PhD in Mathematics

The requirements for obtaining an PhD in Mathematics can be found on the associated page of the BU Bulletin.

The requirements for the PhD are as follows:

• 16 semester courses numbered 500 or above, of which at least 8 must be numbered 700 or above. The courses are chosen in consultation with the Director of Graduate Studies or the thesis advisor and must be passed with a grade of B- or higher. Furthermore there is a “breadth” requirement affecting the choice of courses (see below).
• The preliminary exam (or “MA Comprehensive Exam”), required of all PhD students, even those entering with a master’s degree.
• The oral PhD qualifying exam.
• The doctoral thesis and thesis defense.

Again, the reference to 16 semester courses should be interpreted as 64 credits, and students can claim credit for courses taken elsewhere if transfer credit is awarded by GRS. The external courses would then count as if they had been taken at BU. Thus they could count toward the course requirement and they could also count as topics for the student’s qualifying exam (see below).

• Courses: The courses mentioned on the BU Bulletin page can be chosen from the graduate courses we offer here.
• Breadth Requirement for Courses: The 16 semester courses taken by each PhD student must satisfy the following “breadth requirement:” 2 semester courses must be chosen from each of 3 of the following 7 subject areas:
  • Analysis (examples include MA 711, MA 713, and MA 717)
  • PDEs and Dynamical Systems (examples include MA 771, MA 775, and MA 776)
  • Algebra and Number Theory (examples include MA 741, MA 742, and MA 743)
  • Topology (examples include MA 721, MA 722, and MA 727)
  • Geometry (examples include MA 725, MA 731, and MA 745)
  • Probability and Stochastic Processes (examples include MA 779, MA 780, and MA 783)
  • Applied Mathematics (examples include MA 750, MA 751, and MA 770)

  In one of the 3 subject areas, the 2 courses could be any courses numbered 700 or above; thus they could be 900-level directed study courses. In the other two subject areas only courses numbered 700 to 899 would count toward the breadth requirement.

Here are some examples of ways in which students might choose to fulfill the breadth requirement:

• • A student planning to work in dynamical systems might take 711 and 717 from I, 771 and 775 from II, and 721 and 722 from IV and then choose to be examined on 711 from I, 771 and 775 from II, and 721 from IV.
  • A student planning to work in number theory might take 711 and 713 from I, 741 and 743 from III, 731 and 745 from V and then choose to be examined on 713 from I, 741 and 743 from III, and 745 from V.
  • A student planning to work in geometry might take 711 and 713 from I, 721 and 722 from IV, and 731 and 745 from V and then choose to be examined on 713 from I, 721 and 722 from IV, and 731 from V.
  • A student planning to work in probability and stochastic processes might take 711 and 717 from I, 775 and 776 from II, and 779 and 780 from VI and then choose to be examined on 711 from I, 776 from II and 779 and 780 from VI.
  • A student planning to work in applied math might take 711 and 717 from I, 775 and 776 from II, and 750 and 751 from VII and then choose to be examined on 717 from I, 776 from II, and 750 and 751 from VII.

• Comprehensive Examination:
  • The Master’s Comprehensive Exam has the same format for MA students and for PhD students, but for the latter cohort the designation Preliminary Exam is more descriptive. To prepare for the exam, the student must first of all choose an exam advisor, who may be a faculty member, a postdoctoral fellow, or a research faculty member and who in the case of a PhD student could become, but would not necessarily become, the student’s thesis advisor. The exam has a written component and an oral component. The written component consists of an expository paper of typically fifteen to twenty-five pages on which the student works over a period of a few months under the guidance of the advisor. The topic of the expository paper is chosen by the student in consultation with the advisor. The topic should be appropriate to a 700 level course but not something that is routinely covered in an existing 700 level course, and a high standard of mathematical exposition is expected.
  • On completion of the paper, the student takes an oral exam given by a committee of three, one of whom is the advisor and at least one of whom is a faculty member. The oral exam consists of a presentation by the student on the expository paper followed by questioning by the committee members. The student is strongly advised to provide a near-final draft of the expository paper to committee members at least 3 weeks in advance of the oral exam, and the committee can refuse to hold the oral exam at the previously scheduled time if they have not received the draft sufficiently far in advance.
  • A student who does not pass does not perform satisfactorily on the MA Comprehensive Exam may make a second attempt, but all students are expected to pass the exam no later than the end of the summer following their second year. For this reason, students are strongly encouraged not to wait until the last minute to take the exam, in case a repeat exam is necessary. A student could be asked to repeat the exam because either the written exposition or the performance on the oral component of the exam was judged inadequate, but in making a second attempt, the student would not be expected to choose a whole new topic and write a whole new paper. Rather, the point would be to rewrite the paper thoroughly or to display better mastery of the material on the oral exam. As with the PhD qualifying exam, the committee for the MA Comprehensive Exam has considerable leeway in prescribing the parameters of a second attempt.
• Oral Qualifying Examination: The topics for the PhD qualifying exam correspond to the two semester courses taken by the student from one of the 3 subject areas and one semester course each taken by the student from the other two subject areas. In addition, the exam begins with a presentation by the student on some specialized topic relevant to the proposed thesis research. A student who does not perform satisfactorily on the qualifying exam may make a second attempt, but all PhD students are expected to pass the exam no later than the end of the summer following their third year. The qualifying exam is given by a committee of three or possibly four examiners, one of whom is the candidate’s thesis advisor, and at least two of the committee members must be regular faculty rather than postdocs or research faculty. The committee has considerable discretion with regard to the “second attempt” in case the candidate’s first attempt at passing the exam is found to be inadequate. For example, if the committee finds that only certain sections of the exam were unsatisfactory then it can stipulate that the second attempt should cover only those unsatisfactory portions, or it could even substitute a problem set or an expository submission in place of the unsatisfactory parts of the exam.
• Dissertation and Final Oral Examination: This follows the GRS General Requirements for the Doctor of Philosophy Degree.

Admissions information can be found on the BU Arts and Sciences PhD Admissions website.

Our department funds our PhD students through a combination of University fellowships, teaching fellowships, and faculty research grants. More information will be provided to admitted students.

Please reach out to us directly at mathdept@math.bu.edu if you have further questions.