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  Aug 17, 2017
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Graduate Record 2014-2015 [ARCHIVED RECORD]


Return to: Graduate School of Arts & Sciences: Departments/Programs  

Degree Requirements

Programs of Study

The Department of Mathematics administers programs leading to the degree of Master of Arts, Master of Science, and Doctor of Philosophy. These programs provide diverse opportunities for advanced study and research in algebra, analysis, topology, and mathematical physics.

The Master of Arts and Master of Science degrees are normally completed within two years, though in some cases, these degrees can be completed in one calendar year (two semesters and a summer session). The M.A. and M.S. programs differ in course requirements and in examinations. The M.S. degree requires specific courses in algebra, analysis, and topology. In contrast, the course requirements for the M.A. degree are more flexible and based on individual needs (see below). The M.A. candidate has two options, one requiring an expository paper for a thesis, and the other substituting additional course work in place of a thesis.

The Doctor of Philosophy degree is normally completed within five years. Candidates for the Ph.D. must fulfill certain course requirements and examinations beyond the master’s level. The most important addition is the Ph.D. dissertation, which is based on original research performed under the supervision of a faculty member.

All full-time graduate students are required, as part of their program, to gain teaching experience by assisting the instruction of undergraduate courses.

Master of Arts Degree

Course Requirements

  1. Thesis option: 30 credits of courses approved by the graduate committee at the 5000 level or above (some courses from other departments and thesis research can count towards the 30 credits).
  2. Non-thesis option: 30 credits of courses at the 5000 level or above (no reading or research courses), which must include  MATH 5330  (or replacements from among 7310  , 7320  , 7340  , 7751  or MATH 7752  ), and cannot include more than 9 credits from other departments.


(option (a) only): The master’s thesis is an expository paper written under the supervision of a faculty advisor.


A passing grade on the final master’s exam (or one part of the general examination); specific content of the exam will be decided by the examiners and discussed with the student in advance. The candidate must be a registered student at the time of the exam, and must finish the degree requirements within three years of passing the exam.


Knowledge of a foreign language is not required for a Masters of Arts Degree or a Masters of Science Degree.

Master of Science Degree

The requirements for the M.S. degree are the same as for the M.A. degree, except (1) the program must include MATH 7310 , 7340 , MATH 7751 , 7752 , MATH 7800  and MATH 7820 , and a passing grade is required on one general examination chosen from algebra, analysis or topology. Higher-level substitutes may be approved for the required courses.

Doctor of Philosophy Degree

Course Requirements

72 credits of coursework at the 5000 level or above (which may include 18 credits of non-topical research: MATH 8998 , 8999 , 9998 , 9999 ). A student must do satisfactory work in two semesters of analysis (MATH 7310 , 7340 ), algebra (MATH 7751 , 7752 ), and topology (MATH 7800 , 7820 ), or the equivalent.


Passing grades on two general examinations, chosen from analysis, algebra, and topology, and satisfactory performance on the second-year proficiency examination, are required.

General Examinations

The general exams are written exams, which are set and graded by the graduate committee. They test whether the student has the inventiveness and command of basic material to pursue a Ph.D. degree, and are usually taken in the second year of graduate study. The general exams must be satisfactorily completed by January of the student’s third year.  A student is allowed, at most, a total of four sittings for exams (taking one exam is one sitting).

Second-Year Proficiency Examination

Students take an informal oral examination on material from two or three selected second-year courses. The exact content of the exam is determined by a panel of faculty members in consultation with the student. Its purpose is to gauge the student’s readiness to begin carrying out research in the student’s chosen area. It is normally taken in May of the second year. If any deficiencies are noted, the examining panel will make recommendations on how to fix the detected problems and meet again with the student in August.


Facility in reading mathematical literature in one language (French, German, Russian, Italian, or a substitute acceptable to the department), as demonstrated by an exam administered by the department, in which students are required to translate passages from mathematical works in the given language. The language requirement should generally be satisfied by the end of the fourth year, or by the date of the Ph.D. defense, whichever comes first. Students pursing research in the history of mathematics are required to pass a written translation examination in two foreign languages, typically French and German, although the languages required will depend on the student’s research interests. These language requirements should generally be satisfied by the end of the student’s third year in order to enable the student to do primary source reading in the pertinent languages.

Previous Work:  Ph.D. students entering the program with a Master’s degree from another institution may receive up to 24 hours of credits towards their Ph.D. course requirements.  This is decided on an individual basis by the Graduate Advisor in consultation with the Graduate Committee.  The student must present a copy of their degree and a transcript showing the courses completed for their Master’s degree.

History of Mathematics Program

The requirements for the Ph.D. in the history of mathematics are the same as for the Ph.D. in mathematics, with the exception of the additional language requirements described above. Also, students entering the program must exhibit strong reading comprehension in two foreign languages, preferably French and German. Students interested in pursuing a Ph.D. in history of mathematics should contact Professor Karen Parshall before submitting an application.

Dissertation and Defense

Written under the supervision of the major advisor, the Ph.D. dissertation must contain original contributions to the field of mathematics. The main results of the dissertation are presented at a public oral defense. A committee consisting of the major advisor and three other faculty members (two from within the department and one from outside) must approve the dissertation and defense in order for the dissertation to be considered accepted by the faculty.

Time Limit

The Ph.D. may be completed in as few as three years, but must be completed within seven years. Students are expected to complete coursework, the general examination and the second-year proficiency examinations, by the conclusion of their fifth term of study. Language examinations are not subject to this deadline.

Course Descriptions

The Mathematics Colloquium

The Mathematics Colloquium is held weekly, the sessions being devoted to research activities of students and faculty members, and to reports by visiting mathematicians on current work of interest.