York University: Graduate Program in Mathematics & Statistics

PhD in Mathematics & Statistics

Admissions Requirements

See the section on General Admission Requirements. For admission as a PhD 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.

Applications are considered by the PhD program Committee, which makes its recommendations to the Graduate program Director. The Director will then make a recommendation to the Faculty of Graduate Studies.

Current Master’s students can apply internally by submitting an informal application to the Graduate Program Director indicating the proposed area of study, and, if possible, their proposed dissertation supervisor. The application should also include three official letters of recommendation. There is no application fee.

Degree Requirements

There are four components of the degree requirements for a PhD These are:

course requirement, followed by comprehensive examinations;
requirement in the field of specialization of the student, followed by an oral examination;
a dissertation (thesis), followed by two oral examinations on the dissertation (Program Dissertation Colloquium, and Dissertation Defense);
language requirements.

Course Requirement* and Comprehensive Examinations

Students must successfully complete 24.0 credits (four full courses or equivalent) at the graduate level. The courses must be chosen with the approval of the Program Director. Previous graduate work may be used to meet at most 12.0 credits of this requirement.

Students will declare a specialization in pure mathematics or applied mathematics or statistics, and write comprehensive examinations in subjects which are appropriate to the chosen specialization. In addition, statistics students will complete a statistical consulting requirement.

* Continuing students must petition to the Faculty of Graduate Studies through the Graduate Program in Mathematics and Statistics to be considered under the new regulations for the “Course Requirement” and “Specialization Requirement”.

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:

Analysis [Any two of Complex Analysis (Math 6170), Measure Theory (Math 6280), Functional Analysis I (Math 6461)].
Algebra [Modern Algebra (Math 6120)].
Topology and Geometry [Any two of Group Theory and Geometry (Math 6202), General Topology I (Math 6540), Algebraic Topology I (Math 6550)].
Probability Theory [Stochastic Processes (Math 6602), Probability Models (Math 6604)].
Differential Equations [Ordinary Differential Equations (Math 6340), Partial Differential Equations (Math 6350)].
Numerical Analysis [Advanced Numerical Methods (Math 6651), Numerical Solutions to Partial Differential Equations (Math 6652)].
Mathematical Statistics [Mathematical Statistics (Math 6620), Advanced Mathematical Statistics (Math 6621)].
Statistical Methods [Generalized Linear Models (Math 6622), Applied Statistics I (Math 6630)].
Mathematical Methods [Mathematical Modeling (Math 6931)].

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

Pure mathematics students have two options:
(a) Complete topics 1 and 2. The third topic must be chosen from 3-8, or topic 9 plus one other course chosen from topics 3-8; or (b) Complete topics 1, 4, and 7.

Applied mathematics students must complete three topics, which must include at least two chosen from topics 1,5,6. The third topic must be chosen from 1‑8, or topic 9 plus one other course chosen from topics 1‑8.

Statistics students must complete topics 7 and 8. In addition, statistics students must fulfill a practicum requirement. This requirements consists of two parts. The first part is the completion of MATH 6627 3.0 or an equivalent consulting course from another university, approved by the Graduate Program Director. The second part is the comprehensive exam in consulting as described in the guidelines available in the Graduate Program Office (N519 Ross).

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 PhD program, 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.

Part-time students will have to pass at least 6 credits per year, and will have to complete the comprehensive exams by the end of their third year of enrolment.

Students are required to consult with the Program 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. Comprehensive exams will be closed book in-class exams. Students who are not enrolled in a course but elect to take a comprehensive exam should contact the instructor regarding the time and place of the exam. All comprehensive exams are submitted to the Ph.D. Committee for evaluation.

Specialization Requirement and Dissertation Subject Oral

Students in the doctoral Program must demonstrate depth of knowledge in their field of specialization. The candidate must pass an oral examination (Dissertation Subject Oral), which will occur within one year after the comprehensive examinations have been passed. 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. All members of the student’s Supervisory Committee must be present. Members of the graduate Program 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 be required.

Upon the successful completion of the examination, the tentative 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 with the Faculty of Graduate Studies Thesis and Dissertation Proposal Form (http://www.yorku.ca/grads/policies/ethics.htm), 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 student’s supervisory committee, a description of the dissertation, and a bibliography.

Guidelines for the Preparation and Examination of Dissertations are available at: www.yorku.ca/grads

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). This can be found on the web at:
http://www.pre.ethics.gc.ca/eng/policy‑politique/tcps‑eptc/

Course credits: A student will not receive credit for more than one full integrated course to satisfy thecourse and specialization requirements 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

Dissertation Colloquium
Upon completion of work on the dissertation, the Supervisory Committee, in consultation with the candidate, will set a date (at least 4 weeks prior to the oral) 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 Program 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. Supervisory committee members must be present.

Dissertation Oral Examination
An oral examination (30 minutes presentation and 2 hour question and answer period) 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 Graduate Program Director will recommend the membership of the Examining Committee to the Faculty of Graduate Studies. This recommendation must be accompanied by documentation (forms are available in N519 Ross) as listed in the “Guidelines”, must reach the Faculty of Graduate Studies Office not less 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 Program must demonstrate the ability to read mathematical text in one language other than English. The choice of language must be approved by the student’s 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
student’s dissertation.

Progress Report

During the second year of registration and once a year thereafter, all students enrolled in a PhD program 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.

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 PhD 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.

Supervisory Committees

Upon admission to the doctoral Program, each student will be assigned a tentative supervisor from the Graduate Program. The assignment will be made by the Ph.D. Program Committee. The student will decide upon a study plan in consultation with the tentative supervisor.

Dissertation Supervisory Committee

When a student has successfully written the comprehensive examinations, the tentative supervisor in consultation with the student, will appoint a supervisory committee to be approved by the Ph.D. Program Committee. The student will decide upon a continuing Program of study in consultation with the supervisory committee. A Dissertation Supervisory Committee shall be recommended by the Graduate Program Director to the Dean of Graduate Studies after the student has successfully taken the Dissertation Subject Oral, in accordance with faculty regulations.

A supervisor must be recommended by the graduate Program director for approval by the Dean of Graduate Studies no later than the end of the fifth term of study (end of second term of PhD II). Students will not be able to register in the seventh term of study (the onset of PhD III) unless a supervisor has been approved.

A supervisory committee must be recommended by the appropriate graduate Program director for approval by the Dean of Graduate Studies no later than the end of the eighth term of study (end of second term of PhD III). Students will not be able to register in the tenth term of study (the onset of PhD IV) unless a supervisory committee has been approved.

Dissertation Examining Committee

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

At the final defense, the student will give an oral presentation to the Program (20 minutes presentation and 1½ hours question & answer period), and defend it before an examining committee. In addition to Faculty regulations regarding thesis examination, the thesis candidate gives two talks in a student Colloquium, one outlining work in progress and one presenting the final results. This is done pior to the formal defense.

It is the responsibility of the student/supervisor to ensure that all degree requirements are met.