theoretical mathematics phd

Theoretical Mathematics

Theoretical mathematics is the study of abstract mathematical structures which form the basic framework for the rest of the mathematical sciences. In large part, theoretical mathematics is inspired by intellectual curiosity. Theoretical mathematics provides the tools for scientific discoveries in the future, often in unexpected ways.

Discrete Mathematics

Algebra includes the study of groups, rings, fields, and vector spaces which are the basic frameworks from which to study many other areas of mathematics and science. Combinatorics is the study of counting the number of possibilities that meet a certain criteria. This subject closely relates to probability

Analysis and Partial Differential Equations

Analysis includes operator theory and C*-algebras – the study of infinite dimensional systems of equations,  Fourier analysis – the study of infinite dimensional trigonometric systems, complex analysis – the study of functions of a complex variable, partial differential equations and dynamical systems – the equations which model how physical quantities change with time or space,  and research in theoretical and mathematical physics.

Geometry and Topology

Geometry and topology is the study of how objects in space bend or twist. It also involves categorizing objects based on properties such as the number of holes that are present or their symmetry.

Number Theory

Number theory is the study of integers and fractions (rational numbers) – how they factor and how they arise as solutions to equations.

Algebra is the study of algebraic structures such as groups, rings, and fields, and how they relate to other branches of mathematics including algebraic geometry, combinatorics, and representation theory.

PhD in Mathematics

The PhD in Mathematics provides training in mathematics and its applications to a broad range of disciplines and prepares students for careers in academia or industry. It offers students the opportunity to work with faculty on research over a wide range of theoretical and applied topics.

Degree Requirements

The requirements for obtaining an PhD in Mathematics can be found on the associated page of the BU Bulletin .

• Courses : The courses mentioned on the BU Bulletin page can be chosen from the graduate courses we offer here . Half may be at the MA 500 level or above, but the rest must be at the MA 700 level or above. Students can also request to use courses from other departments to satisfy some of these requirements. Please contact your advisor for more information about which courses can be used in this way. All courses must be passed with a grade of B- or higher.
• Analysis (examples include MA 711, MA 713, and MA 717)
• PDEs and Dynamical Systems (examples include MA 771, MA 775, and MA 776)
• Algebra and Number Theory (examples include MA 741, MA 742, and MA 743)
• Topology (examples include MA 721, MA 722, and MA 727)
• Geometry (examples include MA 725, MA 731, and MA 745)
• Probability and Stochastic Processes (examples include MA 779, MA 780, and MA 783)
• Applied Mathematics (examples include MA 750, MA 751, and MA 770)
• Comprehensive Examination : This exam has both a written and an oral component. The written component consists of an expository paper of typically fifteen to twenty-five pages on which the student works over a period of a few months under the guidance of the advisor. The topic of the expository paper is chosen by the student in consultation with the advisor. On completion of the paper, the student takes an oral exam given by a three-person committee, one of whom is the student’s advisor. The oral exam consists of a presentation by the student on the expository paper followed by questioning by the committee members. A student who does not pass the MA Comprehensive Examination may make a second attempt, but all students are expected to pass the exam no later than the end of the summer following their second year.
• Oral Qualifying Examination: The topics for the PhD oral qualifying exam correspond to the two semester courses taken by the student from one of the 3 subject areas and one semester course each taken by the student from the other two subject areas. In addition, the exam begins with a presentation by the student on some specialized topic relevant to the proposed thesis research. A student who does not pass the qualifying exam may make a second attempt, but all PhD students are expected to pass the exam no later than the end of the summer following their third year.
• Dissertation and Final Oral Examination: This follows the GRS General Requirements for the Doctor of Philosophy Degree .

Admissions information can be found on the BU Arts and Sciences PhD Admissions website .

Financial Aid

Our department funds our PhD students through a combination of University fellowships, teaching fellowships, and faculty research grants. More information will be provided to admitted students.

More Information

Please reach out to us directly at [email protected] if you have further questions.

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Doctor of Philosophy (PhD)

The program of studies for a Math Department PhD student is divided into two main parts:  Pre-  and  Post-Candidacy . Before taking the  Candidacy Exam , students need to fulfill numerous requirements which ensure solid preparation in core mathematical areas as well in their chosen specialization. These include passing the  Qualifying Requirements  as well as fulfilling the  Breadth  and  Foreign Language  requirements. The  Candidacy Exam  is usually taken sometime during the third year, marking the end of the preparatory period and the beginning of research leading ultimately to the  PhD Dissertation . Time from admission to graduation usually averages around 6 years but may vary greatly depending on many factors such as initial preparation level, individual academic progress, complexity of chosen specializations, and strategic thesis and job decisions. Full program requirements can be found in the OSU Department of Mathematics Graduate Program Handbook [pdf]. The  OSU Graduate School  has requirements as well, which can be found in the  Graduate School Handbook .

Headstart Training

Newly admitted graduate students are required to arrive  4 weeks prior to the beginning of the Autumn semester in order to participate in our Headstart Training  teaching preparation program. This is a prerequisite for holding a GTA appointment. Further academic preparation activities are scheduled during this time as well. Exceptions can be made only for students who are offered university fellowships and typically involve a one-year deferment of the teaching portion only. Monetary compensation is provided to all Headstart participants.

Qualifying Requirements

For the Theoretical Track , there are four Qualifying Requirements, corresponding to the content of the four courses Math 6111 , Math 6112 , Math 6211 , and Math 6212 in Abstract Algebra and Real Analysis . Each requirement can be passed by either receiving a grade of A or A- in the respective course, or by receiving a passing grade on a respective (separate) qualifying examination. The Math 6111 and 6211 courses are offered every Autumn Semester and the Math 6112 and 6212 courses are offered every Spring Semester. The four examinations, one for each course, are offered each August (typically in the week before the start of classes) and are open to both incoming and continuing students. Each exam is two hours in length and covers roughly the material of the respective courses.

All four requirements need to be fulfilled by the end of the third semester of study (not including summer). Thus a student has four attempts to fulfill, for example, the 6111-requirement (twice by taking the course, and twice by exam) and three attempts to fulfill the 6112-requirement (once by taking the course, and twice by exam). The 6211 and 6212 requirements are analogous.

Interested students may substitute one of these four requirements with an approved 6000-level year-long course sequence with A or A- grades. This includes all regular full-year 6000-level sequences, namely,  Math 6221-6222, 6251-6252, 6411-6451, 6501-6502, 6601-6602, 6701-6702, and 6801-6802. (For example, the 6112 requirement can be substituted by taking the Math 6411-6451 Differential Equations sequence with A or A- grades in each course). Students should consult the GSC Chair about the use of the 6001-6004 Logic courses. 

Qualifying requirements for the Applied Track combine a mandatory Scientific Computing ( Math 6601 ) course, one of the algebra or analysis courses, and three additional courses from  Math 6602 ,  Math 6411 ,  Math 6451 , and the algebra and analysis courses.

Passing the Qualifying Requirements also entails an increase in stipend, assuming otherwise satisfactory academic progress (see Financial Support ). The outcome of initial exams and coursework will also inform future advising. 

Syllabi and exam materials can be viewed at  https://math.osu.edu/grad/current/phd/quals . 

Advisor and Breadth Requirements 

Upon attaining  Regular PhD status,   students are matched with faculty members who guide them to potential dissertation research ideas.  The primary task of this Dissertation Advisor  is to facilitate their advisee’s development as a mathematician.

The course work required for admission to Candidacy is referred to as our Breadth Requirements . The purpose of these requirements is to ensure that graduates master not only their eventual field of specialization, but also develop the breadth, versatility, and maturity expected from mathematicians working in academic professions that traditionally require a PhD . The requirements are as follows:

• Course Sequences: Complete a year-long sequence from each of  three different mathematical areas (see below)
• All courses must be passed with a grade of B+ or higher
• Course sequences used for the qualifying requirement (such as, for example, 6111-6112) may also be used for the breadth requirement. However, a passed qualifying exam does not count towards a breadth requirement.

PhD students are expect to complete their Breadth Requirements within their first two years from admission. Most students fulfill two breadth sequences  in their first year through qualifying courses and the third in their second year. Timely completion of breadth requirements may influence stipend level and summer support.

All Breadth Requirements must be completed by the time of the Candidacy Exam .

Breadth Requirements Chart

• Math 6111, 6112; Abstract Algebra
• Math 7121, 7122; Number Theory
• Math 7141, 7142; Algebraic Geometry
• Math 7161, 7162; Lie Groups
• Math 6211, 6212; Real Analysis
• Math 7211, 7212; Functional Analysis
• Math 7221, 7222; Ergodic Theory
• Math 6411, 6451; Differential Equations
• Math 7412, 7413; Ordinary Differential Equations
• Math 7452, 7453; Partial Differential Equations
• Math 6701, 6702; Differential Manifolds & Geometry
• Math 6801, 6802; Algebraic Topology
• Math 7711, 7721; Riemannian & Kahler Geometry
• Math 7851, 7852; Differential Topology
• Any Two Math 6001-6004; Advanced Mathematical Logic
• Math 6221, 6222; Complex Analysis
• Math 6251, 6252; Theory of Probability
• Math 6501, 6502; Combinatorics & Graph Theory
• Math 6601, 6602; Numerical Methods in Scientific Computing
• Math 7611, 7612; Computational Partial Differential Equations
• Math 7651, 7652; Applied Complex Variables and Asymptotics

Foreign Language Requirement

The foreign language requirement ensures the ability to read (with the aid of a dictionary) one foreign language chosen from among  French , German , or Russian. It can be fulfilled in one of the following two ways:

Class: Students with little or no prior knowledge of the chosen language can fulfill their language requirement by passing one of the following classes with a grade of B or better:

• French 6571
• German 6101 or German 6102
• Russian 6171 or  Russian 6172  

Exam: Alternatively, a student may pass a translation exam in one of the languages above.

To schedule the exam, please start by contacting the Mathematics Department Language Coordinator:

Dr. Andrzej Derdzinski   ([email protected])

To find out dates and information on the exams, please see below:

French Department Translation Exam Coordinator: Matthew Lang   [email protected] https://frit.osu.edu/graduate/graduate-reading-proficiency-exam/french-reading-proficiency-exam

German Department Translation Exam Coordinator: Natascha Miller [email protected] http://germanic.osu.edu/german-reading-exam

Russian Department Translation Exam Coordinator: Larysa Stepanova [email protected] https://slavic.osu.edu/courses/transfer-credit-and-placement

Specialization & Advisor

An important candidacy requirement is the choice of a dissertation specialization and a Dissertation Advisor . The diligent, timely, and careful pursuit of a future research direction is likely the most important responsibility of a prospective PhD candidate . The student should be fully invested in the choice of specialization, which will impact his/her future academic trajectory more than anything else. There are currently 65+ regular mathematics faculty on the Columbus campus, plus over 20 additional faculty on the branch campuses, who can supervise doctoral dissertations. Consult our current Graduate Faculty List for names, contacts and specializations. Under special circumstances, students can also be advised by faculty outside of the department. The advisor pool in our department is thus as large as that of any department in the country.

There are numerous opportunities for students to get to know potential advisors. This includes having them as teachers in introductory classes, attending the Invitations to Mathematics lecture series, regular research seminars, and colloquia (see Events ), or self-development through academic advisors, peers, and publicly available research information. After narrowing down possible specializations, students typically sample faculty and topics by taking numerous reading courses ( MATH 6193 ) on special topics with a few prospective advisors. These provide introductions to future research areas that are too specialized to be covered in regular courses. The one-on-one teaching of a reading course may also serve as a preview of the advisor / advisee interaction in future thesis work.

The choice of thesis advisor usually evolves out of this process. After student and faculty agree on the thesis advising, the student reports the change from the Preliminary Academic Advisor to the chosen Dissertation Advisor to the Math Graduate Office using the form located outside the Grad Office . 

Master of Science (MS) Degree

Information on how students admitted into the PhD program can earn the MS Degree can be found at  https://math.osu.edu/grad/current/ms .

Candidacy Examination 

For a graduate student to become an official PhD Candidate,  he/she has to pass the Candidacy Exam . The  Candidacy Exam  evaluates the validity and scope of the dissertation proposal, and serves as a forum for critique and guidance towards the successful completion of dissertation research. This exam is regulated by the university's Graduate School and permission from the department to take the exam is subject to the following requirements (for more detailed information on Candidacy see  https://gradsch.osu.edu/handbook/all#7-0 ). These concern the composition of the committee, the written, and the oral portion of the examination:

The committee consists of four regular faculty with graduate P-status, including the advisor of the candidate who serves as the chair. Other committee members can be from other Ohio State departments but have to have graduate P-status in their programs. Additional members, beyond these four, can be added by petition and according to Graduate School rules.  

The written portion consists of a 10-15 page dissertation proposal in which goals, scope, methods, and background of the planned research is outlined. The document has to contain mathematically rigorous statements, needs to be type-set along the usual publishing standards in the field (e.g., LaTeX), and should contain a substantial bibliography that includes all pertinent publications the intended research will be based on. The proposal of the written portion should be submitted to the committee at least ten days before the presentation and oral portion.

The candidate is required to describe his/her proposal in a short presentation of approximately 30 minutes to the committee immediately before the start of the oral portion of the examination. The details of the format are determined by the advisor, including whether the presentation should be public and questioning during the presentation.

Following the presentation there is a two-hour oral examination by the committee. This time has to be completely dedicated to the questioning by the committee and is not allowed to contain further presentations. The questions can focus on the proposal itself and the validity and relevance of the research questions, but can also include skill and knowledge examination of the needed mathematical background, as well as test familiarity with prior research. 

An application for candidacy must be submitted via gradforms.osu.edu at least three weeks before the oral examination. The candidacy examination can be taken at any time during business hours when the university is open -- including summer terms and breaks. The final date on which a candidacy exam can be counted as being within any given semester is the day before the first day of the following semester (for example, a "Spring" exam can be scheduled up until the day before Summer Term begins). All committee members' approval signatures must also be submitted in GradForms prior to the first day of the following term.

All pre-candidacy requirements of the Mathematics Department, as well as all credit and residency requirements of the Graduate School, have to be fulfilled by the end of the term prior to taking the exam. Candidates also need to be enrolled for at least 3 hours at the graduate level during the term of the exam (note: if you schedule your exam in summer, you will need to enroll in 4 total credit hours in order for your summer tuition waiver to apply). Foreign language classes do not count toward the 3 graduate credits required to take a candidacy examination.

Following the exam, the Report on Candidacy  form must be approved on GradForms by all committee members.

Post-Candidacy & Dissertation Research

After the Candidacy Exam , PhD students spend most of their time on research related to their dissertation, under the close supervision of their Dissertation Advisor .

There are some requirements during this time which PhD Candidates must abide by:

• Three-Hour Enrollment: Post-Candidacy students are expected to enroll for exactly 3 credit hours every Autumn and Spring semester. In most cases, this should be 3 credits of  MATH 8999 with their Dissertation Advisor . A 3 credit hour course may be substituted for the 8999 hours with permission of the advisor. Additional credit hours for enrollment are not included in the Graduate Associateship tuition waiver. A student may request to have tuition covered by the department for academically essential courses by a petition to the Graduate Studies Committee . In all other cases, tuition has to be paid for by the student or an external resource. Fellowship recipients will have different guidelines on this matter. 
• Continuous Enrollment: Students who have passed their candidacy examination are required to be enrolled during every Autumn and Spring Semester. There are only exceptions for Summer and formally petitioned  Leaves of Absence . For detailed rules on leaves, see Section VII.7 of the Graduate Handbook .

Final Defense & Graduation

How long one takes to graduate may vary greatly depending upon initial preparation, chosen specialization, difficulty and scope of the research problem, diligence of the candidate, results required to be competitive in the chosen area or job market, and other factors. Requirements to be eligible for graduation include:

• Time Limits: The university allows a maximum of five years from passing the candidacy examination until submission of the  final copy  of the  dissertation . The Math Department however, has the expectation that you can accomplish this in  three years or less . Continuation in the program is contingent on timely and satisfactory progress towards completing a dissertation as determined by the Graduate Studies Committee.

Credit Hours: Students are required to have accumulated 80 graduate credit hours of mathematics courses by the time of graduation. It is possible to substitute some of these with graduate credits from courses outside of the mathematics department, if approved by the advisor and the Graduate Studies Committee . In addition, university rules require that 50 of these credit hours have to be beyond the Master's degree .

Once the Dissertation Advisor deems the Dissertation Doctoral Draft  complete, the candidate needs to assemble a Final Oral Exam Committee . The committee consists of the  Dissertation Advisor  and two additional regular  category P level  faculty members, who will review the draft. The doctoral candidate must submit the  Application for Final Exam form via GradForms no later than  two weeks prior  to the proposed final oral examination date. The approval of the draft is followed by the two-hour   Final Oral Examination (dissertation defense) conducted before the dissertation committee members listed, plus a non-Math representative assigned by the Graduate School . See the  PhD Dissertations link on the Department website for samples of past approved Dissertations.

The department supports the search for academic jobs in several ways. Before graduation, the department provides travel support for students, helps with letters, and circulates job opportunities. After graduation, many former students with can find employment as lecturers with the department while they are looking for permanent jobs, if interim employment is needed.

Program Time Expectations

• The qualifying requirements should be fulfilled by the middle of the second year. The graduate studies committee may decide on conditional continuation at either regular or probationary level in close cases.
• International students should be classroom teaching certified by ESL's Spoken English Program by the beginning of the second year.
• Students are expected to pass the candidacy examination before the Autumn Semester of their fourth year.
• Students are expected to graduate by the end of their sixth year. In cases where this is not possible, the graduate studies committee can be petitioned for a seventh year of financial support at a reduced stipend level.

• Doctor of Philosophy in Mathematics (PhD)
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Mathematicians use theoretical and computational methods to solve a wide range of problems from the most abstract to the very applied. UBC's mathematics graduate students work in many branches of pure and applied mathematics. The PhD program trains students to operate as research mathematicians. The focus of the program is on substantial mathematical research leading to the PhD dissertation. Students also develop their skills in presenting and teaching mathematics and its applications.

For specific program requirements, please refer to the departmental program website

What makes the program unique?

UBC has one of the largest and most vigorous departments of mathematics in Canada. Our faculty routinely win national and international awards for their research and teaching achievements. We have an engaged and sociable cohort of graduate students who are essential members of a broad selection of active research groups. Each group holds a variety of seminars and events that allow graduate students, postdoctoral fellows, visitors and faculty to enjoy regular interaction.

UBC is the headquarters for the Pacific Institute of Mathematical Sciences (PIMS). PIMS hosts a plethora of mathematical events such as conferences and summer schools, greatly enriching the scientific environment in the quantitative sciences at UBC. Our mathematics students are also regular participants at the nearby Banff International Research Station for Mathematical Innovation and Discovery. Finally, our Institute for Applied Mathematics provides options for interdisciplinary studies for PhD students who wish to work in applied and computational mathematics.

I was intrigued and inspired by my professors and advisors to take on the program because of the collaborative aspects with Honeywell. This real-world focus motivated many interesting questions in my research.

Nathan Lawrence

Quick Facts

Program enquiries, admission information & requirements, 1) check eligibility, minimum academic requirements.

The Faculty of Graduate and Postdoctoral Studies establishes the minimum admission requirements common to all applicants, usually a minimum overall average in the B+ range (76% at UBC). The graduate program that you are applying to may have additional requirements. Please review the specific requirements for applicants with credentials from institutions in:

• Canada or the United States
• International countries other than the United States

Each program may set higher academic minimum requirements. Please review the program website carefully to understand the program requirements. Meeting the minimum requirements does not guarantee admission as it is a competitive process.

English Language Test

Applicants from a university outside Canada in which English is not the primary language of instruction must provide results of an English language proficiency examination as part of their application. Tests must have been taken within the last 24 months at the time of submission of your application.

Minimum requirements for the two most common English language proficiency tests to apply to this program are listed below:

TOEFL: Test of English as a Foreign Language - internet-based

Overall score requirement : 100

IELTS: International English Language Testing System

Overall score requirement : 7.0

Other Test Scores

Some programs require additional test scores such as the Graduate Record Examination (GRE) or the Graduate Management Test (GMAT). The requirements for this program are:

The GRE is not required.

2) Meet Deadlines

3) prepare application, transcripts.

All applicants have to submit transcripts from all past post-secondary study. Document submission requirements depend on whether your institution of study is within Canada or outside of Canada.

Letters of Reference

A minimum of three references are required for application to graduate programs at UBC. References should be requested from individuals who are prepared to provide a report on your academic ability and qualifications.

Statement of Interest

Many programs require a statement of interest , sometimes called a "statement of intent", "description of research interests" or something similar.

Supervision

Students in research-based programs usually require a faculty member to function as their thesis supervisor. Please follow the instructions provided by each program whether applicants should contact faculty members.

Instructions regarding thesis supervisor contact for Doctor of Philosophy in Mathematics (PhD)

Citizenship verification.

Permanent Residents of Canada must provide a clear photocopy of both sides of the Permanent Resident card.

4) Apply Online

All applicants must complete an online application form and pay the application fee to be considered for admission to UBC.

Tuition & Financial Support

Financial support.

Applicants to UBC have access to a variety of funding options, including merit-based (i.e. based on your academic performance) and need-based (i.e. based on your financial situation) opportunities.

Program Funding Packages

All full-time students who begin a UBC-Vancouver PhD Mathematics program in September 2018 or later will be provided with a funding package of at least $24,256 for each of the first four years of their PhD. The funding package may consist of any combination of internal or external awards, teaching-related work, research assistantships, and graduate academic assistantships.

Average Funding

• 52 students received Teaching Assistantships. Average TA funding based on 52 students was $13,784.
• 48 students received Research Assistantships. Average RA funding based on 48 students was $11,580.
• 3 students received Academic Assistantships. Average AA funding based on 3 students was $1,814.
• 54 students received internal awards. Average internal award funding based on 54 students was $13,279.
• 4 students received external awards. Average external award funding based on 4 students was $27,083.

Scholarships & awards (merit-based funding)

All applicants are encouraged to review the awards listing to identify potential opportunities to fund their graduate education. The database lists merit-based scholarships and awards and allows for filtering by various criteria, such as domestic vs. international or degree level.

Graduate Research Assistantships (GRA)

Many professors are able to provide Research Assistantships (GRA) from their research grants to support full-time graduate students studying under their supervision. The duties constitute part of the student's graduate degree requirements. A Graduate Research Assistantship is considered a form of fellowship for a period of graduate study and is therefore not covered by a collective agreement. Stipends vary widely, and are dependent on the field of study and the type of research grant from which the assistantship is being funded.

Graduate Teaching Assistantships (GTA)

Graduate programs may have Teaching Assistantships available for registered full-time graduate students. Full teaching assistantships involve 12 hours work per week in preparation, lecturing, or laboratory instruction although many graduate programs offer partial TA appointments at less than 12 hours per week. Teaching assistantship rates are set by collective bargaining between the University and the Teaching Assistants' Union .

Graduate Academic Assistantships (GAA)

Academic Assistantships are employment opportunities to perform work that is relevant to the university or to an individual faculty member, but not to support the student’s graduate research and thesis. Wages are considered regular earnings and when paid monthly, include vacation pay.

Financial aid (need-based funding)

Canadian and US applicants may qualify for governmental loans to finance their studies. Please review eligibility and types of loans .

All students may be able to access private sector or bank loans.

Foreign government scholarships

Many foreign governments provide support to their citizens in pursuing education abroad. International applicants should check the various governmental resources in their home country, such as the Department of Education, for available scholarships.

Working while studying

The possibility to pursue work to supplement income may depend on the demands the program has on students. It should be carefully weighed if work leads to prolonged program durations or whether work placements can be meaningfully embedded into a program.

International students enrolled as full-time students with a valid study permit can work on campus for unlimited hours and work off-campus for no more than 20 hours a week.

A good starting point to explore student jobs is the UBC Work Learn program or a Co-Op placement .

Tax credits and RRSP withdrawals

Students with taxable income in Canada may be able to claim federal or provincial tax credits.

Canadian residents with RRSP accounts may be able to use the Lifelong Learning Plan (LLP) which allows students to withdraw amounts from their registered retirement savings plan (RRSPs) to finance full-time training or education for themselves or their partner.

Please review Filing taxes in Canada on the student services website for more information.

Cost Estimator

Applicants have access to the cost estimator to develop a financial plan that takes into account various income sources and expenses.

Career Outcomes

88 students graduated between 2005 and 2013: 1 is in a non-salaried situation; for 19 we have no data (based on research conducted between Feb-May 2016). For the remaining 68 graduates:

Sample Employers in Higher Education

Sample employers outside higher education, sample job titles outside higher education, phd career outcome survey, career options.

A great majority of our PhD graduates move on to postdoctoral fellowships and faculty positions at universities and research institutes in North America and around the world. However, a significant fraction of students move into careers in industry. Students considering non-academic careers are encouraged to complete an industrial internship (for instance through the Mitacs Accelerate program - headquartered at UBC) during their studies.

Enrolment, Duration & Other Stats

These statistics show data for the Doctor of Philosophy in Mathematics (PhD). Data are separated for each degree program combination. You may view data for other degree options in the respective program profile.

ENROLMENT DATA

Completion rates & times, upcoming doctoral exams, monday, 27 may 2024 - 12:30pm - room 203, thursday, 30 may 2024 - 10:00am - 203, mathematics building, 1984 mathematics road, friday, 5 july 2024 - 9:00am - room 200.

• Research Supervisors

Advice and insights from UBC Faculty on reaching out to supervisors

These videos contain some general advice from faculty across UBC on finding and reaching out to a supervisor. They are not program specific.

This list shows faculty members with full supervisory privileges who are affiliated with this program. It is not a comprehensive list of all potential supervisors as faculty from other programs or faculty members without full supervisory privileges can request approvals to supervise graduate students in this program.

• Adem, Alejandro (Cohomology of finite groups, orbifolds, stringy topology, algebra, sporadic simple group, group actions, arithmetic groups, K-theory, homotopy theory, spaces of homomorphisms)
• Angel, Omer (Probability theory, percolation, random graphs, random walks, particle processes, scaling limits)
• Bachmann, Sven (Mathematics and statistics; Mathematical Analysis; quantum phenomena; Mathematical physics; Quantum statistical physics; Topological states of matter)
• Balmforth, Neil (Fluid mechanics, nonlinear dynamics and applied partial differential equations)
• Behrend, Kai (Moduli spaces, Gromov-Witten invariants, string theory, Donaldson-Thomas invariants, Euler characteristics, categorification)
• Bennett, Michael (Number Theory, Diophantine Approximation and Classical Analysis)
• Bryan, Jim (Algebraic and differential geometry; Algebraic geometry, moduli spaces, enumerative invariants related to theoretical physics.)
• Cautis, Sabin (Mathematics and statistics; Geometry)
• Chau, Albert (Differential Geometry and Partial Differential Equations)
• Chen, Jingyi (Algebraic and differential geometry; Differential Geometry, Partial Differential Equations)
• Colliander, James (hamiltonian dynamical systems; partial differential equations; harmonic analysis)
• Coombs, Daniel (Mathematical biology; Cellular immunology; Complex physical systems; Epidemiology (except nutritional and veterinary epidemiology); Cell Signaling and Infectious and Immune Diseases; Cell biophysics; Disease models; Epidemiology; Immune cell signalling; Mathematics)
• Cytrynbaum, Eric (Bacterial cell division, Microtubule and cellular organization, Wave propagation in excitable media)
• Dao Duc, Khanh (Genomics; Mathematical biology; Neurocognitive patterns and neural networks; Agricultural spatial analysis and modelling; combine mathematical,computational and statistical tools to study fundamental biological processes; regulation and determinants of gene expression and translation; Machine Learning for Biological Imaging and Microscopy; Database development and management; Biological and Artificial Neural Networks for geometric representation)
• Doebeli, Michael Walter (Mathematical ecology and evolution, evolution of diversity, adaptive speciation, evolution of cooperation, game theory, experimental evolution in microorganisms)
• Feng, James (Chemical engineering; Mathematics and statistics; Biophysics; Complex fluids; Fluid mechanics; Mathematical biology)
• Fraser, Ailana (Differential Geometry, Geometric Analysis)
• Friedlander, Michael (numerical optimization, numerical linear algebra, scientific computing, Scientific computing)
• Frigaard, Ian (Fluid mechanics (visco-plastic fluids))
• Ghioca, Dragos (Drinfeld modules, isotrivial semiabelian varieties, Lehmer inequality)
• Gordon, Julia Yulia (Representation theory of p-adic groups and motivic integration; Trace Formula and its applications)
• Gustafson, Stephen James (Mathematics and statistics; Mathematical Analysis; Differential Equation; Global and Non-Linear Analysis; Mathematical physics; Nonlinear partial differential equations; Nonlinear waves; Topological solitons)
• Hauert, Christoph (Mathematics and statistics; Modelization and Simulation; Evolution and Phylogenesis; Biological Behavior; dynamical systems; evolution; game theory; social dilemmas; stochastic processes)
• Hermon, Jonathan (probability theory; Markov chains and the cutoff phenomenon; particle systems; percolation)
• Holmes-Cerfon, Miranda (Mathematical modelling and simulation; Computational methods in statistics; Numerical analysis; Thermodynamics and statistical physics)

Doctoral Citations

Sample thesis submissions.

• Free boundary minimal submanifolds in geodesic balls of simply connected space forms
• On a completion of cohomological functors generalizing Tate cohomology
• Distribution of integral points on varieties
• Effective and explicit S-unit equations with many terms
• Classifying space for commutativity and unordered flag manifolds
• Finite-size scaling of a few statistical physics models in high dimensions
• Residual supersingular Iwasawa theory and μ-invariants for Zₚ²-extensions
• Numerical methods for biological flows laden with deformable capsules and solid particles
• The construction of blow-up solutions for some evolution equations
• Topics in discrete analysis
• Inviscid damping phenomena in some fluid models
• Gibbs measures and factor codes in symbolic dynamics
• Deep reinforcement learning agents for industrial control system design
• Structure-preserving numerical schemes for phase field models
• Enumerative geometry problems for Calabi-Yau manifolds with an action

Related Programs

Same specialization.

• Master of Science in Mathematics (MSc)

At the UBC Okanagan Campus

Further information, specialization.

Mathematicians use theoretical and computational methods to solve a wide range of problems from the most abstract to the very applied. UBC's mathematics graduate students work in many branches of pure and applied mathematics.

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Program website, faculty overview, academic unit, program identifier, classification, social media channels, supervisor search.

Departments/Programs may update graduate degree program details through the Faculty & Staff portal. To update contact details for application inquiries, please use this form .

Nicholas Richardson

Having grown up outside of Toronto and completed my undergrad and master's degree at the University of Waterloo, I was ready to change the scenery and go study somewhere else. I joke that is it the farthest I could move without leaving Canada, but more truthfully it was the campus that felt "right...

Gabriel Currier

I quite like the kind of math that people do here, and enjoy working with my supervisors. The campus is also a beautiful place and the graduate student community is pretty laid back and friendly.

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The reputation of the university and mathematics department, the alignment of my research interests with my advisor’s expertise, and my love for Canada!

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Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
• Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
• Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
• Complete a seminar, giving a talk of at least one-hour duration
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
• Meet the University residence requirement of two years or four semesters

Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members.  All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

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About the Department of Applied Mathematics and Theoretical Physics

The Department of Applied Mathematics and Theoretical Physics (DAMTP) is one of two Mathematics Departments at the University of Cambridge, the other being the Department of Pure Mathematics and Mathematical Statistics (DPMMS). The two Departments together constitute the Faculty of Mathematics , and are responsible for the teaching of Mathematics and its applications within the Mathematical Tripos.

5 courses offered in the Department of Applied Mathematics and Theoretical Physics

Applied mathematics and theoretical physics - phd.

This is a three to four-year research programme culminating in submission and examination of a thesis containing substantial original work. PhD students carry out their research under the guidance of a supervisor, and research projects are available from a wide range of subjects studied within the Department. Students admitted for a PhD will normally have completed preparatory study at a level comparable to the Cambridge Part III (MMath/MASt) course. A significant number of our PhD students secure post-doctoral positions at institutions around the world and become leading researchers in their fields.

More Information

Mathematics - MPhil - Closed

The MPhil is offered by the Faculty of Mathematics as a full-time period of research and introduces students to research skills and specialist knowledge. Its main aims are:

• to give students with relevant experience at first-degree level the opportunity to carry out focused research in the discipline under supervision; and
• to give students the opportunity to acquire or develop skills and expertise relevant to their research interests. 

Mathematics (Applied Mathematics) - MASt - Closed

This course is an application stream for the Master of Advanced Study (MASt) in Mathematics; students should apply to only one of the application streams for this course.

This course, commonly referred to as Part III, is a nine-month taught masters course in mathematics. It is excellent preparation for mathematical research and it is also a valuable course in mathematics and its applications for those who want further training before taking posts in industry, teaching, or research establishments.

Students admitted from outside Cambridge to Part III study towards the Master of Advanced Study (MASt). Students continuing from the Cambridge Mathematical Tripos for a fourth-year study towards the Master of Mathematics (MMath). The requirements and course structure for Part III are the same for all students irrespective of whether they are studying for the MASt or MMath degree, or whether they applied through the Applied Mathematics (MASA), Pure Mathematics (MASP), Mathematical Statistics (MASS), or Theoretical Physics (MASTH) application stream.

Mathematics (Theoretical Physics) - MASt - Closed

Quantitative climate and environmental science - mphil - closed.

The MPhil in Quantitative Climate and Environmental Sciences is a 10-month cross-departmental programme in the School of the Physical Sciences which aims to provide education of the highest quality in the analysis and modelling of Earth's climate and environment at a master’s level. The programme covers a range of skills required for the acquisition and assessment of laboratory and field data, and for the understanding through quantitative modelling of climate and environmental processes. 

6 courses also advertised in the Department of Applied Mathematics and Theoretical Physics

Antarctic studies - phd.

From the British Antarctic Survey

This PhD course takes place under the joint supervision of a research scientist at the British Antarctic Survey (BAS) and a University supervisor. Students may be based at BAS but will be registered for their degree with one of the partnering departments: Archaeology & Anthropology, Land Economy, Plant Sciences, Zoology, Earth Sciences, Geography and Scott Polar Research Institute, Applied Mathematics & Theoretical Physics, Chemistry, Engineering, Computer Science and Technology.

Biological Sciences BBSRC DTP - PhD - Closed

From the School of the Biological Sciences

The Cambridge Biosciences DTP is a four year fully-funded PhD programme that aims to create highly skilled and employable people. The programme offers training across 23 University Departments/Institutes and 3 Partner Institutes providing access to a wide range of research areas related to the strategic themes of the BBSRC. We offer three types of DTP studentships:

• DTP Standard

During the programme, DTP Standard and Targeted students will undertake two ten-week rotations in different labs before commencing their PhD. They will receive training in a variety of areas including but not limited to statistics, programming, ethics, data analysis, scientific writing and public engagement. Students will also undertake a 12-week internship (PIPS).

iCase students are not required to undertake rotations but may do so if they feel that this training would be useful. They must undertake a placement with their Industrial Partner for a minimum of three months and a maximum of 18 months.

Students will be expected to submit their thesis at the end of the fourth year.

Part-time study, whilst not the norm, may be viable, depending on the project, and will be considered on a case by case basis so please discuss this option with your proposed supervisor before making an application for this mode of study.

Computational Methods for Materials Science CDT - MPhil + PhD

From the Department of Physics

The development of new materials lies at the heart of many of the technological challenges we currently face, for example creating advanced materials for energy generation. Computational modelling plays an increasingly important role in the understanding, development and optimisation of new materials.

This four-year doctoral training programme on computational methods for material modelling aims to train scientists not only in the use of existing modelling methods but also in the underlying computational and mathematical techniques. This will allow students to develop and enhance existing methods, for instance by introducing new capabilities and functionalities, and also to create innovative new software tools for materials modelling in industrial and academic research.

The first year of the doctoral training programme is provided by the existing MPhil course in Scientific Computing, which has research and taught elements, as well as additional training elements. The final three years consist of a PhD research project, with a student-led choice of projects offered by researchers closely associated with the CDT. ( https://ljc.group.cam.ac.uk ) 

Data Intensive Science - MPhil

The MPhil in Data Intensive Science is a 10-month cross-departmental programme in the School of the Physical Sciences which aims to provide education of the highest quality at the master’s level. The programme covers the full range of skills required for modern data-driven science. The course covers material from the fields of machine learning and AI, statistical data analysis, research and high performance computing, and the application of these topics to scientific research frontiers.  

The course structure has been designed in collaboration with our leading researchers and industrial partners to provide students with the theoretical knowledge, practical experience, and transferable skills required to undertake world-leading data-intensive scientific research. Students will gain the broad set of skills required for scientific data analysis, covering traditional statistical techniques as well as modern machine learning approaches.  Both the theoretical underpinnings and practical implementation of these techniques will be taught, with the later aspect including training on software development best practice and the principles of Open Science. The course also aims to provide students with direct experience applying these methods to current research problems in specific scientific fields. Students who have completed the course will be equipped to undertake research on data-intensive scientific projects. Beyond academic disciplines, students will be well prepared for a career as a data science professional in a broad range of commercial sectors.

This course will equip students with all the skills required for modern scientific data analysis, enabling them to participate in large experimental or observational programmes using the latest statistical and machine learning tools deployed on leading-edge computer architectures. These computational and statistical skills will also be directly applicable to data-driven problem-solving in industry.

Planetary Science and Life in the Universe - MPhil

From the Institute of Astronomy

The MPhil in Planetary Sciences and Life in the Universe is a 10-month cross-departmental programme designed to deliver outstanding postgraduate level training in the search for life’s origins on Earth and its discovery on planets beyond Earth.

The course structure has been designed by leading scientists to provide students with the theoretical knowledge, practical experience, and transferable skills required to undertake world-leading research in Planetary Sciences and Life in the Universe. Graduating students will be equipped with the discipline specific-specialisations and skills of a masters course, whilst gaining understanding in how the core areas that bridge PSLU fields form the cross-disciplinary foundation of this exciting new frontier.

Graduates of the course will gain valuable skills rooted in the study of the physics, chemistry, mathematics, and biology of planetary science and life in the universe. Transferrable skills training is delivered through the three group-based projects running over the year: these provide a unique opportunity for students to gain experience of leadership, collaboration, and written and oral communication.  The training provided will be an outstanding foundation for PhD research in planetary science, exoplanetary science, Earth system science, planetary astrophysics, astrobiology and allied disciplines, or for the wide range of careers where analytical skills, excellent communication, and experience of leading collaborations are key.

Scientific Computing - MPhil

The MPhil programme in Scientific Computing provides world-class education on high performance computing and advanced algorithms for numerical simulation at continuum and atomic-scale levels. The course trains early-career scientists in the use of existing computational software and in the underlying components of the simulation pipeline, from mathematical models of physical systems and advanced numerical algorithms for their discretisation, to object-oriented programming and methods for high-performance computing for deployment in contemporary massively parallel computers.  As a result, course graduates have rigorous research skills and are formidably well-equipped to proceed to doctoral research or directly into employment. The highly transferable skills in algorithm development and high-performance computing make our graduates extremely employable in all sectors of industry, commerce and finance.

The MPhil in Scientific Computing is suitable for graduates from any discipline of natural sciences, technology or engineering, who have good mathematical and computational skills.  

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Professor colm-cille caulfield head of department.

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• General Relativity and Cosmology
• High Energy Physics
• Quantum Information
• Mathematical Biology
• Fluid and Solid Mechanics
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• Applied and Computational Analysis

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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.

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Research Programmes

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The Faculty of Mathematics offers three doctoral (PhD) and one MPhil research programmes.

Select a course below to visit the University’s Course Directory where you can read about the structure of the programmes, fees and maintenance costs, entry requirements and key deadlines.

Research Areas and Potential Supervisors

Determining whether your interests and ambitions align with our research and expertise is a vital part of the application and admissions process. When we receive your formal application, we will consider the information you provide on your research interests carefully, alongside other factors such as your academic suitability and potential, how you compare to other applicants in the field, and whether we have a suitable academic supervisor with the capacity to take on new students.

We are committed to widening participation in mathematical research at Cambridge. We welcome and encourage applications from people from groups underrepresented in postgraduate study.

Before making an application to study with us we recommend you:

• Investigate our areas of research and consider how they fit with your interests and ambitions.

A list of broad research areas is provided below, together with links to further information. Your interests may span more than one area. On your application form you will be asked to indicate at least one broad area of interest. This is to help us direct your application to the most suitable group of people to review it.

• Identify 2 or 3 appropriate supervisor(s) with whom you might work.

The information linked below will take you to lists of supervisors working in each broad research area, with an indication of their availability. You are encouraged to make informal contact with potential supervisors prior to making an application. Initial contact should be made by email. In your email we recommend you provide a concise explanation of your areas of interest, how your research interests align with the supervisor(s) research, and that you highlight any relevant work you have done in this area. We recommend that you attach an up-to-date CV. The purpose of this contact is to enquire on supervisor capacity and willingness to supervise, and to see if there is a good fit between your interests and theirs.

If you haven’t had a response to an informal enquiry, you are still welcome to apply and list the individual concerned on your application form, although you may also wish to consider other options.

• Give some thought to your intended research and why you want to study with us.

On your application form you will be asked to submit a short research summary, details of your research experience and your reasons for applying to undertake a PhD/MPhil with us. Whilst you are not expected to submit a detailed research proposal at any stage of the process, we do want to know that you have considered the areas of research that you wish to pursue.

Research areas

Click on a research area to find out more about available supervisors and their research:

Please note that a  large majority of the successful applicants for PhD studentships with  the High Energy Physics, and General Relativity & Cosmology (GR) groups   will have taken Part III of the Mathematical Tripos.

Funding Opportunities

Each Department works hard to secure funding for as many offer holders as possible, either from within its own funds, in collaboration with funding partners, or via the University Postgraduate Funding Competition. However, funding is not guaranteed via these routes, and you should investigate funding opportunities early in the process to be sure that you can meet advertised deadlines.

All application deadlines are 23:59pm (midnight) UK time on the stated date. So that your application can be given full consideration please apply by the following deadlines:

Note for PhD applicants:

We will accept applications for an October start up until the general University deadline in May, but your chances of obtaining funding are significantly reduced. In addition, space limitations may mean that late applications cannot be considered (i.e., the most appropriate supervisor may already have committed to taking other students).

Only in exceptional circumstances will we consider admission to a later start date in the academic year (i.e., January or April). If you intend to apply for a later start date please contact us at [email protected] so we can advise you on the feasibility of your plan.

Note for MPhil applicants:

We will accept applications until the general University deadline in February, but you will not be considered for funding. In addition, space limitations may mean that late applications cannot be considered (i.e., the most appropriate supervisor may already have committed to taking other students).

Most interviews are expected to take place in the second half of January.

The purpose of the interview is to try to ascertain the extent of the applicant's relevant knowledge and experience, and to gauge whether their interests and abilities align with the research of the potential supervisor and/or research group. It will most likely consist of a discussion of your background and motivations for applying to the course, as well as some questions on relevant topics.

Not all applicants will be selected for interview.

If you are selected for interview, you will be contacted by email at the address you provided on your application. The email should confirm:

• the location of the interview (it may be in-person or on-line dependent upon interviewer availability, your distance from Cambridge, as well as individual preferences),
• the interview format and whether you should prepare anything specific in advance,
• the approximate duration of the interview,
• who you will be meeting.

Prior to interview you may declare a disability, serious health problem or caring responsibility which may require reasonable adjustments for the interview to be made.

Due to interviewer availability and the tight admissions timetable, we can usually only rearrange the time and date of your interview under exceptional circumstances.

Decision timeline

Both DAMTP and DPMMS make most of their PhD/MPhil admissions decisions for October entry in January and early February, and you should not expect to receive a decision on your application before mid-February (even if you apply much earlier). We expect to have made decisions on all applications by mid-July. The Department makes every effort to take decisions on applications at the earliest opportunity. In some cases, however, it may take some time for a decision to be made. Applications may need to be viewed by several potential supervisors before a final decision can be reached.

To consider your application formally we must receive a complete application form, together with all supporting documents, by the deadline.

Communication of outcomes

You will be notified of the formal outcome of your application via the Applicant Portal.

Following an interview, you can normally expect to receive notification of the outcome within a week or two.

If you are successful, the University’s Postgraduate Admissions Office will issue a formal offer of admission which will outline all your conditions. As processing times can vary, we may also contact you informally to notify you of our decision.

We do not provide formal feedback to applicants who are unsuccessful at either the application or interview stage.

Take a look at our frequently asked questions for PhD applicants.

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A graduate degree in mathematics can help students hone their skills in a specialty area, from algebra and number theory to discrete mathematics and combinatorics. These are the best graduate-level math programs. Each school's score reflects its average rating on a scale from 1 (marginal) to 5 (outstanding), based on a survey of academics at peer institutions. Read the methodology »

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Mathematical Physics Ph.D. Degree

Doctor of philosophy (ph.d.) in mathematical physics.

This program offers advanced graduate training in the overlapping areas of mathematics, theoretical physics, and their applications from a unified point of view and promotes research in this field.

General supervision of the program is controlled by the Interdepartmental Graduate Committee on Mathematical Physics. While no master’s degree is offered, you may qualify for a master’s degree in mathematics or physics during the course of study. Our students usually enter the program at the beginning of the second year of graduate study in Mathematics or Physics.

 Learn more in our Student Portal Learn how to apply

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PhD in Mathematical Sciences

When you pursue a PhD in Mathematical Sciences at WPI, you join a vibrant community of faculty members, postdoctoral researchers, and students who are creating new knowledge and applying their expertise to solve to complex, real-world problems.

Value Proposition Description

We accept about five new doctoral students in our PhD in mathematical sciences each year, all of whom are fully funded, and maintain a team of about 25 PhD candidates. You will be encouraged to collaborate with researchers in the department and across campus as you make important discoveries in an area of interest, from theoretical mathematics to applications in diverse areas like data science, electrical and computer engineering, bioinformatics and computational biology, and biomedical engineering.

Our flexible curriculum enables you to tailor programs of research and study to your professional goals. You will select courses in mathematical sciences and other disciplines and may choose to complete a PhD project with an external sponsor that allows you to connect your theoretical knowledge with relevant applications. We also offer professional development training to enhance your skills in teaching, grant writing, interviewing, and the job search process.

As you pursue your PhD in mathematical sciences, you will be expected to pass two general comprehensive exams by the end of your first year of study and a series of preliminary exams before registering for your dissertation research.

Research for PhD in Mathematical Sciences

Research in our PhD in mathematical sciences program at WPI plays a vital role in solving complex problems facing our world. You will work alongside faculty and student researchers who are involved in a wealth of diverse projects, both fundamental and applied, in these areas:

• Discrete mathematics, combinatorics, and graph theory
• Analysis and differential equations
• Computational mathematics
• Statistics and stochastic analysis

Our strong interdisciplinary culture also offers exciting possibilities to interact with researchers in other disciplines, including physics, engineering, and the life sciences.

In addition, as a PhD in mathematical sciences student, you will have access to a modern and diverse line-up of computing facilities and industry-standard software programs, including SAS, R, MATLAB, Python, and Turing, our high-performance computer cluster.

Faculty Profiles

The evolution of defects in materials present very interesting mathematical challenges.  My research focuses on improving mathematical models for material defects and advancing mathematical methods for studying them. Of particular interest are the growing lower-dimensional surfaces found in fracture mechanics.  There are many open questions here, and my projects involve postdocs, graduate students, and undergraduates, as well as other mathematicians and collaborators from other fields.

I am a mathematical physicist working on the development and application of mathematical methods to atmospheric research and satellites orbits. As part of this research, I am also developing new methods for the use of symmetry principles to solve differential equations. I have taught a broad spectrum of applied math courses on the undergraduate and graduate levels.

Research is a core part of the undergraduate experience at WPI. As Associate Dean of Undergraduate Studies, Prof. Suzanne Weekes increases the focus on the undergraduate research enterprise at the university to continue to advance WPI’s mission to create, to discover, and to convey knowledge at the frontiers of academic inquiry for the betterment of society.

Bill Martin's goal is to find mathematical research projects that lie between beautiful and powerful mathematical theory, on the one hand, and pressing technological applications, on the other. This effort requires one to keep abreast of both mathematical developments and applications in computer science and engineering. Professor Martin's mathematical research is in the area of algebraic combinatorics, where tools from linear and abstract algebra are applied to problems in discrete math.

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Looking for More of a Statistics Focus? Earn a PhD in Statistics Instead.

Maybe you’re interested in specializing in a statistics PhD rather than mathematical sciences? Our PhD in statistics challenges students to use critical thinking and problem solving to identify patterns in big data. Choose from six statistics and mathematics courses ranging from time series analysis to theoretic probability theory when you pursue a PhD in statistics at WPI.

Do You Need Your Master’s First? Explore Our Mathematics MS Programs.

Whether you have a passion for a specific area of mathematics like finance or prefer a broader scope of study, our mathematics master’s programs can help you gain the knowledge you need to get ahead. If you’re looking for a broad background in mathematics, our master’s in applied mathematics may be a good fit for you. Maybe you’re interested in how to solve complex problems as it pertains to business and industry? A master’s in industrial mathematics is the way to go. This MS program provides students the business background needed to tackle real-world industry problems. Do you have a gift for working with big data and tackling challenges? Be sure to explore our master’s in applied statistics , which dives into crucial statistical methods with the flexibility to select courses that align with your career aspirations. Are you interested in becoming a financial analyst who assesses data findings to steer business decisions? Our master’s in financial mathematics will provide a solid understanding in computational methods and credit risk modeling.

Just Starting to Carve Out Your College Career? Explore a BS in Mathematical Sciences.

If you’re an incoming undergrad interested in mastering mathematical theory to solve real-world problems, our BS in mathematical sciences could be the degree for you. Our bachelor’s explores pure mathematics, applied mathematics, and statistics to give you a breadth of expertise. 

Theoretical Computer Science

This field comprises two sub-fields: the theory of algorithms, which involves the design and analysis of computational procedures; and complexity theory, which involves efforts to prove that no efficient algorithms exist in certain cases, and which investigates the classification system for computational tasks. Time, memory, randomness and parallelism are typical measures of computational effort.

Theoretical computer science is a natural bridge between mathematics and computer science, and both fields have benefited from the connection. The field is very active, with exciting breakthroughs and intriguing challenges. The P =? NP problem is one of the seven of the Clay Millennium Problems. The recent polynomial time primality algorithm received a Clay Math research award.

MIT has been the leading center for theoretical computer science for several decades. A strong group of EECS Department faculty also works in this field and runs joint activities with the Mathematics faculty through CSAIL. The RSA cryptosystem and Akamai Technologies are two important success stories that were developed by Mathematics and EECS Department faculty.

Our group investigates active areas such as quantum computation, approximation algorithms, algorithms in number theory, distributed computing and complexity theory.

Department Members in This Field

• Michel Goemans Algorithms, Combinatorics & Optimization
• Jonathan Kelner Theoretical Computer Science
• Tom Leighton Theoretical Computer Science, Combinatorics
• Ankur Moitra Theoretical Computer Science, Machine Learning
• Elchanan Mossel Probability, Algorithms and Inference
• Peter Shor Quantum Computation, Quantum Information
• Michael Sipser Algorithms, Complexity Theory

Instructors & Postdocs

• Mitali Bafna Theoretical Computer Science
• Omri Ben-Eliezer Algorithms and theoretical computer science
• Manik Dhar Combinatorics, Theoretical Computer Science
• Jason Gaitonde Algorithms, Learning Theory, Probability Theory, Networks
• Alexander Poremba Quantum computation, cryptography
• Yihui Quek Quantum Computing, Complexity Theory, Quantum Noise and Error Correction
• Michael Simkin Probabilistic combinatorics, random graphs, and random processes
• Anirudh Sridhar Statistical inference, network cascades, graph algorithms, graph matching

Graduate Students*

• Anna Brandenberger
• Andrey Khesin Quantum Computing
• Jonathan Lu
• Yuchong Pan
• Jonathan Rodriguez Figueroa
• Xinyu (Norah) Tan Quantum computing, quantum information, coding theory
• Neekon Vafa
• Lichen Zhang Theoretical computer science, machine learning and data science
• Kai Zhe Zheng

*Only a partial list of graduate students

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Research degrees

Our internationally renowned department fosters an innovative and intimate research environment. We welcome PhD students and post-doctoral fellows from around the world working on a broad range of topics. Postgraduate research students can study for their degree in any of the Department's research themes .

Our researchers also work in interdisciplinary research collaborations within King’s including with computer scientists, physicists and researchers in the Faculty of Life Sciences and Medicine . They also work externally with industrial partners such as EDF research, the Pensions Policy Institute and Unilever, as well as with the Francis Crick Institute. 

Applied mathematics research: disordered systems/financial mathematics/probability.

Applied Mathematics Research: Disorder Systems/Financial Mathematics/Probability MPhil / PhD from the Department of Mathematics at King's College London

View course

Applied Mathematics Research: Theoretical Physics

Applied Mathematics Research: Theoretical Physics MPhil / PhD from the Department of Mathematics at King's College London.The Theoretical Physics Group in the Department of Mathematics is at the international forefront of research and offers PhD's in string and M-theory, black holes, conformal field theory, supersymmetry, integrability, and other fundamental branches of modern theoretical physics.

Pure Mathematics Research

Pure Mathematics MPhil / PhD from the Department of Mathematics at King's College London.

Statistics Research

Statistics PhD from the Department of Mathematics at King's College London.

Applying for a PhD

The Department of Mathematics welcomes applications and research proposals in any areas of interest in the Department of Mathematics, which can be explored through the  Department’s research groups . Prospective students are also encouraged to explore potential supervisors and their research areas. 

Applicants should hold, or expect to gain, a first or upper second class degree in mathematics or a relevant related subject, and be exceptionally motivated for research.

We offer PhD projects under a range of different research themes for the majority of our postgraduate degrees. Applicants can explore the various projects available and apply for them. However, the list is not exhaustive and potential applicants can alternatively identify and contact appropriate potential supervisors to outline their academic background and research interests or to propose their own PhD project ideas.

There are a number of funding schemes available. Some cover both stipend (to cover living costs) and tuition fees, while others may cover fees only. The main funding opportunities for the Department are listed below. More options are listed on the King's  postgraduate research funding pages  or FindAPhD.com .

We suggest prospective students apply by 1 February to ensure full consideration for our available funding opportunities. All scholarships, bursaries or other awards are offered on a competitive basis. Some King's funding opportunities may have earlier deadlines, such as the King's-China Scholarship programme.

• Faculty studentships

The Faculty of Natural, Mathematical & Engineering Sciences offers studentships in Mathematics, funded for 3.5 years, with a bursary starting at the standard research council rate, and which will cover the full cost of Home and overseas tuition fees.

• ESPRC Doctoral Training Grant and EPSRC Mathematical Sciences Doctoral Training Partnership Studentships

All eligible PhD applicants are automatically considered for these awards when they apply to the Department, no separate application form is required. All academic staff working in the mathematical sciences are eligible to supervise these projects; preference might be given to supervisors who are early-career researchers.

• Science & Technology Facilities Council 

All applicants to the PhD in Theoretical Physics are automatically considered for these awards when they apply to the Department of Mathematics, no separate application form is required. Eligibility criteria may apply.

• London School of Geometry and Number Theory 

This Centre for Doctoral Training has a separate application process, and is run jointly by King's, UCL and Imperial College London. Students can also apply to the Pure Mathematics PhD programme at King's, f ind out more  on their website . 

Centre for Doctoral Training in Digital Twins for Healthcare

DT4Health is an innovative PhD programme that cuts across the Health Faculties and the Faculty of Natural, Mathematical & Engineering Sciences. The programme offers postgraduate researchers fully funded positions with the aim of training the next generation of leaders in healthcare technology to improve healthcare systems using the cutting-edge framework of Digital Twins. Applicants can  find out more on the website . 

• Centre for Doctoral Training on Multiscale Models for Life

This cross disciplinary CDT based in the Faculty of Dentistry, Oral & Craniofacial Science offers support for 3.5 years including a stipend at the current UKRI rate, home rate tuition fees, research expenses and support for training and career enhancement.

It is open to those with an interest and aptitude for interdisciplinary research with a background in life sciences and physics, chemistry, maths, computation or engineering. 

• King's-China Scholarship Council

King's-China Scholarship Council PhD Scholarship programme (K-CSC) is open to students from China. Details of this programme can be found on our  website .

For further information on postgraduate research funding and scholarships please visit the  Centre for Doctoral Studies pages .

Applications should be be submitted via the  King’s College London online application portal.  When applying, please select the degree programme you are applying to and mention your intended Supervisor and their project in the 'Research Proposal' area.

Any written submissions required by the supervisor/department/research group should be submitted in one document. If you are applying to a specific research group, or if your supervisor belongs to a specific research group, you must include the name of the research group at the top of this document.

• Find out more
• London School of Geometry & Number Theory

This four-year PhD programme comprises a largely taught first year followed by a three-year research project, with approximately 50 supervisors available across Imperial College London, King's College London & University College London.

Centres for Doctoral Training

DT4Health is an innovative PhD program located at King's, in the heart of London.

Multiscale Models for Life (MM4L) Centre for Doctoral Training

Training future research leaders adept at bridging the gap between in vivo life sciences and…

NMES Graduate School

The Graduate School in the Faculty of Natural, Mathematical & Engineering Sciences is home to all PhD students studying in the Faculty. From training and funding opportunities, to career development and campus events, it offers a range of services to support the needs and interests of our graduate students, supporting them to achieve their academic and professional goals.

Courses for PhD students

We participate in the London Graduate School in Mathematical Finance, a consortium of the mathematical finance groups of Birkbeck, Brunel, Cass Business School, Imperial, King's, LSE and UCL. It provides a programme of advanced courses in mathematical finance. We also participate in the London Taught Course Centre which offers advanced courses for PhD students in mathematics, including several courses that are relevant for research in analysis and statistics.

• London Taught Course Centre

Discover our research

Discover our research groups in the Department of Mathematics.

Net Zero Centre

Fostering an interdisciplinary environment for research in net zero.

Centre for the Physical Science of Life

Transforming our understanding of life through the innovative power of physical science

Find out more about the Department

Find out more about the Department of Mathematics.

Discover our research in the Department of Mathematics.

Meet the Department of Mathematics, King's College London.

Dissertations

Most Harvard PhD dissertations from 2012 forward are available online in DASH , Harvard’s central open-access repository and are linked below. Many older dissertations can be found on ProQuest Dissertation and Theses Search which many university libraries subscribe to.

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Ph.D. programme in Mathematical Sciences

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Programme overview

The mission of this PhD programme is to combine specialised theoretical training in the Mathematical Sciences with the application of methodological approaches from Mathematics and Statistics to the treatment of problems from the applied sciences.

The goal is to train highly qualified scholars capable of conducting academic research in fundamental sciences, engineering, architecture, economics, finance, biomedicine, or industrial research in bio- and nanotechnologies, the pharmaceutical industry, healthcare, and governance.

The PhD programme in Mathematical Sciences rests on two pillars: a high-level mathematical education is fundamental to tackle theoretical and applied problems with the proper methodological rigour; Mathematics plays a prominent role in avant-garde technological contexts, which are a source of challenging interdisciplinary problems of relevant interest to the Mathematical Sciences.

Training and research activities span the following main subject areas:

ALGEBRA AND GEOMETRY

• Cryptography and number theory. Cryptography and Number Theory, focusing on Diophantine equations, continued fractions, linear recurrent sequences, group actions, pseudo-random number generators, digital signatures, cryptanalysis, and cryptographic applications to blockchain.
• Geometry and topology. The study of Geometry and Topology from an algebraic, differential, computational, and applied point of view, with a special focus on projective varieties, tensors, vector bundles, theory of subvarieties, geometric flows and geometry of PDEs, and topological aspects of artificial intelligence.

MATHEMATICAL ANALYSIS

• Dynamical system analysis. The analysis and control of network dynamics, with applications to engineering, economic, financial, and biological systems; the study of multi-agent systems of players with strategies: well-posedness results, mean-field limits, and their stochastic formulation.
• Harmonic and functional analysis. The study of functional inequalities, singular integrals and functional calculi for Laplacians and sub-Laplacians on manifolds, groups, and graphs.
• Variational models and PDEs. The study of the variational properties of nonlinear quantum graphs and hybrid with a geometric structure, in order to model quantum devices for applications in atomtronics and other quantum technologies; the study of uncertainty principles for Fourier-like transforms arising in harmonic and complex analysis and mathematical physics, from a measure-theoretic and variational perspective; the regularity theory for solutions to weighted parabolic PDEs and applications; the study of variational models for continuum mechanics, with particular applications to membranes and thin structures.

NUMERICAL ANALYSIS

• PDEs in unbounded domains. Innovative and efficient numerical methodologies for solving PDEs in unbounded domains, focusing on the coupling of boundary and virtual element methods and its application to wave propagation problems.
• Simulation of multi-scale coupled problems. Treatment of coupled multi-scale problems in complex domains through innovative domain decomposition techniques and polygonal/polyhedral discretizations.
• Numerical methods for highly complex PDEs. Strategies for enhancing fast numerical solution and optimization of highly complex models governed by PDEs: Physics Informed Neural Networks, Model Order Reduction, and High Performance Computing.

MATHEMATICAL PHYSICS

• Mathematical methods for multi-agent systems. Agent-based models – stochastic particle systems and their continuum limits, Monte Carlo simulation algorithms; continuum models – differential equations, integro-differential equations, collisional kinetic equations, qualitative characterization of their solutions, hydrodynamic limits, asymptotic descriptions, numerical simulations; applications to biology, ecology, medicine, econophysics, sociophysics.
• Continuum mechanics. Mathematical models in biology and medicine; Fluid mechanics, Solid mechanics, Biomechanics, Mechanobiology, Morphoelasticity; micro-swimmers and their control; applications in biomedicine (e.g. tumour growth, angiogenesis, cell migration, mechanical response of soft tissues); variational methods and differential geometry in continuum mechanics; nonholonomic mechanical systems and their control; mathematical modelling of metamaterials, gradient materials, and materials of fractional order; mathematical and mechanical models of cell migration; biomimetics models in particle swarm optimization.
• Nonequilibrium statistical mechanics. Models of anomalous transport of matter, energy etc.; perturbative and exact response theory for systems subjected to perturbations, which are described by dynamical systems or by stochastic processes; applications to physics (e.g. universality and phase transitions away from equilibrium), to biology (e.g. population dynamics, including bacteria), to nanotechnology (e.g. sensors and transport in nanotubes), to climate and environmental problems (e.g. characterization of the Earth temperature in space and time).

PROBABILITY AND STATISTICS, OPERATIONAL RESEARCH

• Probability. Stochastic processes, dependence models and their applications to biology and finance.  Applications to biology focus on  stochastic reaction networks, on their stationary regime, on multi-scale settings, and on model extensions driven by biochemistry ,   while in finance applications focus both on multi-asset derivative pricing and on the modelling and stochastic comparison of portfolios having dependent assets.
• Statistics. Methodological statistics and its fundamental role in data collection, analysis, and interpretation. Statistics as one of the pillars of data science and machine learning. Hierarchical modeling and Bayesian statistics for capturing complex relationships and dependencies among variables. Application in biology, spatial statistics, genetics, clinical trials, design of experiments.
• Operations research. Development of optimization models and solution algorithms for a wide range of applications, including marketing, logistics, production scheduling, and engineering design.  The applied methodologies include: stochastic and robust optimization; dynamic programming, reinforcement learning, and simulation-based optimization; matheuristics for combinatorial optimization, possibly integrated with machine learning.

Thanks to the interplay between theoretical, foundational, and applied aspects of the research in Mathematics and Statistics, the PhD programme in Mathematical Sciences provides a training specifically focused on the creation of innovative research approaches. These are essential to adequately support innovation and technological transfer processes, which are more and more demanding in terms of original theoretical and methodological strategies to manage the increasing complexity of socio-economical paradigms.

Key information

Type of programme:, department:, coordinator:.

TOSIN ANDREA

Vice coordinator:

BERCHIO ELVISE

Admissions:

Titolo contacts, titolo student office.