# Degree Requirements

In order to obtain a graduate degree in Mathematics, a student must have completed courses in the following topics at the senior undergraduate level or above (note that these requirements do not apply to Institute of Applied Mathematics students):

- Real Analysis (including metric spaces, Lebesgue measure and Integration)
- Real Analysis (Math 420 and 421, or Math 507 and 510)

- Complex Analysis
- Complex Analysis (Math 440 or 508)

- Abstract Algebra
- Abstract Algebra (Math 422 and 423, or Math 501 and 502)

- A mathematical topic different from those above
- Applied Analysis (Math 400 and 401)
- Probability and Stochastic Processes (Math 418 and 419)
- Theory of Differential Equations (Math 416 and 417)
- Differential Geometry (Math 424 and 425)
- Algebraic Topology (Math 426 and 427)
- Calculations of Variations and Optimal Control (Math 402 and 403)

Most students will have completed these four requirements before starting their graduate programs. Students admitted in spite of a gap in their background will be required to address it during the first year of the program.

To complete the Master's (M.Sc.) program, a student must . . .

- Satisfy the
**prerequisites**described above. - Earn at least 30 credits from MATH courses numbered 400 or higher, at most 6 credits of which can come from MATH courses numbered 400-499. These 30 credits must include either 3 credits for MATH 589 (M.Sc. Major Essay) or 6 credits for MATH 549 (Thesis for Master's Degree).

Mathematics students pursuing a Master's degree through the Institute of Applied Mathematics face slightly different requirements: details are provided on the IAM web page.

Credit may be given for equivalent courses taken before the student started a graduate degree program at UBC, provided those courses were not counted toward another degree.

Students may also request that credit be given for appropriate courses in departments at UBC other than Mathematics.

To complete the PhD program, a student must . . .

- Satisfy the
**prerequisites**described above. - Complete
**30 credits**of approved graduate coursework. (Credits from an M.Sc. program, either UBC or equivalent, can be counted towards this requirement. Mathematics courses numbered 399 and below are not eligible.) - Pass the
**qualifying exam**within two years of starting the program. (See below.) - Pass a
**candidacy examination**in the student's field of specialization. The examination syllabus is to be determined by the student's Ph.D. committee with the approval of the Graduate Affairs Committee. Passing this examination gains a student admission to candidacy. - Write a thesis and defend it in a public oral examination administered by the Faculty of Graduate Studies.

**Registration requirements:**

A full-time Ph.D. student must register in at least 12 credits of course work in the first year of their graduate program at UBC if they have not completed a M.Sc. degree or if they were admitted directly to Ph.D. after completion of B.Sc. degree. After being admitted to candidacy, a student working on a thesis may register for Math 649 with the approval of the thesis supervisor.

All PhD students are expected to take at least four graduate-level courses, totaling at least twelve credits, at UBC as part of their program. Courses counted towards the 30 credits of approved coursework (#2 above) can also be counted towards this requirement, if taken at UBC as a Master's student. These courses cannot be reading courses; however, seminar courses with credit will be considered separately as they become available, to determine whether they can be counted in the requirement.

To clarify, if you completed your MSc (30 credits) at UBC, you will not be required to take additional courses for your PhD program. However, your supervisor may recommend specific courses, and students affiliated with the IAM (see below) may need additional courses to satisfy the breadth requirements. IAM students are to follow the requirements listed in the IAM graduate student handbook.

Every doctoral student at UBC must have, by mandate of the Faculty of Graduate Studies, a supervisory committee of at least three members: the research supervisor and two others, typically UBC faculty with a rank of Assistant Professor, Associate Professor, or Professor. PhD students in Mathematics are expected to have a committee in place by the end of the first year of their doctoral program at UBC. The committee is responsible for guiding the student's course selection and research program, for administering the candidacy examination and the final doctoral examination, and for providing other support and advice as needed. You can read about the role of PhD supervisory committees in more detail.

**IAM Students:** for mathematics students in a program sponsored by the Institute of Applied Mathematics (IAM), please instead follow the requirements in the IAM Graduate Student Handbook at the link https://www.iam.ubc.ca/graduate-studies/graduate-student-handbook/.

Students are expected to attend the weekly mathematics colloquium. Mathematics students registered with the IAM are expected to attend the Applied Mathematics colloquia.

The Qualifying Examinations ("Quals") are written tests on undergraduate material. They are offered twice a year:

- on the Tuesday immediately after Labour Day, in early September; (*note: Sept. 2022 exams will be held on Saturday, Sept. 10)
- on the Saturday after classes begin in January.

Precise dates and times for examination sessions will be posted well in advance on the Events page.

The Quals are designed to:

- help students integrate their knowledge of advanced undergraduate mathematics,
- give students and supervisors an early baseline indication of preparedness,
- guide students' course selections in cases where a weakness is identified,
- allow well-prepared students to move into research more rapidly,
- allay potential concerns regarding TA assignments,
- define initial standards clearly and uniformly for prospective and new students.

All PhD students in the Mathematics Department, including those in the IAM, must pass the Quals before their second year ends. (For example, for a PhD Student who started in September 2014, the last chance to write the exam will be January 2016.) Doctoral students are strongly encouraged to attempt the Quals immediately upon their arrival at UBC; those who do not are required to participate in the next scheduled sitting.

MSc students are welcome to take the Quals as well, although it is not an official part of the Master's degree requirements. The supervisor of each MSc student can communicate their personal policies concerning the Quals. Note that a student who passes the Quals during their Master's degree and who is admitted into the PhD program in this department will have already satisfied the PhD Quals requirement.

Both MSc and PhD Students may attempt the Quals any number of times, subject to the deadline for PhD students of passing it before their second year ends.

The Quals consist of three different exams:

- Analysis, which covers real and complex analysis
- Algebra, which covers abstract algebra and linear algebra
- Differential Equations, which covers differential equations and linear algebra

To pass the Quals, a student must achieve a passing grade in two of these three exams, including a passing grade in Analysis. Typically, students in the IAM will take the Analysis and Differential Equations exams, while non-IAM students will take the Analysis and Algebra exams; however, the choice of exams is up to each student.

Each exam lasts three hours. The Analysis exam runs from 9 AM to noon of the Saturday on which the Quals are scheduled; both the Algebra and Differential Equations exams run from 1 PM to 4 PM on the same day. A student may attempt either one or two exams on a given day as they wish; each exam can be passed independently of the others. Students must inform the Graduate Program Coordinator well ahead of time (at least two weeks before the August/September Quals, and before the Christmas break for the January Quals) which exams they plan to attempt in a given sitting, so that the proper exam papers will be present.

Each exam will consist of six problems, split roughly evenly between the subjects covered by the exam. There will usually be significant overlap between the linear algebra problems on the Algebra and Differential Equations exams in any given sitting.

The only two possible marks are "pass" and "fail". The passing mark is usually set at a score of 60%. The Quals committee will make every effort to notify the students and their supervisors of the results within one week of writing the exams. The exam papers themselves can be reviewed with a member of the Quals committee. Exam papers are destroyed after two years.

The Qualifying Examinations resources page contains the official Qualifying Examinations syllabi, as well as past exams and sample problems.

The Candidacy Examination is a mandatory formal event to assess the student's readiness to undertake research at the doctoral level. It probes the student's knowledge, problem-solving, and communication skills through mathematical writing, oral presentation, and interactive discussion.

Students are required to complete their Candidacy Examination in the first 25.5 months of their PhD program. Any requests for extensions must be submitted in writing to the Graduate Chair prior to this deadline; extensions are contingent upon an acceptable reason for delay. By FoGS rule, students who do not advance to Candidacy within 36 months will be required to withdraw from the program. For further information on Candidacy, please refer to the Graduate site.

For mathematics students in a program sponsored by the Institute of Applied Mathematics (IAM) , the IAM's Thesis Proposal Examination replaces this requirement.

Here you can read about the Candidacy Examination format and expectations for Math students in more detail.

General information about Candidacy Exams from the Faculty of Graduate Studies is provided at Candidacy Examination format and expectations.