Combinatorics & Optimization

Admission Requirements

It is essential that the application for admission into the PhD program contains evidence of research ability or potential.

The minimum requirements for admission to the PhD program normally include:

• a Master's degree in combinatorics and optimization, or in a closely related field, with a minimum A average in Master's level coursework;
• completion of a Master's thesis;
• three letters of reference, normally from academic sources; proof of competency in English (if applicable).
• A score of at least 580 is required in the Test of English as a Foreign Language (TOEFL). (See Academic Regulations - English Language Proficiency Certification for other acceptable tests of English.)

A student who is enrolled in the MMath program in this department and wishes to continue in the PhD program has to apply for admission into the program.

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 Departmental Graduate Committee. PhD students' progress will be reviewed at least once per year.

The normal period of registration for the PhD degree is six terms from a Master's degree. (See the Academic Regulations - Residence Requirements.) One year of credit may be granted by the Faculty Graduate Committee for work done towards the PhD degree at another institution, provided that the relevance of the previous work to the student's proposed program is clearly established.

Transfer from MMath to PhD 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 candidate has been enrolled in the MMath (Thesis) program for at least two terms.
• The candidate 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 candidate 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.

Applicants for 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.

Degree Requirements

Course Requirements:   

CO 630, CO 642, CO 650, CO 663, CO 681, CO 685

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 course work 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.

Research Skills Seminar - Milestone

Required for the PhD students in C&O unless the student satisfied this requirement as a MMath student at the C&O department.

Comprehensive Examination Requirement

There are two stages in the comprehensive examination. The first stage is a written examination 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 first stage of the comprehensives must be taken within four terms of the student's first registration in the PhD program. The Comprehensive Examination Requirement is satisfied by passing both examinations.

The second stage comprehensive 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. A handout giving more details is available from the Graduate Secretary. This exam should normally be taken within one year of completing the first-stage of the comprehensives. It must be taken at least one full term before the PhD defence is scheduled.

Advisory Committees

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 second stage comprehensive examination, 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.

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 candidate may not put the thesis on display until at least the term following that in which the Lecturing Requirement was successfully completed.

Thesis Requirement

The candidate 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.