The Doctoral Programme (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 upperlevel 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 recommendations 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:
Breadth Requirement and Comprehensive Examinations 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 written comprehensive examinations. To satisfy the breadth requirement, the student must take 24 credits (the equivalent of four full courses) from TABLE 1 (page 10). 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 will be granted by the Director of the Graduate Programme (in consultation with the Ph.D. Programme Committee), if they are deemed equivalent to courses from TABLE 1. A doctoral candidate must pass written comprehensive examinations. These will be given twice a year. The fall session is in late September or early October. The spring session is in late April or early May. Students will have at most two attempts. Detailed lists of topics are available from N519 Ross. Candidates must declare themselves to be in either the applied mathematics, pure mathematics, or statistics streams. Applied mathematics students will write three four hour exams. At least two of these must be chosen from analysis; numerical analysis; and differential equations. The third can be chosen from any of the comprehensive exams given by the programme. Pure mathematics students will write three four hour examinations on the following subjects: (i) analysis; (ii) algebra; (iii) a choice of one of the following: differential equations; probability theory; topology; geometry and discrete mathematics, numerical analysis. Statistics students will write two four hour examinations on the following subjects: (i) mathematical statistics; (ii) linear models and statistical techniques. In addition, statistics students must fulfill 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 6638.03. 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 10). 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 one year after 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 20minute 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 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. Faculty members and graduate students may attend the 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 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. (iii) Residence Requirements There is no formal residence requirement in the Faculty of Graduate Studies, but fulltime students are normally required to pay fulltime fees for two years. Fulltime 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) Deadlines for Meeting Requirements Students are normally expected to take their comprehensive examinations in May of their first year of Ph.D. studies, although this deadline may be extended with permission until the following September. If students are required to repeat all or some of their comprehensive examinations, they must do this at the next scheduled examination period (September or April). 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. (v) 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. 

