthesize knowledge already acquired. For details on arranging your Qualifying Exam, see the exam policies and the individual program pages linked therein.
• Committee Meetings : PhD students' research committees meet according to the guidelines in each area's "Committee Meetings" listing.  For details see the "G3+ Committee Meetings" section of the Policies of the CHD  and the individual program pages linked therein.
• Final Oral Examination (Defense) This public examination devoted to the field of the dissertation is conducted by the student's research committee. It includes, but is not restricted to, a defense of the dissertation itself.  For details of arranging your final oral exam see the  Ph.D. Timeline  page.
• Dissertation Upon successful completion of the qualifying examination, a committee chaired by the research supervisor is constituted to oversee the dissertation research. The dissertation must, in the judgment of the research committee, meet the standards of significant and original research.

Optional additions to the Ph.D. program

Harvard PhD students may choose to pursue these additional aspects:

• a Secondary Field (which is similar to a "minor" subject area).  SEAS offers PhD Secondary Field programs in  Data Science and in  Computational Science and Engineering .   GSAS  lists  secondary fields offered by other programs.
• a Master of Science (S.M.) degree conferred  en route to the Ph.D in one of several of SEAS's subject areas.  For details see here .
• a Teaching Certificate awarded by the Derek Bok Center for Teaching and Learning .

SEAS PhD students may apply to participate in the  Health Sciences and Technology graduate program  with Harvard Medical School and MIT.  Please check with the HST program for details on eligibility (e.g., only students in their G1 year may apply) and the application process.

In Applied Mathematics

• First-Year Exploration
• Areas of Application
• AM & Economics
• How to Declare
• Who are my Advisors?
• Secondary Field
• Senior Thesis
• Research for Course Credit (AM 91R & AM 99R)
• AB/SM Information
• Peer Concentration Advisors (PCA) Program
• Student Organizations
• How to Apply
• PhD Timeline
• PhD Model Program (Course Guidelines)
• Oral Qualifying Examination
• Committee Meetings
• Committee on Higher Degrees
• Research Interest Comparison
• Collaborations
• Cross-Harvard Engagement
• Clubs & Organizations
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• Undergraduate Student Stories

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COLLEGE OF LIBERAL ARTS AND SCIENCES

Department of Mathematics

• Ph.D. in Mathematics
• General Information
• Financial Aid
• Answers to FAQ
• M.S. in Mathematics
• Ph.D. in Mathematics with Thesis in Actuarial Science
• M.S. in Actuarial Science
• M.S. in Applied Financial Mathematics
• M.S. in Applied Financial Mathematics – Actuarial Science
• Course Schedule
• Preliminary Exams

Overview of Graduation Requirements

To graduate with a PhD in Mathematics, a student must satisfy all of the following requirements:

• 45 credits, including 15 doctoral dissertation research credits/GRAD 6950
• If you have a Master’s degree in mathematics at UConn, then 30 credits are required, including 15 doctoral dissertation research credits.
• Pass at least one prelim exam after each semester for the first three semesters of their graduate study.
• Finish all prelim exam requirements by the beginning of the spring semester in their second year.
• Finish all core course requirements by the end of their second year of graduate study,
• Choose a thesis advisor no later than the end of the fall semester of the third year.
• Form your advisory committee :  Major Advisor and at least two Associate Advisors or Co-Major Advisors and at least one Associate Advisor.
• Meant to further the student’s education, scholarship and professional development.
• Material should be in the broad area in which the student intends to write a dissertation, but should not focus on the actual thesis research.  Students present and answer questions about the material they have studied.
• The exam is normally taken at the end of the third year or beginning of the fourth year.
• Submit Dissertation Proposal
• Submit Plan of Study
• Apply for Graduation in Student Admin by the fourth week of the final semester.
• Dissertation Specifications under Dissertation Information
• Thesis templates (for LaTex) are available on our Thesis Formatting page
• Submit Approval Page – Step 7
• Complete Survey of Earned Doctorates – Step 7
• Add Diploma Address in Student Admin and Diploma Info.

Preliminary Examination and Core Course Requirements

Pure Mathematics

• Math 5111/Measure and Integration
• Math 5120/Complex Function Theory
• Math 5210/Abstract Algebra
• Math 5310/Introduction to Geometry and Topology I
• Math 5160/Probability Theory and Stochastic Processes I
• Math 5210/Abstract Algebra I
• Math 5211/Abstract Algebra II
• Math 5260/Mathematical Logic I
• Math 5360/Differential Geometry

Applied Mathematics

• Math 5410/Introduction to Applied Mathematics I
• Math 5510/Numerical Analysis and Approximation Theory I
• Math 5440/Partial Differential Equations
• Math 5520/Finite Element Solution Methods I

Past Prelim Exams

PhD Qualifying Exams

The requirements for the PhD program in Mathematics have changed for students who enter the program starting in Autumn 2023 and later. 

Requirements for the Qualifying Exams

Students who entered the program prior to autumn 2023.

To qualify for the Ph.D. in Mathematics, students must pass two examinations: one in algebra and one in real analysis. 

Students who entered the program in Autumn 2023 or later

To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas: 

• real analysis
• geometry and topology
• applied mathematics

The exams each consist of two parts. Students are given three hours for each part.

Topics Covered on the Exams:

• Algebra Syllabus
• Real Analysis Syllabus
• Geometry and Topology Syllabus
• Applied Mathematics Syllabus

Check out some Past and Practice Qualifying Exams to assist your studying.

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take either or both of the exams in the autumn. If they pass either or both of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing either of the exams.

Students must pass both qualifying exams by the autumn of their second year. Ordinarily first-year students take courses in algebra and real analysis throughout the year to prepare them for the exams. The exams are then taken at the beginning of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Students who started in Autumn 2023 and later

Students must choose and pass two out of the four qualifying exams by the autumn of their second year. Students take courses in algebra, real analysis, geometry and topology, and applied math in the autumn and winter quarters of their first year to prepare them for the exams. The exams are taken during the first week of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Exam Schedule

Unless otherwise noted, the exams will be held each year according to the following schedule:

Autumn Quarter:  The exams are held during the week prior to the first week of the quarter. Spring Quarter:  The exams are held during the first week of the quarter.

The exams are held over two three-hour blocks. The morning block is 9:30am-12:30pm and the afternoon block is 2:00-5:00pm.

For the start date of the current or future years’ quarters please see the  Academic Calendar

Upcoming Exam Dates

Autumn 2024.

Tuesday, September 17: Applied Math , Room 384I and Algebra , Room 384H

Wednesday, September 18: Real Analysis , Room 384H

Thursday, September 19: Geometry and Topology , Room 384H

Graduate Program

Our graduate program is unique from the other top mathematics institutions in the U.S. in that it emphasizes, from the start, independent research. Each year, we have extremely motivated and talented students among our new Ph.D. candidates who, we are proud to say, will become the next generation of leading researchers in their fields. While we urge independent work and research, there exists a real sense of camaraderie among our graduate students. As a result, the atmosphere created is one of excitement and stimulation as well as of mentoring and support. Furthermore, there exists a strong scholarly relationship between the Math Department and the Institute for Advanced Study, located just a short distance from campus, where students can make contact with members there as well as attend the IAS seminar series.  Our program has minimal requirements and maximal research and educational opportunities. We offer a broad variety of advanced research topics courses as well as more introductory level courses in algebra, analysis, and geometry, which help first-year students strengthen their mathematical background and get involved with faculty through basic course work. In addition to the courses, there are several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the Department Colloquium give introductory talks, which allows graduate students to understand the afternoon colloquium more easily; (2) Graduate Student Seminar (GSS), which is organized and presented by graduate students for graduate students, creating a vibrant mathematical interaction among them; and, (3) What’s Happening in Fine Hall (WHIFH) seminar where faculty give talks in their own research areas specifically geared towards graduate students. Working or reading seminars in various research fields are also organized by graduate students each semester. First-year students are set on the fast track of research by choosing two advanced topics of research, beyond having a strong knowledge of three more general subjects: algebra, and real and complex analysis, as part of the required General Examination. It is the hope that one, or both, of the advanced topics will lead to the further discovery of a thesis problem. Students are expected to write a thesis in four years but will be provided an additional year to complete their work if deemed necessary. Most of our Ph.D.'s are successfully launched into academic positions at premier mathematical institutions as well as in industry .

Chenyang Xu

Jill leclair.

Department of Mathematics

Requirements for the ph.d. degree.

In order to qualify for the Mathematics Ph.D., all students are required to:

• Complete eight term courses at the graduate level, at least two with Honors grades.
• Pass qualifying examinations on their general mathematical knowledge;
• Submit a dissertation prospectus;
• Participate in the instruction of undergraduates ;
• Be in residence for at least three years;
• Complete a dissertation that clearly advances understanding of the subject it considers.

All students must also complete any other  Graduate School of Arts and Sciences degree requirements  as they appear in the Programs and Policies bulletin.

The normal time for completion of the Ph.D. program is five to six years. Requirement (1) normally includes basic courses in algebra, analysis, and topology.  Students typically complete the eight-course requirement by the end of their third year.  The Honors grades of (1) must be achieved within the first two years.  A sequence of three qualifying examinations (algebra and number theory, real and complex analysis, topology) is offered each term.  All qualifying examinations must be passed by the end of the second year.  There is no limit to the number of times that students can take the exams, and so they are encouraged to take them as soon as possible.

The dissertation prospectus should be submitted during the third year. 

The thesis is expected to be independent work, done under the guidance of an adviser. This adviser should be contacted not long after the student passes the qualifying examinations. A student is admitted to candidacy after completing requirements (1)–(5) and obtaining an adviser.

In addition to all other requirements, students must successfully complete MATH 991a, Ethical Conduct of Research, prior to the end of their first year of study. This requirement must be met prior to registering for a second year of study.

Master’s Degrees :

The M.Phil. and M.S. degrees are conferred only en route to the Ph.D.; there is no separate master’s program in Mathematics.

M.Phil.   Please refer to the Graduate School Degree Requirements

M.S.   A student must complete six term courses with at least one Honors grade, perform adequately on the general qualifying examination, and be in residence at least one year.

Natural Sciences and Mathematics

Mathematical sciences, doctor of philosophy in mathematics.

The program offers extensive coursework and intensive research experience in theory, methodology, and applications of mathematics (see  degree requirements ). 

• Faculty members with broad and diverse research interests are available to supervise doctoral dissertations .
• Financial support in the form of assistantships, full tuition support, and scholarships and awards are provided. Additional scholarships are available for US citizens and permanent residents.
• Our students, both domestic and international, have a strong record of starting in full-time jobs right after graduation .
• Students have opportunities to participate in active seminar series in  Algebra and Combinatorics ;   Computational Science ;  Geometry, Topology and Dynamical Systems ; and  Nonlinear Analysis and Dynamical Systems ; and the departmental  Colloquium  series.
• To enhance career prospects, students can pursue  Graduate Certificate in Data Science , and possibly use the certificate courses to fulfill the elective requirements.
• NSM Career Success Center  is available to support professional development and experiential learning of students.  
• GRE test score is not required for admission.

More than 85% of our 45 Mathematics PhD graduates since 2020, both domestic and international, secured full-time employment within a few months of receiving their degrees. 

Placement of 2022 & 2023 Mathematics PhD Graduates

See a more complete list  

Assistantships

Graduate Teaching Assistantships are offered to qualified PhD students on a competitive basis. These assistantships include a monthly stipend (currently set at $2,400) along with a full tuition waiver (covering 9 credit hours per term in the Fall and Spring semesters). The assistantship additionally covers the cost of health insurance purchased through the university and most fees. Graduate Research Assistantships for advanced PhD students are also available on some faculty members’ research grants. Typically, assistantship support is provided for five years and encompasses the Summer semester as well.

All admitted students are considered for assistantships; no separate application is necessary. 

Scholarships, Fellowships & Awards

PhD students are additionally supported through the following awards:

• NSM McDermott PhD Admission Fellowship  (for highly qualified new students, offered at the time of admission)
• Dean’s Fellowship  and  EEF Scholarship  (for highly qualified new students who are U.S. citizens and permanent residents, offered at the time of admission)
• Julia Williams Van Ness Merit Scholarship  and  Mei Lein Fellowship
• Outstanding Teaching Assistant of the Year Award
• Dean of Graduate Education Dissertation Research Award
• Best Dissertation Award ,  David Daniel Thesis Award , and  Outstanding Graduate Student Award

Conference Travel Support

NSM Conference Travel Award  and  Betty and Gifford Johnson Travel Award  are available to provide financial support to PhD students to present their research at professional conferences.

• How To Apply
• Frequently Asked Questions
• Scholarships & Awards
• Office of Admissions and Enrollment

Graduate Resources

• Mathematics Research
• Statistics & Actuarial Science Research
• Graduate Advisors
• Mathematics Courses
• Statistics Courses
• Actuarial Science Courses
• Qualifying Exam Archive
• Office of Graduate Education

Ready to start your application?

Before you apply, visit our  How to Apply  page to get familiar with the admission requirements and application process.

[email protected]

Graduate School

Mathematics

General information, program offerings:, director of graduate studies:, graduate program administrator:.

The Department of Mathematics graduate program has minimal requirements and maximal research and educational opportunities. It differentiates itself from other top mathematics institutions in the U.S. in that the curriculum emphasizes, from the start, independent research. Our students are extremely motivated and come from a wide variety of backgrounds. While we urge independent work and research, a real sense of camaraderie exists among our graduate students. As a result, the atmosphere created is one of excitement and stimulation and mentoring and support. There also exists a strong scholarly relationship between the department and the Institute for Advanced Study (IAS), located a short distance from campus. Students can contact IAS members as well as attend the IAS seminar series.

Students are expected to write a dissertation in four years but may be provided an additional year to complete their work if deemed necessary. Each year, our graduates are successfully launched into academic positions at premier mathematical institutions and industry.

Program Offerings

Program offering: ph.d..

The department offers a broad variety of research-related courses as well as introductory (or “bridge”) courses in several areas, which help first-year students strengthen their mathematical background. Students also acquire standard beginning graduate material primarily through independent study and consultations with the faculty and fellow students.

Language(s)

Students must satisfy the language requirement by demonstrating to a member of the mathematics faculty a reasonable ability to read ordinary mathematical texts in one of the following three languages: French, German, or Russian. Students must pass the language test by the end of the first year and before standing for the general exam.

Additional pre-generals requirements

Seminars The department offers numerous seminars on diverse topics in mathematics. Some seminars consist of systematic lectures in a specialized topic; others present reports by students or faculty on recent developments within broader areas. There are regular seminars on topics in algebra, algebraic geometry, analysis, combinatorial group theory, dynamical systems, fluid mechanics, logic, mathematical physics, number theory, topology, and other applied and computational mathematics. Without fees or formalities, students may also attend seminars in the School of Mathematics at the IAS.

The department also facilitates several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the department colloquium will give introductory talks, which allows graduate students to understand the afternoon colloquium more easily; (2) Graduate Student Seminar (GSS), which is organized and presented by graduate students and helps in creating a vibrant mathematical interaction among the graduate students; and, (3) What’s Happening in Fine Hall (WHIFH) seminar, where faculty members present talks in their own research areas specifically geared towards graduate students. Reading seminars are also organized and run by graduate students.

General exam

Beyond needing a strong knowledge of three more general subjects (algebra, and real and complex analysis), first-year students are set on the fast track of research by choosing two advanced research topics as part of their general exam. The two advanced topics are expected to come from distinct major areas of mathematics, and the student’s choice is subject to the approval of the department. Usually, by the second year, students will begin investigations of their own that lead to the doctoral dissertation.

General Exam in Mathematical Physics For a mathematics student interested in mathematical physics, the general exam is adjusted to include mathematical physics as one of the two special topics.

Qualifying for the M.A.

The Master of Arts (M.A.) degree is considered an incidental degree on the way to full Ph.D. candidacy. It is earned once a student successfully passes the language requirement and the general exam, and the faculty recommends it. It may also be awarded to students who, for various reasons, may leave the Ph.D. program, provided that the following requirements are met: passing the language requirement as well as the three general subjects (algebra, and real and complex analysis) of the general exam, and receiving department approval.

During the second, third, and fourth years, graduate students are expected to either grade or teach two sections of an undergraduate course, or the equivalent, each semester. Although students are not required to teach to fulfill department Ph.D. requirements, they are strongly encouraged to do so at least once before graduating. Teaching letters of recommendation are necessary for most postdoctoral applications.

Post-Generals requirements

Selection of a Research Adviser Upon completion of the general exam, the student is expected to choose a thesis adviser.

Dissertation and FPO

Two to three years is usually necessary for the completion of a suitable dissertation. Upon completion and acceptance of the dissertation by the department and Graduate School, the candidate is admitted to the final public oral examination. The dissertation is presented and defended by the candidate.

The Ph.D. is awarded after the candidate’s doctoral dissertation has been accepted and the final public oral examination sustained.

• Igor Rodnianski

Associate Chair

• János Kollár

Director of Graduate Studies

• Lue Pan (associate)
• Chenyang Xu

Director of Undergraduate Studies

• Jennifer M. Johnson (associate)
• Michael Aizenman
• Noga M. Alon
• Manjul Bhargava
• Sun-Yung A. Chang
• Maria Chudnovsky
• Fernando Codá Marques
• Peter Constantin
• Mihalis Dafermos
• Charles L. Fefferman
• David Gabai
• June E. Huh
• Alexandru D. Ionescu
• Nicholas M. Katz
• Sergiu Klainerman
• Peter Steven Ozsváth
• Peter C. Sarnak
• Paul Seymour
• Amit Singer
• Christopher M. Skinner
• Allan M. Sly
• Zoltán Szabó
• Paul C. Yang
• Shou-Wu Zhang

Assistant Professor

• Bjoern Bringmann
• Matija Bucic
• Marc Aurèle Tiberius Gilles
• Jonathan Hanselman
• Susanna Haziot
• Ana Menezes
• Ravi Shankar
• Jacob Shapiro
• Jakub Witaszek
• Ruobing Zhang

Associated Faculty

• John P. Burgess, Philosophy
• René A. Carmona, Oper Res and Financial Eng
• Bernard Chazelle, Computer Science
• Hans P. Halvorson, Philosophy
• William A. Massey, Oper Res and Financial Eng
• Frans Pretorius, Physics
• Robert E. Tarjan, Computer Science
• Ramon van Handel, Oper Res and Financial Eng
• Louis Esser
• Sepehr Hajebi
• Kimoi Kemboi
• Dmitry Krachun
• Anubhav Mukherjee
• Sung Gi Park
• Semon Rezchikov
• Joshua X. Wang
• Mingjia Zhang

University Lecturer

• Jennifer M. Johnson

Senior Lecturer

• Jonathan M. Fickenscher
• Mark W. McConnell
• Fraser M. Binns
• Tatyana Chmutova
• Tatiana K. Howard
• Justin Lacini
• Tristan J. Leger
• Jennifer Li
• Andrew O'Desky
• Stan Palasek
• John T. Sheridan
• Sahana Vasudevan
• David Villalobos
• Liyang Yang
• Bogdan Zavyalov

Visiting Professor

• Bhargav B. Bhatt
• Alex Kontorovich

Visiting Lecturer with Rank of Professor

• Camillo De Lellis
• Helmut H. Hofer
• Aaron Naber
• Akshay Venkatesh

For a full list of faculty members and fellows please visit the department or program website.

Permanent Courses

Courses listed below are graduate-level courses that have been approved by the program’s faculty as well as the Curriculum Subcommittee of the Faculty Committee on the Graduate School as permanent course offerings. Permanent courses may be offered by the department or program on an ongoing basis, depending on curricular needs, scheduling requirements, and student interest. Not listed below are undergraduate courses and one-time-only graduate courses, which may be found for a specific term through the Registrar’s website. Also not listed are graduate-level independent reading and research courses, which may be approved by the Graduate School for individual students.

COS 522 - Computational Complexity (also MAT 578)

Mat 500 - effective mathematical communication, mat 515 - topics in number theory and related analysis, mat 516 - topics in algebraic number theory, mat 517 - topics in arithmetic geometry, mat 518 - topics in automorphic forms, mat 519 - topics in number theory, mat 520 - functional analysis, mat 522 - introduction to pde (also apc 522), mat 525 - topics in harmonic analysis, mat 526 - topics in geometric analysis, mat 527 - topics in differential equations, mat 528 - topics in nonlinear analysis, mat 529 - topics in analysis, mat 531 - introduction to riemann surfaces, mat 547 - topics in algebraic geometry, mat 549 - topics in algebra, mat 550 - differential geometry, mat 555 - topics in differential geometry, mat 558 - topics in conformal and cauchy-rieman (cr) geometry, mat 559 - topics in geometry, mat 560 - algebraic topology, mat 566 - topics in differential topology, mat 567 - topics in low dimensional topology, mat 568 - topics in knot theory, mat 569 - topics in topology, mat 572 - topics in combinatorial optimization (also apc 572), mat 577 - topics in combinatorics, mat 579 - topics in discrete mathematics, mat 585 - mathematical analysis of massive data sets (also apc 520), mat 586 - computational methods in cryo-electron microscopy (also apc 511/mol 511/qcb 513), mat 587 - topics in ergodic theory, mat 589 - topics in probability, statistics and dynamics, mat 595 - topics in mathematical physics (also phy 508), mat 599 - extramural summer research project, phy 521 - introduction to mathematical physics (also mat 597).

Department of Mathematics

Phd requirements.

• Prospective Graduate Student FAQ
• Graduate Board Oral Exam
• Graduate Courses

Qualifying Exams

• Recent PhD Theses
• Graduate Awards

Course Requirements

Mathematics PhD candidates must show satisfactory work in Algebra (110.601-602), Real Variables (110.605), Complex Variables (110.607), and one additional non-seminar mathematics graduate course in their first year. The first-year algebra and analysis requirement can be satisfied by passing the corresponding written qualifying exam in September of the first year; these students must complete at least two courses each semester. In addition, PhD candidates must take Algebraic Topology (110.615) and Riemannian Geometry (110.645) by their second year. Students having sufficient background can substitute an advanced topology course for 110.615, or an advanced geometry course for 110.645 with the permission of the instructor.

Candidates must show satisfactory work in at least two mathematics graduate courses each semester of their second year, and if they have not passed their oral qualifying exam, in the first semester of their third year.

Teaching Seminar

Candidates must take, attend, participate in, and pass the course 110.771 (GTA Teaching Seminar). The seminar is an important part of the preparation for classroom teaching, and thus an essential part of mathematics graduate education. The seminar is generally required in a student’s first year at JHU. A student supported by an external fellowship may delay taking the seminar until the spring before they are required to TA (but may not postpone the seminar entirely).

Candidates must pass written qualifying exams by the beginning of their second year in Analysis (Real & Complex) and in Algebra. Exams are scheduled for September and May of each academic year, and the dates are announced well in advance. More information as well as old exams and syllabi can be found on the Qualifying Exams page .

Candidates must pass an oral qualifying examination in the student’s chosen area of research by April 10th of the third year. The topics of the exam are chosen in consultation with the faculty member who has agreed (provisionally) to be the student’s thesis advisor, who will also be involved in administering the exam.

PhD Dissertation

Candidates must produce a written dissertation based upon independent and original research. After completion of the thesis research, the student will defend the dissertation by means of the  Graduate Board Oral exam . The exam must be held at least three weeks before the Graduate Board deadline the candidate wishes to meet.

Our PhD program does not have a foreign language requirement.

The MA Degree

Although the Department of Mathematics does not admit students seeking a terminal MA degree, students in the PhD program may earn an MA degree.

MA candidates must complete:

• Four graduate courses given by the Hopkins Department of Mathematics
• Two additional courses at the graduate or 400 level, other than 110.401, 110.405 and 110.415, given by the Hopkins Department of Mathematics, or with the permission of the graduate program director, graduate mathematics courses given by other departments or universities.

All courses used to satisfy the requirements must be completed with a grade of B- or better. (Advanced graduate courses completed with a grade of P can also be used to satisfy the requirements.)

The department offers programs covering a broad range of topics leading to the Doctor of Philosophy and the Doctor of Science degrees (the student chooses which to receive; they are functionally equivalent). Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 125 Ph.D. students, about 2/3 are in Pure Mathematics, 1/3 in Applied Mathematics.

The two programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, mathematics of data, and the theory of computation. In addition, many mathematically-oriented courses are offered by other departments. Students in Applied Mathematics are especially encouraged to take courses in engineering and scientific subjects related to their research.

All students pursue research under the supervision of the faculty , and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

Degree Requirements

Degree requirements consist of:

• Oral qualifying exam
• Classroom teaching
• Original thesis and defense

Prospective students are invited to consult the graduate career timeline for more information, and to read about the application procedure .

Graduate Co-Chairs

Graduate Student Issues, math graduate admissions

Jonathan Kelner , Davesh Maulik , and Zhiwei Yun

Study Postgraduate

Phd in mathematics (2025 entry).

Course code

6 October 2025

3-4 years full-time

Qualification

Mathematics Institute

University of Warwick

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Virtual tour

Find out more about our Mathematics PhD at Warwick

The PhD in Mathematics offers an intellectually stimulating and dynamic research course. Study at the University of Warwick's Mathematics Institute, an international centre of research excellence, ranked 3rd for research power and 3rd for the number of 4* research outputs in REF 2021 (amongst UK universities).

Course overview

Mathematics at Warwick covers the full spectrum of mathematics and its applications. The Mathematics Postgraduate Degrees are appropriate for students with a strong and broad mathematical background who wish to engage in advanced mathematical techniques and attack mathematical research problems in their postgraduate work.

All students are required to undergo training in Year One and are encouraged to make use of further training opportunities available in subsequent years. Training ranges from gaining a broader knowledge of mathematics through taught modules, seminars and workshops, to enhancing your professional and transferrable skills. Our PhD students undertake high quality original research and are being well-prepared for a career, either in academia or elsewhere.

Teaching and learning

Students are required to complete a series of modules in their first year (from a very wide selection of bespoke modules), with assessment including an oral examination component. Upon the successful completion of these modules, students are required to complete a research project before being formally upgraded to a PhD at the end of the first year.

Training will be supplemented with attendance to seminars, cohort building activities, and additional transferable skills training.

General entry requirements

Minimum requirements.

First Class Honours undergraduate integrated Master's (4-year) degree from a UK university in Mathematics or a science degree with high mathematical content, or the equivalent qualification and grade from a non-UK university.

Alternatively, applicants who have a Bachelor's degree AND a Distinction in a postgraduate Master's degree would be considered.

English language requirements

You can find out more about our English language requirements Link opens in a new window . This course requires the following:

• IELTS overall score of 6.5, minimum component scores not below 6.0

International qualifications

We welcome applications from students with other internationally recognised qualifications.

For more information, please visit the international entry requirements page Link opens in a new window .

Additional requirements

There are no additional entry requirements for this course.

Our research

The mathematics department covers a wide range of research areas in mathematics and its applications.

You may also wish to explore the research interest of current Warwick academics .

Find a supervisor

The 'Find A Supervisor' link below will allow you to explore the research interests of academics within the department. Please include in your application the names of potential supervisors, with interests aligned with yours, or people you would like to work with.

The mathematics department, unlike some other departments, does not require students to make any arrangements with any potential supervisors before applying, though of course you are welcome to contact them directly and discuss your interests and any potential projects they may offer.

Mathematics staff directory Link opens in a new window  

Research proposals

Warwick academics can supervise thesis research in a wide range of topics in mathematics and its applications. In addition, all PhD projects are expected to be co-supervised by an external partner, bringing knowledge, data and experience to the modelling of real-world problems.

The mathematics department covers a wide range of research areas Link opens in a new window in mathematics and its applications.

Research proposal guidance

In many cases, applicants have not settled on a specific area or project and have rather broad interests. If that is the case, this document can be used to describe your interests, your background on those areas, and what attracts you to those topics for example. This will help in assessing your application and help us determine potential supervisors and projects that you might be interested in.

Find out more about writing a research proposal.

Tuition fees

Tuition fees are payable for each year of your course at the start of the academic year, or at the start of your course, if later. Academic fees cover the cost of tuition, examinations and registration and some student amenities.

Find your research course fees

Fee Status Guidance

We carry out an initial fee status assessment based on the information you provide in your application. Students will be classified as Home or Overseas fee status. Your fee status determines tuition fees, and what financial support and scholarships may be available. If you receive an offer, your fee status will be clearly stated alongside the tuition fee information.

Do you need your fee classification to be reviewed?

If you believe that your fee status has been classified incorrectly, you can complete a fee status assessment questionnaire. Please follow the instructions in your offer information and provide the documents needed to reassess your status.

Find out more about how universities assess fee status

Additional course costs

As well as tuition fees and living expenses, some courses may require you to cover the cost of field trips or costs associated with travel abroad.

For departmental specific costs, please see the Modules tab on the course web page for the list of core and optional core modules with hyperlinks to our  Module Catalogue  (please visit the Department’s website if the Module Catalogue hyperlinks are not provided).

Associated costs can be found on the Study tab for each module listed in the Module Catalogue (please note most of the module content applies to 2022/23 year of study). Information about module department specific costs should be considered in conjunction with the more general costs below:

• Core text books
• Printer credits
• Dissertation binding
• Robe hire for your degree ceremony

Scholarships and bursaries

Scholarships and financial support.

Find out about the different funding routes available, including; postgraduate loans, scholarships, fee awards and academic department bursaries.

Living costs

Find out more about the cost of living as a postgraduate student at the University of Warwick.

Mathematics Funding Opportunities

Find out more about the various funding opportunities that are available in our department.

Mathematics at Warwick

Our challenging Mathematics degrees will harness your strong mathematical ability and commitment, enabling you to explore your passion for mathematics.

Find out more about us on our website Link opens in a new window

Our courses

• Mathematics (Diploma plus MSc)
• Mathematics (MSc)
• Mathematics (PhD)

How to apply

The application process for courses that start in September and October 2025 will open on 2 October 2024.

For research courses that start in September and October 2025 the application deadline for students who require a visa to study in the UK is 2 August 2025. This should allow sufficient time to complete the admissions process and to obtain a visa to study in the UK.

How to apply for a postgraduate research course  

After you’ve applied

Find out how we process your application.

Applicant Portal

Track your application and update your details.

Admissions statement

See Warwick’s postgraduate admissions policy.

Join a live chat

Ask questions and engage with Warwick.

Explore ways to connect with us

We understand how important it is to visit and explore your future university before you apply. That's why we have put together a range of online and in-person options to help you discover more about your course, visit campus, and get a sense of postgraduate life at Warwick. Our events offer includes:

• Warwick hosted events
• Postgraduate Fairs
• Talk and Tours
• Department events

Why Warwick

Discover why Warwick is one of the best universities in the UK and renowned globally.

9th in the UK

of the UK's best universities (The Guardian University Guide 2024, The Times and Sunday Times Good University Guide 2024.)

6th Most Targeted University

by the UK's top 100 graduate employers (The Graduate Market in 2024, High Fliers Research Ltd.)

69th in the World

out of 1,500 institutions across 104 locations (QS World University Rankings 2025.)

10th in the UK

for our 'Graduate Prospects' score. (The Times and Sunday Times Good University Guide 2024.)

Gold Standard Teaching

Gold, the highest possible rating across all three categories (student experience, student outcomes, and overall) (Teaching Excellence Framework 2023)

Page updates

We may have revised the information on this page since publication. See the edits we have made and content history .

About the information on this page

This information is applicable for 2025 entry. Given the interval between the publication of courses and enrolment, some of the information may change. It is important to check our website before you apply. Please read our terms and conditions to find out more.

DPhil in Mathematics

• Entry requirements
• Funding and Costs

College preference

• How to Apply

About the course

The DPhil in Mathematics is an advanced research degree which provides the opportunity to investigate a project in depth and write a thesis which makes a significant contribution in the field.  You will gain a wide range of research and other skills as well as in-depth knowledge and expertise in your chosen field, whilst studying in a beautiful, modern setting. 

During your study at Oxford, you can share in the excitement of contributing to research in one or more of the many topics studied by Oxford mathematicians. The department’s research covers the entire spectrum of mathematics, with subject areas including:

• algebra (primarily group theory and representation theory)
• number theory
• algebraic geometry
• differential geometry
• complex manifolds
• global analysis
• partial differential equations
• functional analysis
• stochastic analysis
• dynamical systems
• mathematical logic
• optimisation
• combinatorial theory
• quantum theory
• string theory
• mathematical biology and ecology
• mathematical modelling
• fluid and continuum mechanics
• mathematical and computational finance
• numerical analysis
• history of mathematics
• mathematics applied to problems in earth sciences, materials science and finance
• machine learning and data science
• network science.

More information about the Research Groups in the Mathematical Institute  can be found on the department's website. 

You will be asked to outline your research interests when you apply by listing at least one but no more than three of the fields of research listed above on your application form. Full instructions for completing this section of the application form can be found in the How to apply section of this page.

You will be expected to acquire transferable skills as part of your training, which will require you to attend courses, lectures, workshops and colloquia. You will be expected to complete some broadening training during your studies, comprising attendance at lectures, relevant seminars and colloquia. You will have the opportunity to develop other valuable skills and to contribute to the teaching work of the department, both by marking students’ work and later by leading classes of students.

The department offers extensive support to students, from regular skills training and career development sessions to a variety of social events in a welcoming and inclusive atmosphere. You will have the opportunity to interact with fellow students and other members of your research groups, and more widely across the department. The department is committed to offering you the best supervision and to providing a stimulating research environment.

The course is full-time and requires attendance in Oxford. Full-time students are subject to the University's Residence requirements.

Undertaking the course is regarded as equivalent to working full-time hours and may also sometimes require some additional hours. The minimum period of registration for the DPhil is six terms but in practice you may need nine terms at least.

Provision exists for students on some courses to undertake their research in a ‘well-founded laboratory’ outside of the University. This may require travel to and attendance at a site that is not located in Oxford. Where known, existing collaborations will be outlined on this page. Please read the course information carefully, including the additional information about course fees and costs. 

Resources to support your study

As a graduate student, you will have access to the University's wide range of world-class resources including libraries, museums, galleries, digital resources and IT services.

The Bodleian Libraries is the largest library system in the UK. It includes the main Bodleian Library and libraries across Oxford, including major research libraries and faculty, department and institute libraries. Together, the Libraries hold more than 13 million printed items, provide access to e-journals, and contain outstanding special collections including rare books and manuscripts, classical papyri, maps, music, art and printed ephemera.

The University's IT Services is available to all students to support with core university IT systems and tools, as well as many other services and facilities. IT Services also offers a range of IT learning courses for students, to support with learning and research.

The Mathematical Institute's home is the purpose-built Andrew Wiles Building, opened in 2013. This provides ample teaching facilities for lectures, classes and seminars. Each research student is allocated an office in the Andrew Wiles Building that they will share with three or four other students: each student has their own desk, with a computer.

The Mathematical Institute provides IT support, and students can use the department's Whitehead Library, with an extensive range of books and journals.

In addition to the common room, where graduate students regularly gather for coffee and other social occasions, there is also a café in the Andrew Wiles Building.

Supervision

You will be invited to suggest a specific supervisor or supervisors in your application, and your preferences will be considered in allocating you a supervisor (which will be done before your arrival). The allocation of graduate supervision for this course is the responsibility of the Mathematical Institute and it is not always possible to accommodate the preferences of incoming graduate students to work with a particular member of staff. Under exceptional circumstances a supervisor may be found outside the Mathematical Institute. 

You will have the opportunity to meet individually with your supervisor.

All students will be initially admitted to the status of Probationer Research Student (PRS). Within four terms of admission as a PRS student you will be expected to apply for transfer of status from Probationer Research Student to DPhil status.

A successful transfer of status from PRS to DPhil status will require satisfactory attendance and the submission of a thesis. Students who are successful at transfer will also be expected to apply for and gain confirmation of DPhil status within nine terms of admission, to verify that their work continues to be on track. Both milestones normally involve an interview with two assessors (other than your supervisor) and therefore provide important experience for the final oral examination.

You will be expected to teach at least two sets of classes before transfer of status and a further three additional sets before confirmation of status. You will be expected to acquire transferable skills as part of your training, this normally involves attendance at workshops, seminars and colloquia.

You will be expected to submit a substantial original thesis which should not exceed 200 pages after three or, at most, four years from the date of admission. To be successfully awarded a Doctor of Philosophy in Mathematics you will need to defend your thesis orally (viva voce) in front of two appointed examiners.

Graduate destinations

The department, working alongside the university’s Careers Service, supports graduate students as they move from the DPhil to the next stage of their career. Many graduate students stay in academia, by taking up a postdoctoral position, and many move into employment in a range of industries and sectors where the expertise and skills developed during the DPhil are highly valued.

Changes to this course and your supervision

The University will seek to deliver this course in accordance with the description set out in this course page. However, there may be situations in which it is desirable or necessary for the University to make changes in course provision, either before or after registration. The safety of students, staff and visitors is paramount and major changes to delivery or services may have to be made if a pandemic, epidemic or local health emergency occurs. In addition, in certain circumstances, for example due to visa difficulties or because the health needs of students cannot be met, it may be necessary to make adjustments to course requirements for international study.

Where possible your academic supervisor will not change for the duration of your course. However, it may be necessary to assign a new academic supervisor during the course of study or before registration for reasons which might include illness, sabbatical leave, parental leave or change in employment.

For further information please see our page on changes to courses and the provisions of the student contract regarding changes to courses.

Entry requirements for entry in 2025-26

Proven and potential academic excellence.

The requirements described below are specific to this course and apply only in the year of entry that is shown. You can use our interactive tool to help you  evaluate whether your application is likely to be competitive .

We know that factors such as socio-economic circumstances and school performance can make it difficult for students to demonstrate their full potential. This course is taking part in an initiative to use contextual data to help us to better understand your achievements in the context of your individual background. For further details, please refer to the information about improving access to graduate study in the How to apply section of this page.

Please be aware that any studentships that are linked to this course may have different or additional requirements and you should read any studentship information carefully before applying. Contextual data may also be used in the assessment of studentships. 

Degree-level qualifications

As a minimum, applicants should hold or be predicted to achieve the following UK qualifications or their equivalent:

• a first-class undergraduate degree with honours in mathematics or a related discipline.

A previous master's degree is not required, though the requirement for a first-class undergraduate degree with honours can be alternatively demonstrated by strong performance in a master's degree.

For applicants with a bachelor's degree from the USA, the minimum overall GPA that is normally required to meet the undergraduate-level requirement is 3.7 out of 4.0.

If your degree is not from the UK or another country specified above, visit our International Qualifications page for guidance on the qualifications and grades that would usually be considered to meet the University’s minimum entry requirements.

GRE General Test scores

No Graduate Record Examination (GRE) or GMAT scores are sought.

Other qualifications, evidence of excellence and relevant experience

• Research or working experience in the proposed research area may be an advantage.
• Publications are not expected but may be used as evidence of excellence.

English language proficiency

This course requires proficiency in English at the University's  standard level . If your first language is not English, you may need to provide evidence that you meet this requirement. The minimum scores required to meet the University's standard level are detailed in the table below.

*Previously known as the Cambridge Certificate of Advanced English or Cambridge English: Advanced (CAE) † Previously known as the Cambridge Certificate of Proficiency in English or Cambridge English: Proficiency (CPE)

Your test must have been taken no more than two years before the start date of your course. Our Application Guide provides further information about the English language test requirement .

Declaring extenuating circumstances

If your ability to meet the entry requirements has been affected by the COVID-19 pandemic (eg you were awarded an unclassified/ungraded degree) or any other exceptional personal circumstance (eg other illness or bereavement), please refer to the guidance on extenuating circumstances in the Application Guide for information about how to declare this so that your application can be considered appropriately.

You will need to register three referees who can give an informed view of your academic ability and suitability for the course. The  How to apply  section of this page provides details of the types of reference that are required in support of your application for this course and how these will be assessed.

Supporting documents

You will be required to supply supporting documents with your application. The  How to apply  section of this page provides details of the supporting documents that are required as part of your application for this course and how these will be assessed.

Performance at interview

Technical interviews are normally held as part of the admissions process.

If invited you can expect to be interviewed by at least two people and for the interview to last around 30 minutes. The interview will take place remotely.

It is expected that interviews will take place around three to five weeks after an application deadline.

Offer conditions for successful applications

If you receive an offer of a place at Oxford, your offer will outline any conditions that you need to satisfy and any actions you need to take, together with any associated deadlines. These may include academic conditions, such as achieving a specific final grade in your current degree course. These conditions will usually depend on your individual academic circumstances and may vary between applicants. Our ' After you apply ' pages provide more information about offers and conditions . 

In addition to any academic conditions which are set, you will also be required to meet the following requirements:

Financial Declaration

If you are offered a place, you will be required to complete a  Financial Declaration  in order to meet your financial condition of admission.

Disclosure of criminal convictions

In accordance with the University’s obligations towards students and staff, we will ask you to declare any  relevant, unspent criminal convictions  before you can take up a place at Oxford.

Other factors governing whether places can be offered

The following factors will also govern whether candidates can be offered places:

• the ability of the University to provide the appropriate supervision for your studies, as outlined under the 'Supervision' heading in the About section of this page;
• the ability of the University to provide appropriate support for your studies (eg through the provision of facilities, resources, teaching and/or research opportunities); and
• minimum and maximum limits to the numbers of students who may be admitted to the University's taught and research programmes.

Academic Technology Approval Scheme (ATAS)

Some postgraduate research students in science, engineering and technology subjects will need an Academic Technology Approval Scheme (ATAS) certificate prior to applying for a  Student visa (under the Student Route) . For some courses, the requirement to apply for an ATAS certificate may depend on your research area.

Mathematics

Mathematics has been studied in Oxford since the University was first established in the 12th century. The Mathematical Institute aims to preserve and expand mathematical culture through excellence in teaching and research.

The Mathematical Institute offers a wide range of graduate courses, including both taught master’s courses and research degrees. Research and teaching cover the spectrum of pure and applied mathematics with researchers working in fields including:

• combinatorics
• mathematical physics
• mathematical finance
• mathematical biology networks
• numerical analysis.

The Mathematical Institute is proud to have received an Athena SWAN silver award renewal in 2021, reflecting its commitment to promoting diversity and to creating a working environment in which students and staff alike can achieve their full potential.

Graduate students are an integral part of the department, interacting with each other and with academic staff as part of a vibrant community that strives to further mathematical study. As a graduate student at Oxford you will benefit from excellent resources, extensive training opportunities and supportive guidance from your supervisor or course director.

The Mathematical Institute has strong ties with other University departments including Computer Science, Statistics and Physics, teaching several courses jointly. Strong links with industrial and other partners are also central to the department.

The institute’s home is the purpose-built Andrew Wiles Building, opened in 2013. This provides ample teaching facilities for lectures, classes and seminars. The Mathematical Institute provides six lecture theatres and six classrooms.

Graduate students have access to the department common room, where members of the department regularly gather for coffee and other social occasions, and the mezzanine level of the Andrew Wiles Building houses a café and teaching spaces.

View all courses   View taught courses View research courses

The Mathematical Institute's home is the purpose-built Andrew Wiles Building, opened in 2013. This provides ample teaching facilities for lectures, classes and seminars. Each research student is allocated an office in the Andrew Wiles Building that they will share with 3 or 4 other students: each student has their own desk, with a computer. The Mathematical Institute provides IT support, and students can use the department's Whitehead Library, with an extensive range of books and journals.

For entry in the 2025-26 academic year, the collegiate University expects to offer over 1,000 full or partial graduate scholarships across a wide range of graduate courses.

If you apply by the January deadline shown on this page and receive a course offer, your application will then be considered for Oxford scholarships. For the majority of Oxford scholarships, your application will automatically be assessed against the eligibility criteria, without needing to make a separate application. There are further Oxford scholarships available which have additional eligibility criteria and where you are required to submit a separate application. Most scholarships are awarded on the basis of academic merit and/or potential.

To ensure that you are considered for Oxford scholarships that require a separate application, for which you may be eligible,  use our fees, funding and scholarship search tool  to identify these opportunities and find out how to apply. Alongside Oxford scholarships, you should also consider other opportunities for which you may be eligible including  a range of external funding ,  loan schemes for postgraduate study  and any other scholarships which may also still be available after the January deadline as listed on  our fees, funding and scholarship search tool .

Details of college-specific funding opportunities can also be found on individual college websites:

Select from the list:

Please refer to the College preference section of this page to identify which of the colleges listed above accept students for this course.

For the majority of college scholarships, it doesn’t matter which college, if any, you state a preference for in your application. If another college is able to offer you a scholarship, your application can be moved to that college if you accept the scholarship. Some college scholarships may require you to state a preference for that college when you apply, so check the eligibility requirements carefully.

Annual fees for entry in 2025-26 

Information about course fees

Course fees are payable each year, for the duration of your fee liability (your fee liability is the length of time for which you are required to pay course fees). For courses lasting longer than one year, please be aware that fees will usually increase annually. For details, please see our guidance on changes to fees and charges .

Course fees cover your teaching as well as other academic services and facilities provided to support your studies. Unless specified in the additional information section below, course fees do not cover your accommodation, residential costs or other living costs. They also don’t cover any additional costs and charges that are outlined in the additional information below.

Continuation charges

Following the period of fee liability , you may also be required to pay a University continuation charge and a college continuation charge. The University and college continuation charges are shown on the Continuation charges page.

Where can I find further information about fees?

The Fees and Funding  section of this website provides further information about course fees , including information about fee status and eligibility  and your length of fee liability .

Additional information

There are no compulsory elements of this course that entail additional costs beyond fees (or, after fee liability ends, continuation charges) and living costs. However, please note that, depending on your choice of research topic and the research required to complete it, you may incur additional expenses, such as travel expenses, research expenses, and field trips. You will need to meet these additional costs, although you may be able to apply for small grants from your department and/or college to help you cover some of these expenses.

Living costs

In addition to your course fees and any additional course-specific costs, you will need to ensure that you have adequate funds to support your living costs for the duration of your course.

Living costs for full-time study

For the 2025-26 academic year, the range of likely living costs for a single, full-time student is between £1,425 and £2,035 for each month spent in Oxford. We provide the cost per month so you can multiply up by the number of months you expect to live in Oxford. Depending on your circumstances, you may also need to budget for the  costs of a student visa and immigration health surcharge and/or living costs for family members or other dependants that you plan to bring with you to Oxford (assuming that dependant visa eligibility criteria are met).

Further information about living costs

The current economic climate and high national rate of inflation make it very hard to estimate potential changes to the cost of living over the next few years. For study in Oxford beyond the 2025-26 academic year, it is suggested that you budget for potential increases in living expenses of around 4% each year – although this rate may vary depending on the national economic situation. For further information, please consult our more detailed information about living costs , which includes a breakdown of likely living costs in Oxford for items such as food, accommodation and study costs.

Students enrolled on this course will belong to both a department/faculty and a college. Please note that ‘college’ and ‘colleges’ refers to all 43 of the University’s colleges, including those designated as societies and permanent private halls (PPHs). 

If you apply for a place on this course you will have the option to express a preference for one of the colleges listed below, or you can ask us to find a college for you. Before deciding, we suggest that you read our brief  introduction to the college system at Oxford  and our  advice about expressing a college preference . 

If you are a current Oxford student and you would like to remain at your current Oxford college, you should check whether it is listed below. If it is, you should indicate this preference when you apply. If not, you should contact your college office to ask whether they would be willing to make an exception. Further information about staying at your current college can be found in our Application Guide. 

The following colleges accept students on the DPhil in Mathematics:

• Balliol College
• Brasenose College
• Christ Church
• Corpus Christi College
• Exeter College
• Green Templeton College
• Hertford College
• Jesus College
• Keble College
• Kellogg College
• Lady Margaret Hall
• Linacre College
• Lincoln College
• Magdalen College
• Mansfield College
• Merton College
• New College
• Oriel College
• Pembroke College
• The Queen's College
• Reuben College
• St Anne's College
• St Catherine's College
• St Cross College
• St Edmund Hall
• St Hilda's College
• St Hugh's College
• St John's College
• St Peter's College
• Somerville College
• Trinity College
• University College
• Wadham College
• Wolfson College
• Worcester College
• Wycliffe Hall

Before you apply

Our guide to getting started provides general advice on how to prepare for and start your application. You can use our interactive tool to help you evaluate whether your application is likely to be competitive .

If it is important for you to have your application considered under a particular deadline – eg under the January deadline in order to be considered for Oxford scholarships – we recommend that you aim to complete and submit your application at least two weeks in advance . Check the deadlines on this page and the information about deadlines and when to apply in our Application Guide.

Application fee waivers

An application fee of £20 is payable for each application to this course. Application fee waivers are available for the following applicants who meet the eligibility criteria:

• applicants from low-income countries;
• refugees and displaced persons; 
• UK applicants from low-income backgrounds; and 
• applicants who applied for our Graduate Access Programmes in the past two years and met the eligibility criteria.

You are encouraged to  check whether you're eligible for an application fee waiver  before you apply.

Readmission for current Oxford graduate taught students

If you're currently studying for an Oxford graduate taught course and apply to this course with no break in your studies, you may be eligible to apply to this course as a readmission applicant. The application fee will be waived for an eligible application of this type. Check whether you're eligible to apply for readmission .

Application fee waivers for eligible associated courses

If you apply to this course and are considering applying (or have already applied) to any of the associated courses listed below , you can request an application fee waiver so that you only need to pay one application fee. We recommend that you use your application fee waiver to apply only for eligible courses that are closely related in research area to this one.

For full details about how to request an application fee waiver, please select the course you are interested in from the list below and refer to the equivalent section of its course page.

The following associated courses are taking part in this application fee waiver scheme:

• Autonomous Intelligent Machines and Systems , EPSRC CDT
• Biomedical and Clinical Sciences (Oxford-GSK) , DPhil
• Cancer Science (Biological background) , CDT
• Cancer Science (Clinician) , DPhil
• Cancer Science (Intercalation) , DPhil
• Cancer Science (Maths/Physics background) , CDT
• Chemical Synthesis for a Healthy Planet , CDT
• Engineering Biology , BBSRC and EPSRC CDT
• Fusion Power , EPSRC CDT
• Genomic Medicine and Statistics , DPhil
• Healthcare Data Science , EPSRC CDT
• Inflammatory and Musculoskeletal Disease , DPhil
• Inorganic Materials for Advanced Manufacturing , EPSRC CDT
• Intelligent Earth , UKRI CDT in AI for the Environment
• Materials 4.0 , EPSRC CDT
• Mathematics of Random Systems: Analysis, Modelling and Algorithms , CDT
• Neuroscience (1+3), DPhil
• Quantum Informatics , EPSRC CDT
• Robotics and AI for Net Zero , EPSRC CDT
• Statistics and Machine Learning , EPSRC CDT
• Superconductivity: Enabling Transformative Technologies , EPSRC CDT

Do I need to contact anyone before I apply?

You do not need to make contact with the department before you apply but you are encouraged to visit the relevant departmental webpages to read any further information about your chosen course.

However, if you would like to speak to an academic member of staff involved in your preferred area of research, you may get in touch with them directly using the contact details on the Mathematical Institute's webpages.

Improving access to graduate study

This course is taking part in initiatives to improve the selection procedure for graduate applications, to ensure that all candidates are evaluated fairly. It is taking part in the 'Close the Gap' project which aims to improve access to doctoral study.

Socio-economic data (where it has been provided in the application form) and your contextual statement (if you choose to provide one) will be used as part of an initiative to contextualise applications at the different stages of the selection process.

Completing your application

You should refer to the information below when completing the application form, paying attention to the specific requirements for the supporting documents .

For this course, the application form will include questions that collect information that would usually be included in a CV/résumé. You should not upload a separate document. If a separate CV/résumé is uploaded, it will be removed from your application .

If any document does not meet the specification, including the stipulated word count, your application may be considered incomplete and not assessed by the academic department. Expand each section to show further details.

Proposed field and title of research project

Under the section titled 'Field and title of research project', you are strongly encouraged to name at least one but no more than three research groups that you would like your application to be seen by.  More information about the Research Groups in the Mathematical Institute  can be found on the department's website.

You should not use this field to type out a full research proposal. You will be able to upload your research supporting materials separately if they are required (as described below).

Proposed supervisor

If known, under 'Proposed supervisor name' enter the name of the academic(s) whom you would like to supervise your research. Otherwise, leave this field blank.

You can enter up to three names and you should list them in order of preference or indicate equal preference. 

Referees: Three overall, academic preferred

Whilst you must register three referees, the department may start the assessment of your application if two of the three references are submitted by the course deadline and your application is otherwise complete. Please note that you may still be required to ensure your third referee supplies a reference for consideration.

Your references should generally be academic, though up to one professional reference will be accepted.

Your references should describe your intellectual ability, academic achievement, motivation, and aptitude for advanced research.

Official transcript(s)

Your transcripts should give detailed information of the individual grades received in your university-level qualifications to date. You should only upload official documents issued by your institution and any transcript not in English should be accompanied by a certified translation.

More information about the transcript requirement is available in the Application Guide.

Contextual statement

If you wish to provide a contextual statement with your application, you may also submit an additional statement to provide contextual information on your socio-economic background or personal circumstances in support of your application.

Submit a contextual statement

It is not necessary to anonymise this document, as we recognise that it may be necessary for you to disclose certain information in your statement. This statement will not be used as part of the initial academic assessment of applications at shortlisting, but may be used in combination with socio-economic data to provide contextual information during decision-making processes.

Please note, this statement is in addition to  completing the 'Extenuating circumstances’ section of the standard application form .

You can find  more information about the contextual statement  on our page that provides details of the continuing pilot programme to improve the assessment procedure for graduate applications.

Statement of purpose/personal statement: A maximum of 800 words

Your statement should be written in English and explain your reasons for applying for the course at Oxford, your relevant experience and education, and the specific areas that interest you and/or you intend to specialise in. This will be assessed for evidence of motivation for and understanding of the proposed area of study and whether a suitable supervisor can be found.

If possible, please ensure that the word count is clearly displayed on the document.

Start or continue your application

You can start or return to an application using the relevant link below. As you complete the form, please  refer to the requirements above  and  consult our Application Guide for advice .

Apply Continue application

After you've submitted your application

Your application (including the supporting documents outlined above) will be assessed against the entry requirements detailed on this course page. Whether or not you have secured funding will  not  be taken into consideration when your application is assessed. You can  find out more about our shortlisting and selection process  in our detailed guide to what happens next.

Find out how to manage your application after submission , using our Applicant Self-Service tool.

ADMISSION STATUS

Open to applications for entry in 2025-26

12:00 midday UK time on:

Wednesday 8 January 2025

Latest deadline for most Oxford scholarships

Tuesday 4 March 2025

Final application deadline for entry in 2025-26

*Three-year average (applications for entry in 2022-23 to 2024-25)

Further information and enquiries

This course is offered by the Mathematical Institute

• Course page on the Mathematical Institute's website
• Academic and research staff
• Departmental research  
• Mathematical, Physical and Life Sciences
• Residence requirements for full-time courses
• Postgraduate applicant privacy policy

Course-related enquiries

Advice about contacting the department can be found in the How to apply section of this page

✉ [email protected] ☎ +44 (0)1865 615210

Application-process enquiries

Application guide

IIT Bombay Department of Mathematics

The m.sc mathematics programme, ph.d admissions.

• You are here:

Admission to Ph.D programme in Mathematics and Statistics

2016 ). With state-of-the art infrastructural facilities and a sound research base, IITB Mathematics offers a Ph.D. programme in a wide variety of areas. The program leading to the Ph.D. degree involves a course credit requirement and a research project leading to thesis submission. All that you need to know about admissions to our PhD program can be found in the information brochure for PhD admissions, which can be found at the following link. ----- A special welcome to all who wish to pursue a career in Mathematics and Statistics research. The Department of Mathematics, IITB offers a Ph.D. program in a wide variety of areas. To know more about the research interests of faculty members, please visit the page here . The program leading to the Ph.D. degree involves a course credit requirement, clearing of qualifier examinations and a research project leading to thesis submission. For more details, follow one of the links below

A special welcome to all who wish to pursue a career in Mathematics and Statistics research. The Department of Mathematics, IITB offers Ph.D. program in the areas of Mathematics and Statistics. Admission to the PhD program is based on a written test and interview. There are separate written tests and interviews for students in Mathematics and Statistics. The syllabus is given below. Students are required to choose one option specifying either Mathematics or Statistics.

To know more about the research interests of faculty members, please visit the page here . The program leading to the Ph.D. degree involves a course credit requirement, clearing of qualifier examinations and a research project leading to thesis submission. For more details, follow one of the links below

Syllabus for Entrance exam can be found here Past question papers can be found here

Syllabus for Mathematics Entrance Exam

Syllabus for Statistics Entrance Exam

Past Question Papers

PhD Admissions

Choose Colour

Mathematics Education, PHD

On this page:, at a glance: program details.

• Location: Tempe campus
• Second Language Requirement: No

Program Description

Degree Awarded: PHD Mathematics Education

This transdisciplinary PhD program in mathematics education accommodates students from a variety of academic backgrounds. It provides students with a solid foundation in graduate-level mathematics as well as research skills and perspectives that enable them to incorporate mathematics into such core educational areas as:

Conducting individual and collaborative research in the learning and teaching of mathematics is an integral part of the program.

Degree Requirements

Curriculum plan options.

• 84 credit hours, a written comprehensive exam, an oral comprehensive exam, a prospectus and a dissertation

Required Core (12 credit hours) MTE 501 Research in Undergraduate Mathematics Education I (3) MTE 502 Research in Undergraduate Mathematics Education II (3) MTE 503 Research in Undergraduate Mathematics Education Ill (3) MTE 504 Research in Undergraduate Mathematics Education IV (3)

Electives (42 credit hours)

Area Courses (12 credit hours)

Research (6 credit hours) MTE 792 Research (6)

Culminating Experience (12 credit hours) MTE 799 Dissertation (12)

Additional Curriculum Information Four to five graduate-level (500 and above) elective courses from mathematics, cognitive science, psychology, educational technology, philosophy or research should be taken as approved by the advisor.

For the area courses, students are required to take four graduate-level courses from the following areas of interest: mathematics, applied mathematics or statistics. Students should see the academic unit for the approved course list.

Students should see the school's website for information about qualifier and comprehensive examinations based on math coursework.

The doctoral dissertation culminating experience consists of a dissertation prospectus, oral dissertation defense and the submission of a final revised, formatted dissertation document to the Graduate College. Dissertations are composed under chair- and committee-supervised research, including literature review, research, data collection and analysis, and writing.

When approved by the student's supervisory committee and the Graduate College, up to 30 credit hours from a previously awarded master's degree may be used for this program. If students do not have a previously awarded master's degree, the remaining coursework is made up of electives and research.

Admission Requirements

Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and Sciences.

Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics or a closely related area, with exceptionally high grades in advanced coursework in mathematics, from a regionally accredited institution.

Applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in the last 60 hours of their first bachelor's degree program or a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in an applicable master's degree program.

All applicants must submit:

• graduate admission application and application fee
• official transcripts
• statement of education and career goals
• writing sample
• three letters of recommendation
• proof of English proficiency

Additional Application Information An applicant whose native language is not English must provide proof of English proficiency regardless of their current residency.

At least two of the letters of recommendation must be from faculty.

Next Steps to attend ASU

Learn about our programs, apply to a program, visit our campus, application deadlines, learning outcomes.

• Able to complete original research in applied mathematics.
• Able to incorporate mathematical concepts into novel teaching methods.
• Address an original research question in mathematics education.

Career Opportunities

Graduates of the doctoral program in mathematics education have opportunities in Arizona, the U.S. and internationally. Opportunities are typically at research universities and liberal arts colleges, community colleges, and education consulting firms and in roles such as:

• faculty-track academic
• education consultant or analyst
• mathematics professor, instructor or researcher

Program Contact Information

If you have questions related to admission, please click here to request information and an admission specialist will reach out to you directly. For questions regarding faculty or courses, please use the contact information below.

• [email protected]
• 480/965-3951

Applied Physics and Applied Mathematics

PhD in Applied Mathematics

Master core areas of advanced math and their application in areas like photonics, machine learning, climate modeling, or data science.

Situated within a unique department spanning applied mathematics, applied physics, materials science and engineering, and medical physics, the PhD in Applied Mathematics program at Columbia Engineering immerses you in a vibrant, multidisciplinary community that will build your expertise and imagination beyond the traditional boundaries of the field. You’ll join faculty in addressing challenges in areas including optics and photonics, material science, machine learning, data science, imaging science, biology, and climate modeling.

The program has a strong track record in research and training, with outstanding placement of PhD students. Recent graduates have accepted tenure-track/tenured faculty positions at Carnegie Mellon, Columbia, Drexel, Purdue, Tsinghua, UC Santa Cruz, Utah, and Washington, as well as private sector jobs in leading financial and technology companies.

Why Earn your PhD in Applied Mathematics at Columbia?

Columbia gives you a rigorous Ivy League education in the heart of a vibrant global city for unmatched opportunities and impact.

As a student, you’ll benefit from:

• New York City Join top talent in one of the world’s most exciting and influential cities. Students choose Columbia Engineering over MIT, Berkeley, and others because of the New York City ecosystem of research and enterprise that can’t be found anywhere else.   
• An Environment of Innovation Our faculty lead the world in innovative research, generating new knowledge, patents, and startups at a staggering rate. When you work with the very best faculty, you acquire the experience and habits of mind to follow in their footsteps for leadership in the field. What you discover, make, and do here will be the best work you have ever done — but only the beginning of the best work you will do.  
• Broad Research Opportunities Find a place to make your mark. In addition to multiple areas of ongoing research within the department, faculty also have strong connections with interdisciplinary laboratories and research centers including the  Data Science Institute and  Learning the Earth with Artificial Intelligence and Physics , an NSF Science and Technology Center.  
• Columbia University With a PhD from Columbia, you’ll join one of the world’s most international, accomplished networks of researchers and alumni. Your Ivy League credential will open doors wherever you go and the relationships you build here will accelerate your career.

See Full Program Details

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OpenAI Unveils New ChatGPT That Can Reason Through Math and Science

Driven by new technology called OpenAI o1, the chatbot can test various strategies and try to identify mistakes as it tackles complex tasks.

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By Cade Metz

Reporting from San Francisco

Online chatbots like ChatGPT from OpenAI and Gemini from Google sometimes struggle with simple math problems . The computer code they generate is often buggy and incomplete. From time to time, they even make stuff up .

On Thursday, OpenAI unveiled a new version of ChatGPT that could alleviate these flaws. The company said the chatbot, underpinned by new artificial intelligence technology called OpenAI o1, could “reason” through tasks involving math, coding and science.

“With previous models like ChatGPT, you ask them a question and they immediately start responding,” said Jakub Pachocki, OpenAI’s chief scientist. “This model can take its time. It can think through the problem — in English — and try to break it down and look for angles in an effort to provide the best answer.”

In a demonstration for The New York Times, Dr. Pachocki and Szymon Sidor, an OpenAI technical fellow, showed the chatbot solving an acrostic, a kind of word puzzle that is significantly more complex than an ordinary crossword puzzle. The chatbot also answered a Ph.D.-level chemistry question and diagnosed an illness based on a detailed report about a patient’s symptoms and history.

The new technology is part of a wider effort to build A.I. that can reason through complex tasks. Companies like Google and Meta are building similar technologies, while Microsoft and its subsidiary GitHub are working to incorporate OpenAI’s new system into their products.

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