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Ph.D. Degree Requirements

The requirements for the Ph.D. fall into five categories: coursework, examinations (Preliminary, Language, and Comprehensive), teaching, a dissertation, and a residency requirement of two consecutive semesters with full-time registration.

Coursework: Students must complete at least 50 credit hours at the graduate level, including the first-year core curriculum, and receive a grade of B- or higher in at least 44 of these. The first-year program consists of core courses in Algebra (MATH8806, MATH8807), Geometry/Topology (MATH8808, MATH8809), Real Analysis (MATH8810), Complex Analysis (MATH8811), Graduate Teaching Seminar (MATH8890), and Graduate Research Seminar (MATH8892). The second year is devoted to electives. Students with prior comparable coursework may be exempted from one or more of the first-year graduate courses, upon approval of the Assistant Chair for Graduate Programs. Up to 18 credit hours of the coursework requirement may be waived for students with prior graduate work, upon approval of the Assistant Chair for Graduate Programs.

Examinations: There are two types of examinations: Preliminary and Comprehensive.

Preliminary: All students must take preliminary exams in two of the three following subjects: Real and Complex Analysis; Algebra; Geometry and Topology. These exams cover the material in the core first-year courses and are taken at the end of May following the first year. They may also be taken in the fall by advanced entering first year students or by second year students who have not already passed all their preliminary exams. Preliminary exams are graded as follows: Ph.D. pass, M.A. pass, or fail. Students are strongly encouraged to complete two preliminary examinations at the Ph.D. pass level by the start of their second year. The following two rules apply: (1) students may retake each preliminary exam once; (2) students must pass two preliminary examinations at the Ph.D. pass level by the end of their second year in order to continue in the program after their second year. Exceptions to these two rules require the approval of the Chair, who will consult the Assistant Chair for Graduate Programs, and the student’s instructors before arriving at a decision. Exceptions will be granted when there is clear evidence of potential to complete a degree in a timely way or for special circumstances such as extended illness. A student with advanced preparation may choose to take one or more of the preliminary examinations upon entering °¬¿ÉÖ±²¥ College. In such a case, the examination would not count as one of the student’s two attempts and failure of the examination would have no negative consequences. Students who wish to take an examination upon entrance should notify the Assistant Chair for Graduate Programs in writing by mid-July that they intend to do so.

Doctoral Comprehensive: After passing the preliminary exams at the Ph.D. level, the student requests that a department °¬¿ÉÖ±²¥member (tenured or tenure-track) serves as his/her research advisor. Upon agreement of the °¬¿ÉÖ±²¥member, the student, in consultation with the research advisor, forms a Comprehensive Examination Committee, consisting of the research advisor and at least two other members. Two members must be from °¬¿ÉÖ±²¥ College; the chair must be a tenured or tenure-track member of the departmental faculty. Committee composition is subject to departmental approval. The comprehensive exam can be taken any time after the Comprehensive Examination Committee has been chosen. We recommend that it be taken as soon as possible, to allow time for dissertation research. The doctoral comprehensive exam consists of a research topic and one secondary topic, chosen by the student in consultation with the student’s Committee. Typically, these are based on topics courses or independent study completed by the student in the second and third years. The comprehensive exam has both a written and an oral component. In the written part the student is given a week to solve problems or answer questions in the chosen areas. The Committee then reads the student’s work and questions the student about it in the oral exam, which may range into related areas. After the oral exam, the Committee grades the entire comprehensive exam as Pass with Distinction, Pass, or Fail. A student who fails the comprehensive exam may take it one additional time, but not sooner than the following semester.

Teaching: In addition to their responsibilities as teaching assistants and teaching fellows, students participate in the required Graduate Teaching Seminar in the fall semester of their first two years. The first-year teaching seminar is for teaching assistants, covering their responsibilities both to their students and their supervisors and providing guidance on leading a classroom for the first time. The second-year teaching seminar is for teaching fellows, providing guidance for teaching their own classes.

Dissertation: Upon satisfactory performance on the Language and Doctoral Comprehensive Examinations, the student is eligible to be admitted to candidacy for the Ph.D. To be admitted, the student formally constitutes a Dissertation Committee which is then approved by the Assistant Chair for Graduate Programs. The Comprehensive Examination Committee will typically become the student’s Dissertation Committee; any changes of membership require the department’s approval. At this point the student begins research for the doctoral dissertation. The dissertation must consist of original scholarly work. The Dissertation Committee will read and evaluate the completed dissertation and conduct an oral examination, at which the dissertation is defended in a public meeting. The dissertation is accepted when endorsed on the official title page by the Dissertation Committee after the oral examination. After ensuring that the format of the accepted dissertation conforms to °¬¿ÉÖ±²¥ College requirements, the student submits the dissertation to the University.

Qualified students accepted to the program are offered financial aid stipends and tuition remission. It is anticipated that support will be provided for five years of study, given reasonable progress toward the degree and acceptable performance of Teaching Assistant/Fellow duties.

More information about the Ph.D. program can be found at the graduate program menu option at bc.edu/math.

Master of Arts Degree

Students enrolled in the Ph.D. program who receive at least an M.A. pass in two of the three preliminary exams and pass at least 30 credits of mathematics graduate courses, including at least five semesters of the first-year graduate courses in Real and Complex Analysis, Algebra, and Geometry/Topology, will receive an M.A. degree. Advanced undergraduate courses may count towards the 30 credits in meeting the M.A. requirements with the permission of the Assistant Chair for Graduate Programs. Students who skip a first-year course because of advanced preparation may substitute more advanced courses in any area, with the permission of the Assistant Chair for Graduate Programs. A student may receive an M.A. and continue on to a Ph.D. provided the student meets the Ph.D. requirements above.

Master of Science in Teaching Program

The Master of Science in Teaching (M.S.T.) program is administered through the Lynch School of Education and Human Development in cooperation with the Department of Mathematics. Application for the program is made to the Lynch School, and students must be accepted by the Lynch School and approved by the Department of Mathematics.

This program is designed either for experienced teachers or for prospective teachers. It is a two-year program that consists of 46 credits, of which 31 are in Education and 15 are in Mathematics. All master’s programs leading to certification in secondary education include practical experiences in addition to coursework. Students seeking certification in Massachusetts are required to pass the Massachusetts Educators Certification Test. Degree candidates draw up an overall plan of study with joint advisement from the Assistant Chair for Graduate Programs in Mathematics and the advisor for the M.S.T. program in the Lynch School. For further information on the M.S.T., refer to the Master’s Programs in Secondary Teaching in the Lynch School section of the University Catalog or call the Office of Graduate Admissions, Lynch School, at 617-552-4214.

Of the 15 credits for the mathematics component of the M.S.T., candidates are required to complete MATH8810–8811 Real and Complex Analysis, which should be completed in the first year. The other credits must be earned in MATH courses at or above the 4000-level. Because of certification requirements, unless approved equivalents have been taken previously, these required courses should include the following:

• MATH4451 Euclidean and Non-Euclidean Geometry
• MATH4426–4427 Probability and Mathematical Statistics

Some exposure to the use of computers in mathematics that may be accomplished by any Computer Science major course beyond Computer Science I.

Other courses particularly well-suited for this program are MATH4430 Number Theory and MATH4475 History of Mathematics. M.S.T. candidates must also pass an oral comprehensive examination and submit a brief expository paper in some area of mathematics.