PhD Program

The Department of Pure Mathematics is one of five separate and unique departments in the Faculty of Mathematics. The department offers graduate programs leading to the degrees Master's of Mathematics (MMath) and Doctor of Philosophy (PhD).

The PhD program consists of course work, comprehensive examinations and a thesis embodying results of original research. From a Master's degree, it is expected that the PhD program will be completed within four years (12 terms). Continued funding is conditional on being in good academic standing and satisfactory performance of teaching assistant duties.

PhD program admission requirements

To be considered for admission into the PhD program, applicants normally require a Master's degree in Mathematics with a minimum B+ (78%) average. Exceptions may be made for students with an Honours Bachelor degree who demonstrate a very high level of background preparation and research potential.

Specific PhD admission requirements are as follows:

  • a Master's degree (or equivalent) with at least a B+ standing (78%);

  • three letters of reference which attest to the student's research potential. We recommend these be from academic sources;

  • a one page personal statement;

  • proof of competency in English (if applicable). A Test of English as a Foreign Language (TOEFL) (or equivalent) score of at least 580 (paper-based) and 4.0 on the TWE, 237 (computer-based) and 4.0 on the TWE, or 92-93 (Internet-based) with a minimum score of at least 25 in the Speaking and Writing sections. (See English Language Proficiency in the Graduate Studies Calendar for other acceptable tests.)

The deadline for applications to the PhD program is January 15 for all applicants. However, we often begin making offers in late December, so getting your application in by mid-December is recommended.  Please refer to the application section of this website for more information.

Admission to the PhD program is highly competitive and many more students apply than are admitted. Those admitted usually significantly exceed the minimum admission requirement.

PhD degree requirements

Course requirement

This program requires a minimum of four term graduate courses (with unit weight of .50) for those entering the PhD program with a Master's Degree. The candidate must obtain an average of at least 70% in these courses. Three of the required courses must be Pure mathematics (PMATH) graduate courses numbered in the 800's and 900's. None of the four required courses can be joint graduate/undergraduate courses or reading courses. Up to three course credits may be granted by the Graduate Committee for work completed towards the PhD degree at another institution provided that the relevance of the previous work to the student's proposed program is clearly established.

Those entering the PhD program with a Bachelor's degree normally must also satisfy the course requirements of an MMath degree in addition to those of the PhD program.

PhD comprehensive examination 1:

Satisfactory performance in two written comprehensive examinations is required; one in algebra and one in analysis and topology.

The syllabus is based on the material covered in the University of Waterloo's third and fourth-year undergraduate courses. The Graduate Committee normally offers the analysis/topology exam each May and the algebra exam each January. Normally students must attempt both exams within one year of their registration in the PhD program, and both exams must be successfully completed within two years.

Outlines of topics, references and previous comprehensive examinations can be downloaded here.

PhD comprehensive examination 2

Satisfactory performance in the oral comprehensive examination is required. The oral comprehensive examination demonstrates an in-depth knowledge in an area chosen by the candidate and his or her supervisor, usually the general area in which the thesis will be written. It should be normally completed within two years of registration in the PhD program. At most two attempts are permitted. There must be at least two examiners, including the supervisor and another expert in the area of the exam, normally a faculty member or postdoctoral fellow from a university mathematics department. The examining committee and the area to be examined must be approved by the Graduate Officer.

PhD lecturing requirement

Regular participation in a at least one departmental seminar is expected. In addition, the candidate must present at least two talks in a departmental seminar. Signatures from “faculty members in attendance” may be obtained from any appointed Professor (including Adjunct professors) or any Postdoctoral Fellow in the Department of Pure Mathematics.

PhD thesis requirement

A thesis embodying the results of original research. This is the most important requirement!

The thesis must be of a standard that warrants publication in the research literature of the field. The thesis must be acceptable to a committee approved by the Graduate Committee consisting of the student's supervisor and four other professors; one of whom must be from another department, and one of whom must be an independent external examiner familiar with the student's research field. The student is required to defend the thesis at an oral examination.

Thesis information