Doctor of Philosophy (PhD) in Combinatorics and Optimization

The program information below was valid for the spring 2023 term (May 1, 2023 - August 31, 2023). This is the archived version; the most up-to-date program information is available through the current Graduate Studies Academic Calendar.

The Graduate Studies Academic Calendar is updated 3 times per year, at the start of each academic term (January 1, May 1, September 1). Graduate Studies Academic Calendars from previous terms can be found in the archives.

• Admit term(s) 
  • Fall
  • Winter
  • Spring
• Delivery mode 
  • On-campus
• Length of program 
  • Students are required to complete the program in accordance with the University program time limits.
• Program type 
  • Doctoral
  • Research
• Registration option(s) 
  • Full-time
  • Part-time
• Study option(s) 
  • Thesis

• Minimum requirements 
  • A Master's degree in combinatorics and optimization, or in a closely related field, with a minimum 89% average in Master's level coursework.
  • Completion of a master's thesis.
  • It is essential that the application for admission into the PhD program contains evidence of research ability or potential.
  • Students in the PhD program are regarded as being on probation during their first year in the Department, and their performance during this first year determines whether they are allowed to continue in the program. In particular, failure in any one course, or an unsatisfactory performance in the comprehensive examination, automatically results in a review of the student's progress by the Department Graduate Committee. PhD students' progress will be reviewed at least once per year.
  • A student who is enrolled in the Master of Mathematics (MMath) program in the Department of Combinatorics and Optimization and wishes to continue in the PhD program has to apply for admission into the program. In exceptional cases, a graduate student enrolled in a MMath (Thesis) program in the Department of Combinatorics and Optimization may, through the Graduate Officer and with the consent of the Supervisor, petition the Graduate Committee to be transferred into a PhD program. The guidelines for such a transfer are as follows:
    • The student has been enrolled in the MMath (Thesis) program for at least two terms.
    • The student has made considerable progress in the research project (of the type that would warrant the MMath degree) and is committed to carrying the project to completion in a PhD program.
    • The student gives a seminar presentation of the work carried out so far, and answers related questions to the satisfaction of an examining committee consisting of the supervisor and two other faculty members.
  • Students applying to the PhD program who hold a Master's degree from another university may, in some cases, be admitted initially into the MMath program. In such cases the Graduate Committee will decide, within three terms, whether to transfer the student into the PhD program.
  • Applicants educated outside of Canada, including those from the United States, are recommended to submit scores from the Graduate Record Examinations (GRE) subject test in mathematics. Neither the GRE general test nor the GRE subject test is required.
• Application materials 
  • Supplementary information form
  • Transcript(s)
• References 
  • Number of references:  3
  • Type of references: 

    normally from academic sources.

• English language proficiency (ELP) (if applicable)

Thesis option:

Graduate Academic Integrity Module (Graduate AIM)

Courses 

• Students must complete 8 courses, including 4 core courses and at least 1 other CO course. The remaining 3 courses must be graduate courses in the Faculty of Mathematics, or courses approved by the CO Graduate Committee.
• At least 6 courses should normally be completed within the first 6 terms.
• Core courses:
  • CO 630 Algebraic Enumeration
  • CO 642 Graph Theory
  • CO 650 Combinatorial Optimization
  • CO 663 Convex Optimization and Analysis
  • CO 681 Quantum Information Processing
  • CO 685 The Mathematics of Public-Key Cryptography
• If students have credit for a course deemed equivalent to a particular core course by the Department Graduate Committee, then that part of the core requirement may be waived.
• The Department may require additional coursework in cases where this is judged to be necessary; for instance, when a student is admitted to the PhD program without having been granted credit for a Master's degree.

Link(s) to courses

• Combinatorics and Optimization (CO) courses

Graduate Studies Research Skills Seminar

• Required for PhD students unless the student satisfied this requirement as a MMath student at the Department of Combinatorics and Optimization.

PhD Lecturing Requirement

• Every PhD student will be required to lecture under supervision during the program of studies. If a PhD student gives a scheduled course on a regular basis, the same two faculty members will attend three of the lectures and make a confidential, constructive critique of the student's performance to the student.
• The PhD Lecturing Requirement should normally be completed within the first eight terms of the student’s PhD program. Students may not put their thesis on display until at least the term following that in which the Lecturing Requirement was successfully completed.

PhD Comprehensive Examination

• This requirement consists of 2 written examinations covering the fundamentals of combinatorics and optimization. These are usually offered once a year, in the spring term. The student must write one exam from two of the following three categories:
  • Combinatorial Enumeration, Graph Theory
  • Continuous Optimization, Discrete Optimization
  • Cryptography, Quantum Computing
• The choice of exams is made by the student, in consultation with their supervisor. The exams must be taken within the first four terms of the student’s PhD program.
• The PhD Comprehensive Examination requirement is satisfied by passing both examinations.

PhD Thesis Proposal

• The PhD Thesis Proposal is an oral exam at which the student is expected to give a brief description of the questions they propose to work on for the PhD and a summary of the main results in this area. This exam should normally be taken within the first six terms of the student’s PhD program. The student should provide a short written version of their thesis proposal to their committee one week before the oral presentation. The PhD Thesis Proposal requirement is satisfied by successful completion of this exam.
• Advisory Committee: each student has an Advisory Committee, which normally consists of the student's supervisor and two other department members with expertise in the area of the student's research interests. The Advisory Committee acts as the examining committee at the student's PhD Thesis Proposal, and is usually formed at this time. The members of the advisory committee will also usually act as examiners at the student's PhD defence. The Advisory Committee is selected by the Graduate Officer, who will consult the student and their supervisor.

PhD Thesis

• Students must prepare a thesis, embodying the results of original research, of a standard that would warrant publication in a research journal of the field. The thesis must be acceptable to the student's supervisor, to two professors in the Department and one professor outside the Department, and to an external examiner familiar with the student's research field. The student is required to defend the thesis at an oral examination. This requirement is met when the thesis has been successfully defended and accepted.