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The Doctoral Programme

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(i)  Admissions Requirements

See the section on General Admission Requirements. For admission as a Ph.D. student, students must have completed an acceptable Master's degree or must have completed one year of comparable work, with a B+ average (high second class) or better. The admission process is very selective and not all students meeting this requirement will be admitted.

Applicants should obtain at least three letters of recommendation by academics who know them well. They should provide a list of upper-level undergraduate and graduate courses in mathematics and statistics, including the syllabus, the name and author of the textbook used, and the name of the instructor.

Students who are currently enrolled as Master's students at York and who wish to enter the Ph.D. programme should arrange for the three letters of recommendation and submit a letter indicating their desire to enter the programme, their proposed field of specialization, and, if possible, their proposed dissertation supervisor.

Applications are considered by the Ph.D. Programme Committee, which makes its recom-mendations to the Graduate Programme Director.  The Director will then make a recommendation to the Faculty of Graduate Studies.

(ii)   Degree Requirements

There are four components of the degree requirements for a Ph.D. These are:

(a) breadth requirements, followed by comprehensive examinations;

(b) depth requirements in the field of specialization of the student, followed by an oral examination;

(c) a dissertation (thesis), followed by two oral examinations on the dissertation (Programme Dissertation Colloquium, and Dissertation Defense);

(d) language requirements.

Breadth Requirement and Comprehensive Exam-inations

Students in the doctoral programme must demonstrate that they have breadth of knowledge in mathematics/statistics outside their field of specialization.  This breadth must be achieved by taking courses from a wide range of fields according to the following specific requirements, and must be demonstrated by passing comprehensive examinations.

To satisfy the breadth requirement, the student must take 24 credits (the equivalent of four full courses) from TABLE 1 (page 11).  A student who plans to specialize in Foundations or Algebra and Geometry must take at least 9 credits from Section A, a student who plans to specialize in Analysis must take at least 9 credits from Section B, and a student who plans to specialize in Probability or Statistical Theory must take at least 9 credits from Section C. Furthermore, a student who is not specializing in Statistical Theory may apply no more than 12 credits from any one of the three Sections A,B,C toward the fulfilment of the breadth requirement. A student specializing in Statistical Theory must take at least 6 credits outside Section C and at least 9 credits from Section C.

A student may have taken graduate courses prior to admission to the programme.  Advanced standing for up to 24 such credits may be recommended by the Director of the Graduate Programme (in consultation with the Ph.D. Programme Committee) to the Faculty of Graduate Studies for approval, if they are deemed equivalent to courses from TABLE 1.

A doctoral candidate must satisfy their comprehensive exam requirement by completing a certain set of comprehensive topics. Each such topic will be completed by satisfactory performance on a set of final examinations in certain prescribed courses. Students need not enrol in the course nor attend lectures in order to write the exam for comprehensive credit. The comprehensive topics, and the course examinations needed for each are as follows:

1. Analysis [Any two of Complex Analysis (Math 6170), Measure Theory (Math 6280), Functional Analysis I (Math 6461)].

2. Algebra [Modern Algebra (Math 6120)].

3. Topology and Geometry [Any two of Group Theory and Geometry (Math 6202), General Topology I (Math 6540), Algebraic Topology I (Math 6550)].

4. Probability Theory [Stochastic Processes (Math 6602), Probability Models (Math 6604)].

5. Differential Equations [Ordinary Differential Equations (Math 6340), Partial Differential Equations (Math 6350)].

6. Numerical Analysis [Advanced Numerical Methods (Math 6651), Numerical Solutions to Partial Differential Equations (Math 6652)].

7. Mathematical Statistics [Mathematical Statistics (Math 6620), Advanced Mathematical Statistics (Math 6621)].

8. Statistical Methods [Generalized Linear Models (Math 6622), Applied Statistics I (Math 6630)].

Candidates must declare themselves to be in one of these three streams: applied mathematics, pure mathematics, or statistics streams.

Pure mathematics students must complete topics 1, 2, and any one of the remaining ones.

Applied mathematics students must complete three topics, which must include at least two chosen from topics 1, 5, 6.

Statistics students must complete topics 7, 8. In addition, statistics students must fulfil a practicum requirement. They will act as a statistical consultant for three or four small projects and prepare a report for the clients, prepare and lead discussions in the statistical consulting service regular meetings; complete a project involving the analysis of a substantial real data problem. Part, but not all of this requirement will be met by completing Math 6627.

All of the above named courses will count as 3 credits, except for Modern Algebra (Math 6120) which will count as 6 credits. During the first year of enrolment in the Ph.D. programme, pure and applied students will have to pass sufficient credits of exams so that they have no more than 9 credits of remaining exams to pass in order to complete the requirements. Statistics students will have to pass either 9 credits of exams or 6 credits of exams plus the practicum. All students will then have to complete the comprehensive exams by the end of their second year of enrolment.

Students are required to consult with the Programme Director to make their course and exam selections. In certain extreme cases of difficulty due to scheduling, the Ph.D. Committee will designate certain other courses as substitutes, arrange for reading courses, or modify the timing requirements. All comprehensive exams are submitted to the Ph.D. Committee for evaluation.

Depth Requirement and Dissertation Subject Oral

Students in the doctoral programme must achieve and demonstrate depth of knowledge in their field of specialization.  To satisfy the depth requirement the student must take at least 18 credits in the chosen field of specialization.  The student's tentative supervisory committee will propose the courses which constitute these 18 credits, and this proposal must be approved by the Graduate Programme Director. Normally, no student will take more than 9 credits of these courses from any one faculty member. Some examples of courses which might be used to fulfill this requirement are given in Table 2 (page 11). Courses from TABLE 1 may be acceptable if they were not used to satisfy the breadth requirement and if they belong to the student's field of specialization.

The candidate's depth of knowledge will be demonstrated during an oral examination (Dissertation Subject Oral).  This will usually occur within one year of passing the comprehensive examinations.  In preparation for this examination, the student shall, in consultation with the tentative supervisory committee, decide on a dissertation subject and a syllabus of materials.  The syllabus of materials shall consist of those theoretical results, techniques, examples, etc. in the student's area which are deemed most likely by the tentative supervisory committee to be useful in research on the dissertation subject.

The tentative supervisory committee must approve the dissertation subject and agree that a command of the syllabus of materials will enable the student to pursue original research in that subject.  A date for the examination will be set by the tentative supervisory committee in consultation with the candidate.

The Dissertation Subject Oral shall consist of a 20-minute oral presentation of the dissertation subject and a question period, up to one hour in length.  Members of the graduate programme may attend the examination and may ask questions on the presentation or on the syllabus of materials.

At the end of the question period, the tentative supervisory committee shall judge the examination as successful or unsuccessful.  In the latter case, the student may try again after additional study.  If a student decides to change the dissertation subject then an examination in the new subject will normally be required.

Upon the successful completion of the examination, the Dissertation Supervisory Committee will recommend approval of the candidate's research proposal to the Faculty of Graduate Studies. Dissertation proposals must be forwarded for approval to the Dean of Graduate Studies not less than six months prior to the date set for the oral examination of the completed dissertation. The Dissertation proposal shall consist of a listing of the studentís supervisory committee, a description of the dissertation, and a bibliography.

A copy of the Guidelines for the Preparation and Examination of Dissertations is available in N519 Ross or the Faculty of Graduate Studies Office, 283 York Lanes.

Graduate students doing Dissertations, in which research involving human participants occurs shall familiarize themselves with York University's policies about the use of human participants and with the SSHRC/NSERC/MRC Tri-Council Policy Statement "Ethical Conduct for Research involving Humans" (August, 1998). These can be found at the Office of Research Administration, S414 Ross Building or on the web at

http://www.sshrc.ca/english/programinfo/policies/Intro01.htm.

All Dissertation proposals must be submitted to the Faculty of Graduate Studies with a copy of the Faculty of Graduate Studies Human Participants Research Form attached. The Faculty of Graduate Studies Human Participants Research form is available in N519 Ross.

Course credits: A student will not receive credit for more than one full integrated course to satisfy the breadth and depth requirement towards the Ph.D. degree. Students may not take or receive credit for an integrated course at the graduate level if they took it at York or elsewhere at the undergraduate level.

Dissertation Evaluation

1.  Dissertation Colloquium

Upon completion of work on the dissertation, the Supervisory Committee, in consultation with the candidate, will set a date for a preliminary examination thereof (Dissertation Colloquium).

The examination will consist of an oral presentation of the dissertation, of at most one hour's duration, and a question period, up to one hour in length.  Members of the Graduate Programme in Mathematics and Statistics may attend the examination and may ask questions related to the candidate's dissertation.  At the end of the question period the Supervisory Committee shall judge the examination.  In the case of failure, a detailed rationale must be given to the candidate.  The candidate may repeat the examination, but only after an interval of at least one month.

2.  Dissertation Oral Examination

An oral examination on the candidate's dissertation will be conducted according to Faculty regulations.  See "Guidelines for Preparation and Examination of Theses and Dissertations" for details. The Gradauate Programme Director will recommend the membership of the Examining Committee to the Faculty of Gradaute Studies. This recommendation must be accompanied by one copy of the dissertation, and along with other documentation as listed in the "Guidelines", must reach the Faculty of Graduate Studies Office not less than than four weeks before the date set for the oral. This deadline is strictly enforced by the Faculty of Graduate Studies.

Faculty members and graduate students may attend the oral examination.  They may, at the discretion of the Chair of the Examining Committee, participate in the questioning, but only members of the Examining Committee may be present for the evaluation and for the vote at the conclusion of the examination.

Language Requirements

A candidate in the doctoral programme must demonstrate the ability to read mathematical text in one language other than English. The choice of language must be approved by the studentís supervisory committee.

The language should be a language in which a significant amount of research of mathematics is published, particularly in fields related to that of the studentís dissertation.

(iii)  Residence Requirements

There is no formal residence requirement in the Faculty of Graduate Studies, but full-time students are normally required to pay full-time fees for two years.

Full-time students may not be absent from the campus without the permission of the Director for more than four weeks of any term in which they are registered.

(iv) Progress Report

During the second year of registration and once a year thereafter, all students enrolled in a Ph.D. programme are required to complete an annual research progress report detailing the achievements of the previous year and the objectives for the next year. Permission to continue to register in the program depends on a satisfactory report. Report on Progress forms are available in N519 Ross.

(v)  Deadlines for Meeting Requirements

Students are normally expected to take most of their comprehensive examinations in their first year and are required to complete the exams by May of their second year of Ph.D. studies. The Dissertation Subject Oral should be taken within one year of passing the comprehensive examinations. The Dissertation itself should be completed within two years of the Disseration Subject Oral, although one additional year may be allowed by permission.

(vi)  Supervisory Committees

Student's Advisory Committee

Upon admission to the doctoral programme, each student  will    be   assigned   three  advisors  from  the

Graduate Programme.  The assignment will be made by the Ph.D. Programme Committee in consultation with the student.  If the student has already decided on a field of specialization, then at least two of the advisors will be chosen from that field.  Otherwise, at least one advisor will be chosen from each field in which the student indicates interest.  The student will decide upon a study plan in consultation with the advisors.

Dissertation Supervisory Committee

When a student has successfully written the comprehensive examinations, the Ph.D. Programme Committee, in consultation with the student, will appoint a tentative supervisory committee.  This committee will consist of at least three members of the Graduate Programme, at least two of whom are in the student's field of specialization and at least one of whom is not.  The student will decide upon a continuing programme of study in consultation with the tentative supervisory committee.  A Dissertation Supervisory Committee shall be recommended by the Graduate Programme Director to the Dean of Graduate Studies after the student has successfully taken the Dissertation Subject Oral, in accordance with faculty regulations.

Dissertation Examining Committee

A Dissertation Examining Committee will be appointed according to Faculty regulations.